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A study of nutation dampers with application to wind induced oscillations Welt, François 1988

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A S T U D Y OF NUTATION D A M P E R S WITH APPLICATION TO WIND INDUCED OSCILLATIONS b y F R A N C O I S W E L T B . S c , E c o l e P o l y t e c h n i q u e d e M o n t r e a l , 1 9 7 9 M . A . S c , T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1 9 8 3 A 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 T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y i n T H E F A C U L T Y O F G R A D U A T E S T U D Y 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 W e a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A J a n u a r y 1 9 8 8 @ F r a n c o i s W e l t , 1 9 8 8 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of NlECviAuicftL £kJCiMEE-RitJ Q The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date Jfrv] .28 m% A B S T R A C T ii Energy dissipation due to sloshing liquid in torus shaped nutation dampers is studied using the potential flow model with nonlinear free surface conditions in conjunction with the boundary layer correction. Special consideration is given to the case of resonant interactions which were found to yield interesting damping characteristics. An extensive test program with the dampers undergoing steady-state oscillatory translation is then undertaken to establish the optimal damper parameters. Low liquid heights and large diameter ratios with the system operating at the liquid sloshing resonance are shown to result in increased damping, while low Reynolds numbers and presence of baffles tend to reduce the peak efficiency by restricting the action of the free surface. Tests with two-dimensional as well as three-dimensional models in laminar flow and boundary layer wind tunnels suggest that the dampers can successfully control both the vortex resonance and galloping types of instabilities. Applicability of the concept to vertically oscillating structures such as transmission lines is also demonstrated with dampers undergoing a rotational motion about their horizontal axis. iii T A B L E O F C O N T E N T S Chapter Page 1 INTRODUCTION 1 1.1 Preliminary Considerations 1 1.2 Literature Survey 5 1.3 Scope of the Investigation 9 2 AN APPROXIMATE ANALYTICAL APPROACH TO PREDICT ENERGY DISSIPATION 12 2.1 Preliminary Remarks 12 2.2 Potential Flow Solution 12 2.2.1 Basic Equations 12 2.2.2 Linear Solution 14 2.2.3 Nonlinear, Nonresonant Solution 15 2.2.4 Nonlinear, Resonant Solution 16 2.2.4.1 No Interactions 17 2.2.4.2 Resonant Interactions 19 2.2.5 Properties of the Potential Function 21 2.2.5.1 Variation with Damper Geometry 21 2.2.5.2 Variation with the Excitation 24 2.3 Pressure Forces 29 2.4 Damping Forces 31 2.4.1 Effect of Viscosity 31 2.4.2 Energy Dissipation and Reduced Damping Ratio 33 2.4.3 Energy Ratio Er,i 37 3 EXPERIMENTAL DETERMINATION OF DAMPER CHARACTERISTICS 41 3.1 Preliminary Remarks 41 3.2 Test Arrangement and Models 41 3.3 Flow Visualization 44 3.4 Added Mass and Reduced Damping Ratio 50 3.4.1 General Procedure 50 3.4.2 Results and Discussion 53 3.4.3 Comparison with Free Oscillation Tests 77 3.5 Concluding Comments 81 4 WIND INDUCED INSTABILITY STUDY 84 4.1 General Description 84 4.2 Two-Dimensional Tests in Laminar Flow 84 4.2.1 Preliminary Remarks 84 4.2.2 Test Arrangement and Model Description 85 4.2.3 Calibration Procedure 88 4.2.4 Model Characteristics 88 4.2.5 Results and Discussion 98 4.2.5.1 Vortex Resonance of a Circular Cylinder 98 4.2.5.2 Vortex Resonance and Galloping Response of a Square Cylinder 103 4.2.6 Concluding Comments I l l iv Chapter Page 4.3 Three-Dimensional Tests 115 4.3.1 Preliminary Remarks 115 4.3.2 Test Arrangement and Model Description 115 4.3.3 Model Characteristics 117 4.3.4 Results and Discussion . . . 122 4.3.4.1 Vortex Resonance Response of a Circular Cylinder 122 4.3.4.2 Vortex Resonance and Galloping Response of a Square Cylinder 127 4.3.5 Concluding Comments 134 4.4 Application to Transmission Lines 136 4.4.1 Preliminary Remarks 136 4.4.2 Test Arrangement and Model Description 136 4.4.3 Model Characteristics 139 4.4.4 Results and Discussion 141 4.4.5 Result Summary 145 5 CONCLUSIONS 149 BIBLIOGRAPHY 153 APPENDICES 162 I NONLINEAR FREE SURFACE CONDITION 162 1. Basic Equation 162 2. Perturbation Series Expansion 162 3. Free Surface Equation 163 II NONLINEAR, NONRESONANT POTENTIAL FLOW SOLUTION 164 1. Second Order Terms 164 2. Stability and 3rd Order Equation 165 III NONLINEAR, RESONANT POTENTIAL FLOW SOLUTION 167 1. No Interactions 167 2. Resonant Interactions 168 2.1 Second Order Terms and Detuning Parameters 168 2.2 Third Order Equation 171 2.3 Solution Stability 175 IV ADDED MASS AND DAMPING RATIOS: DETAILS OF T H E ANALYSIS 177 1. Added Mass 177 1.1 General Procedure 177 1.2 Higher Order Terms 178 2. Damping Ratio 179 2.1 Correction Velocity U 2 179 2.1.1 First Order 1 7 9 2.1.2 Second Order 181 2.2 Reduced Damping Ratio 183 V C h a p t e r P a g e 2 . 2 . 1 C o n t r i b u t i o n f r o m R i g i d B o u n d a r y 1 8 3 2 . 2 . 2 F r e e S u r f a c e C o n t r i b u t i o n 1 9 0 2 . 3 E n e r g y R a t i o 1 9 0 V U S E F U L B E S S E L A N D H Y P E R B O L I C F U N C T I O N R E L A T I O N S A N D D E F I N I T I O N S 1 9 3 1 . B e s s e l F u n c t i o n s 1 9 3 1.1 O r t h o n o g a l i t y C o n d i t i o n 1 9 3 1 .2 C r o s s - P r o d u c t I n t e g r a l s 1 9 3 1 .3 S i m p l e C r o s s - P r o d u c t s 1 9 5 2 . H y p e r b o l i c F u n c t i o n s 1 9 6 2 . 1 D e f i n i t i o n s a n d C r o s s - P r o d u c t I n t e g r a l s 1 9 6 2 . 2 S i m p l e C r o s s - P r o d u c t s 1 9 7 2 . 3 C o m b i n a t i o n s o f H y p e r b o l i c a n d B e s s e l F u n c t i o n C r o s s - P r o d u c t s 1 9 7 V I M E C H A N I C A L A N A L O G Y O F A L I N E A R S Y S T E M 1 9 8 1 . O n e - D e g r e e - o f - F r e e d o m 1 9 8 2 . T w o - D e g r e e - o f - F r e e d o m S y s t e m 1 9 9 V I I W I N D - I N D U C E D O S C I L L A T I O N A M P L I T U D E C A L C U L A T I O N 2 0 2 1 . G a l l o p i n g T h e o r y w i t h E q u i v a l e n t D a m p i n g 2 0 2 2 . V o r t e x R e s o n a n c e o f a F u l l S c a l e C h i m n e y F i t t e d w i t h N u t a t i o n D a m p e r s 2 0 2 vi L I S T O F T A B L E S T a b l e s P a g e I D e t a i l s o f t h e d a m p e r m o d e l s u s e d i n t h e t e s t p r o g r a m 4 4 I I P h y s i c a l d e s c r i p t i o n o f t h e t w o - d i m e n s i o n a l a e r o d y n a m i c m o d e l s 8 7 I I I P h y s i c a l d e s c r i p t i o n o f t h e t h r e e - d i m e n s i o n a l a e r o d y n a m i c m o d e l s . . . 1 1 7 vii L I S T O F F I G U R E S Figure Page 1 Wind-induced instabilities of bluff bodies undergoing: (a) vortex resonance; (b) galloping 2 2 Several typical devices providing: (a) external damping; (b) internal damping 4 3 Geometry of the square section selected for analytical study 13 4 Variation of the linear coefficient F\ with o 22 5 Planar mode coefficients JKi, KK\, Ei, Ei and K2 as affected by the damper geometry 25 6 Nonplanar coefficients —(KK\ — 2K\) and K\jKK\ as functions of h for: (a) a = 0.308; (b) a = 0.608 26 7 Variation of / u , fo\ and £ 2 1 versus u for different damper geometries and amplitudes 28 8 Typical added mass characteristics at low amplitude 31 9 Typical variation of theoretical r}rti versus u> 36 10 Typical curves showing variation of Er,i with frequency and amplitude 39 11 E*i versus CJ as affected by: (a) h\ (b) a 40 12 Test arrangement 42 13 Sketch showing several damper internal configurations: (a) plain; (b) baffles; (c) inner tube 45 14 1st planar mode exhibiting: (a) antisymmetric motion about circumference; (b) variation across f; (c) variation along B 47 15 1st nonplanar mode shape 48 16 Mode (1,2) shown as: (a) in the plane of the excitation; (b) perpendicular to the same plane 48 17 Mode (1,3) shown as: (a) in the plane of the excitation; (b) perpendicular to the same plane 49 viii Figure Page 18 Close-up view of mode (1,4) 49 19 Calibration curves to determine: (a) Mo; (b) slope for the response using both static and dynamic procedures; (c) slope for the excitation; (d) phase angle between response and excitation 52 20 Output signal showing: (a) moving frame displacement; (b) damper support beam deflection; (c) frequency spectrum of the response 55 21 Variation of damping and added mass ratios with frequency for half-full damper#7 56 22 Variation of damping and added mass ratios with frequency for half-full damper#l 57 23 Effect of amplitude on \Ma/Mi\ and rjr>i for half-full damper#l 59 24 Resonant behavior of damper#5 with: (a) h/d = 0.5 and u < 1; (b) h/d = 0.19 and w > 1 61 25 Peak damping ratio as affected by amplitude 63 26 Maximum damping and added mass ratios for various liquid heights at e0/d = 0.046 64 27 Variation of the peak response for damper#l with liquid height at e0/d = 0.105 66 28 Variation of the peak damping ratio for damper#7 with liquid height at €0/d = 0.091 67 29 Frequency spectrum of the response for damper#5 and #6 showing the effect of D/d 68 30 Maximum damping ratios versus amplitude for various D/d 70 31 Damping and added mass ratios as affected by Re for dampers with: (a) D/d = 1.89; (b) D/d = 4.10 71 32 Effect of internal configuration on r\r \ and \Ma/Mi\ versus: (a) coM (b) CJ 75 33 Proposed sloping cross-section 77 34 Apparatus used for free oscillation tests 78 ix F i g u r e P a g e 35 D a m p i n g cha rac te r i s t i c s ve rsus e o / d as o b t a i n e d b y s t eady -s ta te a n d f ree -osc i l l a t i on e x p e r i m e n t s for : (a) d a m p e r w i t h baf f les; (b) p l a i n d a m p e r 79 36 A m p l i t u d e decay for t he ha l f - fu l l d a m p e r # l o s c i l l a t i n g a t CJ = 0.924 w i t h a n i n i t i a l d i s p l a c e m e n t of eo/d = 1.28 ( cha r t r eco rde r T y p e T R 3 2 2 , G u l t o n Indus t r ies ) 81 37 W i n d t u n n e l se t -up fo r t w o - d i m e n s i o n a l tests 86 38 C a l i b r a t i o n c o n s t a n t s used d u r i n g t he tests for : (a) c h a r t r eco rde r ; (b) s p e c t r u m a n a l y s e r 89 39 E f fec t o f e n d p l a te d i m e n s i o n o n Cfy fo r : (a) m o d e l # 2 ; (b) m o d e l # l . . 9 0 40 M a x i m u m d i s p l a c e m e n t s o f m o d e l # 3 u n d e r g o i n g v o r t e x resonance 92 41 S y s t e m d a m p i n g for d i f fe rent e l e c t r o m a g n e t i c d a m p e r se t t i ngs 92 42 Cfy ve rsus a fo r : (a) m o d e l # 2 ; (b) m o d e l # l 94 43 G a l l o p i n g response for (a) m o d e l # 2 w i t h h i g h d a m p i n g ; (b) m o d e l # l w i t h h i g h d a m p i n g ; (c) l o w d a m p i n g 95 44 R e s p o n s e o f t w o - d i m e n s i o n a l square c y l i n d e r w i t h o u t d a m p e r for : (a) m o d e l # l ; (b) m o d e l # 2 96 45 V o r t e x s h e d d i n g e x c i t a t i o n o n t w o - d i m e n s i o n a l m o d e l s s h o w i n g : (a) f r e q u e n c y s p e c t r u m of t he response ; (b) S t r o u h a l n u m b e r fo r t he c i r c u l a r c y l i n d e r ; (c) S t r o u h a l n u m b e r fo r squa re c ross -sec t ions 97 46 V o r t e x resonance response o n m o d e l # 3 s h o w i n g : (a) effect o f l i q u i d he igh t a n d CJ; (b) effect o f i n t e r n a l c o n f i g u r a t i o n ; (c) effect o f d i a m e t e r r a t i o a n d CJ 100 47 V a r i a t i o n of r/r>/ w i t h a m p l i t u d e d u r i n g free osc i l l a t i ons for d a m p e r # l w i t h h/d = 1 /8 a n d 1/4 101 48 C o m p a r i s o n be tween e x p e r i m e n t s a n d p r e d i c t i o n s b a s e d o n t he energy b a l a n c e m e t h o d 103 49 P r e d i c t i o n s o f t he H a r t l e n - C u r r i e m o d e l s h o w i n g : (a) de te r -m i n a t i o n o f ah w i t h e m p t y d a m p e r ; (b) d e t e r m i n a t i o n o f 6^; (c) response w i t h p a r t i a l l y filled d a m p e r a n d Cjo = 0 .3 ; (d) response w i t h Cjo = 0.5 104 X Figure Page 50 Galloping response of model#l showing: (a) effect of frequency; (b) effect of internal configuration 106 51 Galloping response of model#2 showing: (a) effect of liquid height and w; (b) effect of diameter ratio and CJ 107 52 Predicted galloping response of square prisms fitted with nutation dampers for: (a) model#l; (b) model#2 110 53 Hartlen-Currie predictions for model#l with: (a) electromagnetic damping; (b) nutation damping 112 54 Response of a square prism with nutation damper as affected by: (a) end plates; (b) model size 113 55 Wind tunnel set-up for the three-dimensional tests 116 56 Static side force for three-dimensional square prisms as: (a) measured on model#2; (b) on model#l; (c) expressed as a moment coefficient 119 57 Inherent damping ratio for 3-D set-up and two frequencies of excitation 121 58 Strouhal number for the large square cylinder 121 59 Effect of damper position on the response of a square prism 122 60 Vortex resonance response of model#3 as affected by h/d and u> in: (a) laminar flow; (b) turbulent flow 123 61 Effect of internal configuration on the 3-D model response 124 62 Boundary layer velocity profile as recorded during the 3-D tests 125 63 Damping characteristics as affected by liquid heights for nutation dampers undergoing rotation 126 64 Effect of D/d and u on the vortex resonance response of model#3.. .128 65 Vortex resonance response for model#3 without dampers at various frequencies 129 66 Galloping response in 3-D for model#2 with nutation dampers in: (a) laminar flow; (b) turbulent flow 130 67 Galloping response in 3-D for model#l with nutation dampers in: (a) laminar flow; (b) turbulent flow 131 xi F i g u r e P a g e 68 E f fec t o f l o w l i q u i d he igh t s o n the 3 - D g a l l o p i n g response o f m o d e l # 2 132 69 E f f ec t o f D/d a n d CJ o n t he 3 - D g a l l o p i n g response o f m o d e l # 2 133 70 S k e t c h of t he h o r i z o n t a l l y m o u n t e d w i n d t u n n e l se t -up 137 71 H o r i z o n t a l l y m o u n t e d w i n d t u n n e l se t -up s h o w i n g : (a) F r o n t v i e w o f t he o s c i l l a t i n g s y s t e m ; (b) c lose -up v i e w of t he d a m p i n g dev i ce 138 72 E v a l u a t i o n o f the s e c o n d a r y s y s t e m d a m p i n g r a t i o s h o w i n g : (a) c a l i b r a t i o n p r o c e d u r e ; (b) na2 a n d r/ r )/ ve rsus a m p l i t u d e 140 73 R e s p o n s e of t he s y s t e m w i t h o u t d a m p i n g l i q u i d s h o w i n g : (a) effect o f a u x i l i a r y dev i ce ; (b) b e a t i n g m o t i o n 142 74 E f fec t o f l i q u i d he igh t o n the s y s t e m response 143 75 F r e q u e n c y s p e c t r u m of t he response for : (a) / „ « frot; (b) / „ » / r o t 144 76 R e s p o n s e o f t he m o d e l as af fec ted b y t he p a r a m e t e r s : (a) a>2/u>i; (b) mi/m\ 146 77 S k e t c h o f t he d a m p e r a r r a n g e m e n t use fu l t o c o n t r o l r o l l 147 V I - 1 M e c h a n i c a l r e p r e s e n t a t i o n o f a n u t a t i o n d a m p e r 198 VJ.-2 F o r c e s a c t i n g o n t he m o v i n g base 198 V I - 3 One -deg ree -o f - f r eedom s y s t e m cha rac te r i s t i c s a t resonance s h o w i n g : (a) \Ma/Mt\; (b) t | r , i ; (c) Er,i 200 V I - 4 M e c h a n i c a l r e p r e s e n t a t i o n of t he t r a n s m i s s i o n l i ne a r r a n g e m e n t 199 VJ.-5 F o r c e d i a g r a m fo r t he d a m p i n g dev i ce 201 V I I - 1 S tee l c h i m n e y w i t h 6 n u t a t i o n d a m p e r s 204 V I I - 2 S tee l c h i m n e y w i t h n u t a t i o n d a m p e r r i n g 204 xii L I S T O F S Y M B O L S a d a m p e r inner to outer radius r a t i o , Ri/Ro ah free parameter of the H a r t l e n - C u r r i e l ift oscillator m o d e l o i Bessel f u n c t i o n r e l a t i o n defined i n A p p e n d i x III, p. 170 a j dimensionless expression for a\ defined i n A p p e n d i x III, p. 170 5 7 a m p l i t u d e coefficient of the correction v e l o c i t y u 2 defined i n A p -p e n d i x I V , p. 180 A d a m p e r w a l l area An higher order terms of the added mass r a t i o M 0 / M j , n = 1 , . . . , 4 ; also used as coefficients of the p o l y n o m i a l fit for r/rj/ versus e0/d i n C h a p t e r 3 , a n d Cfy versus a i n C h a p t e r 4 At w i n d - t u n n e l test section area AQ a e r o d y n a m i c m o d e l f r o n t a l area AAn higher order terms of the reduced d a m p i n g r a t i o T7 r > j , n = 1 , 2 , 3 AAAn higher order terms of t h e energy r a t i o Erj, n = 1 , 2 , 3 AKn d a m p e r geometry dependent coefficients defined i n A p p e n d i x III, p p . 168-169 b n u t a t i o n d a m p e r baffle w i d t h bh free p a r a m e t e r of the H a r t l e n - C u r r i e l ift oscillator m o d e l &i Bessel f u n c t i o n r e l a t i o n defined i n A p p e n d i x III, p . 171 b\ dimensionless expression for &i defined i n A p p e n d i x III, p. 171 & i i 2 Bessel f u n c t i o n relations defined i n A p p e n d i x III, p p . 180-181 B s t a b i l i t y coefficient of the n o n p l a n a r s o l u t i o n , eq. I H . 4 , p. 168 Bn higher order terms of the added mass r a t i o Ma/Mi, n = 1 , 4 BB3 higher order t e r m of the reduced d a m p i n g r a t i o r)Tti BBB3 higher order term of the energy ratio Er>i C stability coefficient for the nonplanar solution, eq. 111.4, p. 168 Cd> Cdi 5 Cd2 absolute damping coefficient of the one-degree-of-freedom, primary and auxiliary system, respectively Ce equivalent absolute damping coefficient Cfy aeorodynamic static side force coefficient, Fy/[padmLm(V cos a)2/2] Cir aerodynamic lift coefficient normalized as C\r = Ci(U/Ur)2, where Ci is equivalent to C/y for the moving circular cylinder C/o aerodynamic static lift coefficient Cm aerodynamic moment coefficient for 3-D square prisms CKn damper geometry dependent variables defined in Appendix III, pp. 172-173 C4 higher order term of the added mass ratio M 0 / M / defined in Ap-pendix IV, p. 179 C n (A m f ) combined 1st and 2nd kind of Bessel function, d damper cross-section width or diameter di damper inner tube outer diameter dm structure or aerodynamic model cross-section width or diameter dmn 2nd order mode shape coefficients for the potential function $ de-fined in Appendix III, pp. 169-170 d\ 3 characteristic dimensions of a damper's sloping cross-section as de-fined in Fig. 33 D damper outer diameter Dr aerodynamic drag force D\ 1 0 2nd order amplitude coefficients for rjr)i defined in Appendix V, p. 195; also used as 3rd order coefficients for $ in Appendix II (n = 1,2), pp. 165-166 xiv DDi 3rd order coefficient for $ defined in Appendix II, p. 165 DKn damper geometry dependent variables defined in Appendix III, pp. 172, 174 A emn 2nd order mode shape coefficients for the potential function $ de-fined in Appendix III, p. 169-170 e n , e 2 2 amplitude coefficients of the 2nd order component of u 2 as defined in Appendix IV, pp. 181-182 Ed total dissipated energy by the moving fluid per cycle Erj dissipated to total energy ratio, Ed/Et E*j energy ratio accounting for Reynolds number, Er>iVRe Et total energy of the fluid moving with respect to the damper, eq. 37, p. 37 E\,E2 nonlinear coefficients of the 3rd order equation for fu, f 2 i defined in Appendix III, p. 174 / exciting frequency fn fundamental natural frequency of the system fnii fn2 1st and 2nd natural frequency of system in translation, respectively fmn 1st or 2nd order planar mode shape coefficients for the potential function $ frot 1st rolling natural frequency of the system / „ vortex shedding exciting frequency F sloshing force acting on the damper wall due to pressure F* damper geometry dependent function defined in Appendix IV, p. 190 Fg sloshing force transmitted from the fluid to the damper walls Fy aerodynamic static side force with respect to the model, Sp cos a — DT sin a XV Fijki ijp 2 n d o r d e r a m p l i t u d e c o e f f i c i e n t s f o r nr i d e f i n e d i n A p p e n d i x V , p . 1 9 6 Fo i n e r t i a f o r c e g e n e r a t e d b y t h e s y s t e m w i t h o u t d a m p i n g l i q u i d ; a l s o u s e d a s t h e a e r o d y n a m i c f o r c e a c t i n g o n t h e s y s t e m i n A p p e n d i x V I F\ l i n e a r c o e f f i c i e n t o f t h e 3 r d o r d e r n o n l i n e a r e q u a t i o n f o r fu, e q . I I I . 2 , p . 1 6 7 g a c c e l e r a t i o n d u e t o g r a v i t y 9nm h y p e r b o l i c f u n c t i o n r e l a t i o n d e f i n e d i n A p p e n d i x V , p . 1 9 6 G " m h y p e r b o l i c f u n c t i o n c r o s s - p r o d u c t i n t e g r a l s d e f i n e d i n A p p e n d i x V , p . 1 9 6 G l , . . , 5 d a m p e r g e o m e t r y d e p e n d e n t v a r i a b l e s d e f i n e d i n A p p e n d i x I I I , p p . 1 6 7 , 1 7 5 G G " m h y p e r b o l i c f u n c t i o n c r o s s - p r o d u c t i n t e g r a l s d e f i n e d i n A p p e n d i x V , p . 1 9 6 h d a m p e r l i q u i d h e i g h t h d i m e n s i o n l e s s d a m p e r l i q u i d h e i g h t , h/Ro b! n u t a t i o n d a m p e r b a f f l e m i d - h e i g h t p o s i t i o n H f u l l - s c a l e s t r u c t u r e h e i g h t H3,4 d a m p e r g e o m e t r y d e p e n d e n t v a r i a b l e s d e f i n e d i n A p p e n d i x I I I , p . 1 7 5 I t o t a l i n e r t i a f o r t h e r o t a t i n g s y s t e m 11 l i q u i d i n e r t i a f o r s y s t e m i n r o t a t i o n i i 9 B e s s e l f u n c t i o n c r o s s - p r o d u c t i n t e g r a l s d e f i n e d i n A p p e n d i x V , p . 1 9 3 IA B e s s e l f u n c t i o n c r o s s - p r o d u c t i n t e g r a l s d e f i n e d i n A p p e n d i x V , p . 1 9 5 IDD*m B e s s e l f u n c t i o n c r o s s - p r o d u c t i n t e g r a l s d e f i n e d i n A p p e n d i x V , p p . 1 9 3 - 1 9 5 12 c o m b i n a t i o n o f h y p e r b o l i c a n d B e s s e l f u n c t i o n c r o s s - p r o d u c t s d e -fined i n A p p e n d i x V , p . 1 9 7 XVI B e s s e l f u n c t i o n c r o s s - p r o d u c t i n t e g r a l s d e f i n e d i n A p p e n d i x V , p . 1 9 5 B e s s e l f u n c t i o n c r o s s - p r o d u c t i n t e g r a l s d e f i n e d i n A p p e n d i x V , p . 1 9 4 c o m b i n a t i o n s o f h y p e r b o l i c a n d B e s s e l f u n c t i o n c r o s s - p r o d u c t s d e -fined i n A p p e n d i x V , p . 1 9 7 o n e - d e g r e e - o f - f r e e d o m s y s t e m s t i f f n e s s s p r i n g s t i f f n e s s o f t h e m a i n s y s t e m s p r i n g t o r s i o n a l s t i f f n e s s o f t h e s e c o n d a r y s y s t e m c o m b i n a t i o n s o f h y p e r b o l i c a n d B e s s e l f u n c t i o n c r o s s - p r o d u c t s d e -fined i n A p p e n d i x V , p . 1 9 7 n o n l i n e a r c o e f f i c i e n t s o f t h e 3 r d o r d e r e q u a t i o n f o r fu d e f i n e d i n A p p e n d i x I I I , p . 1 6 7 B e s s e l c r o s s - p r o d u c t i n t e g r a l s d e f i n e d i n A p p e n d i x V , p . 1 9 4 s q u a r e r o o t o f R e y n o l d s n u m b e r , i . e . , y/~R~e l e n g t h o f t h e s u p p o r t i n g d a m p e r p l a t f o r m , F i g . V I - 5 c o m b i n a t i o n s o f h y p e r b o l i c a n d B e s s e l f u n c t i o n c r o s s - p r o d u c t s d e -fined i n A p p e n d i x V , p . 1 9 7 d i s t a n c e o f d a m p e r c e n t e r o f g r a v i t y t o p i v o t i n g p o i n t f o r t h e s e c -o n d a r y s y s t e m , F i g . V I - 5 a r m l e n g t h o f t h e p i v o t i n g s y s t e m , F i g . 6 3 a e r o d y n a m i c m o d e l l e n g t h B e s s e l f u n c t i o n c r o s s - p r o d u c t i n t e g r a l s d e f i n e d i n A p p e n d i x V , p . 1 9 4 2 n d o r d e r c o m p o n e n t s f o r rjr>i d e f i n e d i n A p p e n d i x I V , p p . 1 8 4 - 1 8 5 m a s s p e r u n i t l e n g h t o f t h e f u l l - s c a l e s t r u c t u r e d a m p i n g l i q u i d m a s s o f a s i n g l e , f u l l - s c a l e d a m p e r u n i t m a s s o f t h e d a m p e r s u p p o r t i n g p l a t f o r m XVII mi m<i M Ma Mi Mi Mlm> MnmMj,q, MnrnNpqi MnmEp n N2 N N NnmEp Pnm P* • M l Qi Q 2 2 Tin Re Ri m a s s o f t h e p r i m a r y s y s t e m i n t r a n s l a t i o n e q u i v a l e n t m a s s o f t h e s e c o n d a r y s y s t e m t o t a l m a s s o f t h e o s c i l l a t i n g s y s t e m a d d e d m a s s d u e t o s l o s h i n g l i q u i d r e s t o r i n g m o m e n t o f t h e t o r s i o n a l s p r i n g 1 s t m o d a l m a s s o f t h e s t r u c t u r e , e q . V I I . 6 , p . 2 0 2 t o t a l m a s s o f t h e s l o s h i n g l i q u i d 2 n d o r d e r c o m p o n e n t s f o r rjr>i d e f i n e d i n A p p e n d i x I V , p p . 1 8 5 - 1 8 8 v e c t o r n o r m a l t o t h e f r e e s u r f a c e 2 n d o r d e r c o m p o n e n t s f o r rjrti d e f i n e d i n A p p e n d i x I V , p p . 1 8 8 - 1 8 9 p r e s s u r e e x e r t e d b y t h e s l o s h i n g l i q u i d A 3 r d o r d e r c o e f f i c i e n t s o f t h e p o t e n t i a l f u n c t i o n $ , e q . L T . 4 , p . 1 6 6 c o e f f i c i e n t s o f t h e 3 r d o r d e r e q u a t i o n f o r $ d e f i n e d i n A p p e n d i x I I I (» = 1 , 2 ) , p p . 1 7 1 - 1 7 2 d a m p e r i n n e r t u b e h o l e s i z e d i a m e t e r c o e f f i c i e n t s o f t h e 3 r d o r d e r e q u a t i o n f o r $ d e f i n e d i n A p p e n d i x I I I , p . 1 7 3 d a m p e r b a s e d m o v i n g c o o r d i n a t e i n t h e r a d i a l d i r e c t i o n d i m e n s i o n l e s s m o v i n g c o o r d i n a t e , r/Ro A 2 n d o r d e r c o e f f i c i e n t f o r $ a s d e f i n e d i n A p p e n d i x I V , p . 1 9 2 a i r s t r e a m R e y n o l d s n u m b e r , Vd/va; a l s o u s e d a s i n n e r o r o u t e r r a d i u s i n A p p e n d i x I V s l o s h i n g l i q u i d R e y n o l d s n u m b e r , ueR$/vf d a m p e r i n n e r r a d i u s X V 111 Ro s so Si, 5 2 s St SUM1,...,5 t To u' u Uf t ? 2 u Ur Ul V Vnm, VVnm d a m p e r o u t e r r a d i u s s e c o n d a r y t o p r i m a r y m a s s r a t i o , mi/mi ul -&^2(l + s) fii ~ ? i i a n d fii + ? 2 i > r e s p e c t i v e l y a r e a o f t h e s l o s h i n g l i q u i d ' s o u t e r b o u n d a r y a e r o d y n a m i c s i d e f o r c e w i t h r e s p e c t t o t h e flow d i r e c t i o n S t r o u h a l n u m b e r d a m p e r g e o m e t r y d e p e n d e n t v a r i a b l e s d e f i n e d i n A p p e n d i x I I I , p p . 1 6 7 , 1 7 4 t i m e B e s s e l f u n c t i o n c r o s s - p r o d u c t s d e f i n e d i n A p p e n d i x V , p . 1 9 6 m e a s u r e o f t h e p h a s e a n g l e tp0 a s d e f i n e d i n F i g . 1 9 a i r s t r e a m t u r b u l e n t i n t e n s i t y s l o s h i n g l i q u i d v e l o c i t y v e c t o r f r e e s t r e a m v e l o c i t y f o r t h e t u r b u l e n t b o u n d a r y l a y e r p r o f i l e B e s s e l f u n c t i o n c r o s s - p r o d u c t s d e f i n e d i n A p p e n d i x V , p . 1 9 6 c o r r e c t i o n v e l o c i t y d u e t o l i q u i d v i s c o s i t y a i r s t r e a m d i m e n s i o n l e s s v e l o c i t y , V/undm; a l s o u s e d a s t h e a v e r a g e p o t e n t i a l e n e r g y o f t h e s l o s h i n g l i q u i d flow i n e q . 3 7 , p . 3 7 v o r t e x r e s o n a n c e d i m e n s i o n l e s s v e l o c i t y , l/(2irSt) d i m e n s i o n l e s s g a l l o p i n g o n s e t v e l o c i t y , r)rta/(7rAi) m a g n i t u d e o f U 2 v o l u m e o f t h e d a m p i n g l i q u i d B e s s e l f u n c t i o n c r o s s - p r o d u c t s i n t e g r a l s d e f i n e d i n A p p e n d i x V , p . 1 9 6 x i x V air stream velocity —+ V velocity of the damper walls in translation —• Vn component of V in the n direction VS damper geometry dependent variables defined in Appendix IV, p. 192 wnm Bessel function cross-products defined in Appendix V, p. 196 WS damper geometry dependent variables defined in Appendix IV, p. 192 x position of the damper in the direction of the excitation X output voltage variation in x direction y output voltage due to the system response yn output voltage at the nth harmonic of y; also used to describe response of the two-degree of freedom system in Appendix VI (n=l,2), Fig. VI-4 V aerodynamic model tip displacement, normalized by dm Yp dimensionless deflection of the full-scale structure along its height Yi, Y2 amplitude response of the primary and secondary system, respec-tively z position of the liquid along the vertical axis; also used as the vertical axis for the full-scale structure of Appendix VII z dimensionless coordinate, Z/RQ a air stream angle of attack "mn hyperbolic function of the damper liquid height, tanh A n m h (3 excitation and damper geometry dependent variable defined in Ap-pendix III, p. 171 /?i 1st order detuning parameter for A / ? 2 2nd order detuning parameter for Pnmt fin hyperbolic functions defined in Appendix V , p. 197 •7„m 2nd order mode shape coefficient of potential function $ defined in Appendix III, pp. 169-170 A A n m 2nd order component of the potential function $ V gradient operator V n m 2nd order component of the potential function $ e amplitude of the excitation velocity, eou e dimensionless amplitude of the excitation velocity, e/R0 Co amplitude of the displacement excitation Co dimensionless amplitude of the displacement, £O/RQ f n m 1st or 2nd order mode shape coefficients of the potential function $ 77 damping ratio of the oscillatory system, C e / C c , where CC is the system critical damping coefficient r}2 damping ratio of the secondary system rj / free surface elevation of the sloshing liquid f\f dimensionless free surface elevation, rjf/Ro rjrta aerodynamic reduced damping ratio, 4irrj M Pad^Lm VB, VS2 e 0 reduced damping ratio for the nutation damper, 2ueMi maximum value of rjrj over a range of frequency for a given ampli-tude of excitation primary and secondary system inherent damping ratio, respectively angular moving coordinate for the nutation damper rotation of the secondary damping device XXI power coefficient of the exponentially decaying velocity profile for Uz as defined in Appendix IV; also used as the exponent for the stability analysis in Appendices II and III nutation damper eigenvalues, solution of C'n(Xnma) = 0 Bessel function relations defined in Appendix V , p. 193 sloshing liquid absolute viscosity coefficient air stream kinematic viscosity coefficient sloshing liquid kinematic viscosity coefficient 1st order detuning parameter for CJ 2nd order detuning parameter for CJ phase angle between excitation and potential function sinusoidal component of $ volumetric mass of the sloshing liquid volumetric mass of the air stream dimensionless time, u>ut various time scales of the expansion defined in eq. 9, p. 18 potential function for the sloshing liquid flow field nth order in the series solution of $ = 2_. enq®n, q = 1/3,1 potential function relative to the moving coordinates r, 0, z potential function for the damper solid body motion phase angle between the excitation and the cosinusoidal component A of the potential function $ phase angle between excitation and sloshing response n dimensionless potential function, 2nd order component of the potential flow solution $ inherent phase angle between excitation and sloshing response dimensionless exciting frequency, 0Je/u)u excitation angular frequency, 2irf natural frequency of the one-degree-of-freedom system 1st and 2nd natural angular frequency of the two-degree-of-freedom system, 27r/n l and 27r/n 2, respectively ratio of a^/u^ fundamental angular frequency of the primary and secondary sys-tem, respectively sloshing liquid natural frequencies, \l ^ tanh\\\h power coefficient of the exponentially decaying boundary layer cor-rection velocity u 2 A 2nd order coefficient for the potential function <& defined in Appen-dices II, p. 164, and III, pp. 167, 169-170 A C K O W L E D G E M E N T xxiii T h e a u t h o r w i s h e s t o t h a n k D r . V . J . M o d i f o r h i s a s s i s t a n c e a s w e l l a s f o r p r o v i d i n g m u c h o f t h e i n s p i r a t i o n d u r i n g t h e c o u r s e o f t h i s t h e s i s . T h e m o d e l s u s e d d u r i n g t h e e x p e r i m e n t s w e r e c o n s t r u c t e d i n t h e m a c h i n e s h o p a n d s p e c i a l t h a n k s a r e d u e t o t h e t e c h n i c i a n s f o r t h e i r h i g h q u a l i t y w o r k a n d c o o p e r a t i o n . S o m e o f D r . G a r t s h o r e ' s w i n d t u n n e l e q u i p m e n t w a s b o r r o w e d . H i s g e n e r o s i t y i n p e r m i t t i n g i t s u s e i s g r e a t l y a p p r e c i a t e d . T h e f e l l o w g r a d u a t e s t u d e n t s o f t h e D e p a r t m e n t w e r e v e r y h e l p f u l i n t h e r e a l i z a t i o n o f t h i s p r o j e c t t h r o u g h t h e d i s c u s s i o n a n d s h a r i n g o f t e c h n i c a l i n f o r m a t i o n . T h e a u t h o r w a n t s t o e x p r e s s h i s a p p r e c i a t i o n f o r t h e r e s e a r c h a s s i s t a n t s h i p a w a r d e d f r o m t h e N a t u r a l S c i e n c e s a n d E n g i n e e r i n g R e s e a r c h C o u n c i l o f C a n a d a ' s g r a n t t o D r . M o d i . T h e a u t h o r i s a l s o v e r y g r a t e f u l t o L o u i s e f o r h e r m o r a l s u p p o r t . 1. I N T R O D U C T I O N l 1.1 Preliminary Considerations A number of large structures such as smokestacks, tall buildings, bridges and other bluff bodies are known to oscillate under the action of the natural wind. A l -though there are many possible mechanisms for such behavior, it is the relatively low frequency cross-flow response generated by vortex resonance or galloping that has been often identified as the cause for structural damage. Vortex resonance takes place when the frequency of the alternate vortices i n the wake of a struc-ture, being governed by the Strouhal number, coincides with one of the natural frequencies of the structure itself (Fig. l a ) . Large amplitudes can generally be reached under conditions of low inherent damping and favorable wind velocities. More recent occurrences involving tall smokestacks have been reported by Hirsch and Ruscheweyh 1, and C h a u l i a 2 . Conditions creating the presence of an asymme-try i n the wake of a bluff body with the wind having a certain angle of attack may cause galloping. It is a type of self-induced oscillation which takes place when the body is aerodynamically unstable while the excitation is generated by the motion itself, as illustrated i n F i g . 1(b). A classical example is the galloping of sleeted transmission lines under severe icing conditions. Due to their widespread occurrences and the extent of damage, prediction and suppression of wind-induced oscillations have been the object of many studies. A 2 v o r t i c e s p l a n v i e w Fig. 1 W i n d induced instabilities of bluff bodies undergoing: (a) vortex res-onance; (b) galloping 3 common approach to reduce vortex resonance has dealt with the modification of the fluid mechanics responsible for the time dependent excitation and has led to the design of helical strakes, perforated shrouds, slats and other such devices (Fig. 2a). This is also referred to as an addition of external or aerodynamic damping. The concept of strakes has often been used around steel smokestacks and in ocean engineering applications, although the resulting increase in aero or hydrodynamic drag associated with most of these devices is a serious limitation. As the response to wind excitations was also found to be quite sensitive to internal damping, another approach has been the installation of various types of passive devices such as tuned mass or impact dampers, hydraulic dashpots, etc. (Fig. 2b). A tuned mass damper essentially consists of an auxiliary mass attached to the main structure by a simple configuration which provides stiffness and damp-ing. It is optimized to acheive minimum response of the primary system to a known excitation. The Stockbridge damper used on transmission lines is another example of such arrangements, where two heavy weights linked by a cable provide counter-acting motion while energy is dissipated within the cable strands. Tower or bridge trusses can similarly be equipped with secondary masses supported by a rubber stem 3. More sophisticated arrangements of counteracting masses have resulted in the development of active systems. The choice of material, size, and performance independent of the wind direction are of course important considerations in their designs. H e l i c a l S t r a k e s S h r o u d s S l a t s / / / / / M , D a m p e r r r i T u n e d M a s s D a m p e r S t o c k b r i d g e D a m p e r ( b ) H y d r a u l i c D a s h p o t Fig. 2 Several typical devices providing: (a) external damping; (b) internal damping 5 In the same category of devices belongs a relatively simple concept involving the motion of a liquid within a closed container with dissipation of energy through the action of viscous and turbulent stresses. The presence of a free surface permits sig-nificant displacement of the sloshing fluid. This thesis proposes axisymmetric torus shaped containers, also called nutation dampers, as a means to suppress wind-induced oscillations. Motivation for the present investigation came from spacecraft technology where partially filled containers are frequently used to control very long period (90 minutes to around 24 hours) librational motion. As the frequency en-countered in wind-induced instabilities of large structures is relatively low, typically less than 1 Hz, it seemed appropriate to explore applicability of nutation dampers to this class of problems. 1.2 Literature Survey Scruton and Walshe4 made a significant contribution to the suppression of the vortex resonance type of wind-induced oscillations with the concept of helical strakes for structures of circular cross-sections in the 1950's, while Price5 intro-duced the perforated shrouds. A number of other aerodynamic devices, such as the slat configuration6, were subsequently proposed. A comprehensive classification of the devices and comparative assessments were later undertaken by Zdravkovich7, as well as Every, King and Weaver8 who discussed the instabilities of immersed marine cables. Wong and Cox9 drew a less extensive comparison scheme based on systematic wind tunnel tests. However, only a few studies such as the exploratory 6 work by Naudascher et al . 1 0 have attacked the problem of galloping instabilities using this approach. Meanwhile, ways of increasing energy dissipation within structures have received an equal amount of attention. In the 1960's, Reed1 1 investigated the applicability of impact dampers to lightmasts and antennas. The installation of hydraulic dashpots on guyed structures was well illustrated by Den Hartog12, and more recently the ad-dition of viscoelastic material into the walls was modelled analytically by Gasparini et al . 1 3 , with Ogendo et al . 1 4 presenting results on a full-scale steel smokestack foundation. However, it is the tuned mass damper, also called dynamic vibration absorber, that has been most popular with a wide range of practical applications to bridges and towers, as indicated by Wardlaw and Cooper15 as well as Hunt 1 6 . Performance of the device on steel smokestacks was evaluated and compared against that of heli-cal strakes during wind tunnel tests in smooth flow by Ruscheweyh17, and results in turbulent flow were given by Tanaka and Mah 1 8 . Stockbridge dampers, bearing the name of the inventor19, are used extensively for controlling transmission line oscil-lations. Their application to this class of problems is still under investigation2 0 - 2 1. Several analytical schemes have also been developed by Schafer and others 2 2 - 2 4 to predict the damped response of conductors. An extension of the concept of tuned mass dampers has been the introduction of active or semi-active systems including 7 a feedback mechanism to control inertia forces25. They also have been considered for earthquake applications as indicated by Chowdhury et al . 2 6 and Yang 2 7. Hirsch et a l . 2 8 - 3 0 have reviewed this literature at some length. Another interesting development has been to exploit the liquid motion within a closed container to design suitable dampers. Brunner31 studied a full-scale tank containing viscous oil flowing through stacks of perforated plates on a smokestack, and Berlamont32 considered the water tank of a tower fitted with baffles. However, it is Modi et al . 3 3 who first carried out wind tunnel tests to validate the idea. Very recently, Bauer34 proposed utilizing the sloshing motion of two immiscible liquids within a rectangular container, while Kwek3 5 used a tank of water to provide the auxiliary mass with energy dissipation taking place in the shock absorbers support-ing the system. Liquid sloshing has had limited success with ship stability16, however, it has been used extensively to control nutation motion of satellites. Although many stud-ies have dealt with its effect on satellite dynamics 3 6 - 4 1 , relatively little is known about the damper behavior. A few experimental investigations have been reported by several authors 4 2 - 4 3 . Alfriend44 tried to theoretically analyse the flow as a rigid slug moving inside the ring and Tossman45 predicted damping characteristics for a tube fitted with a solid rolling ball. However, one has to turn to the early research efforts at agencies such as NASA or ABMA (which were concerned with fuel-rocket 8 interactions), to find important information about liquid sloshing theory. These contributions are reviewed by Cooper46 and Abramson47. Interests of civil engi-neers such as Jacobsen48 and Housner 4 9 - 5 0 , to predict water tank response under earthquake excitation, have also contributed to the field. Earlier studies were aimed at analytical solutions of a potential function for linearized free surface conditions, with typical work by Graham and Rodriguez51, and Chu 5 2 for rectangular and elliptical containers under harmonic excitation, re-spectively. Bauer53 derived a theory for the straight wall torus. More complicated problems started to be examined, such as the compartimented cylindrical tank54 or the flexible wall interaction based on a variational approach55, sometimes re-quiring numerical procedures 5 6 - 5 8. Equivalent mechanical models also emerged to simplify analysis 5 9 - 6 0 and reached a high degree of sophistication with models by Bauer61 and the pendulum analogy by Sayar62. Meanwhile, nonlinear free surface conditions were included with Hutton's theory63 for circular cylinders in particular, to be followed by Woodward and Bauer's approach64 for the torus case. These formulations were then applied by Abramson65 and Chen 6 6 to derive sloshing gen-erated pressure forces on container walls, and were substantiated by experiments. Recently, Miles 6 7 derived a fairly general theory based on the variational principle proposed by Whitham68 and others 6 9 - 7 0 , and verified Hutton's results for circular cylinders71. He also included a solution procedure for the case of resonant interac-tions encountered in liquid sloshing72. This typically nonlinear behavior has been 9 found in many ocean wave problems77-78, as summarized by Philips76, as well as in other areas of research Consideration for the damping terms came from Ocean Engineering in the 1950's, with Hunt79 and Ursell80 linearizing the momentum equations account-ing for viscosity. Case and Parkinson81 applied the theory to cylindrical containers undergoing small oscillations, while Miles82 modified it to include surface tension ef-fects. The method is frequently used nowadays83-84, with additional consideration often given to nonlinear terms 8 5 - 8 6. Experimental results were obtained by Silviera et al. 8 7 and Stephens et al. 8 8 for circular tanks with and without baffles, respec-tively, while Summer and Stophan89 found damping characteristics for a spherical container based on a dimensional analysis. More recently, torus shaped nutation dampers were investigated during free vibration tests in a preliminary study90 at the University of British Columbia. 1.3 Scope of the Investigation Optimal efficiency of nutation dampers is first sought through a combination of theoretical and experimental procedures aimed at providing a better understanding of the energy dissipation mechanisms during liquid sloshing. Relatively low vis-cosity fluids are investigated using a nonlinear potential flow model in conjunction with the thin boundary layer correction. The associated theory derived by earlier investigators • ' f o r straight wall containers is extended to include a solution 10 f o r t h e r e s o n a n c e o f t h e h i g h e r o r d e r t e r m s f o u n d t o b e p r e s e n t f o r a c e r t a i n c l a s s o f d a m p e r s . T h e m e t h o d t h u s p r e d i c t s t h e p r e s s u r e a n d b o u n d a r y l a y e r d a m p i n g f o r c e s a n d p r o v i d e s i m p o r t a n t i n f o r m a t i o n a b o u t t h e c o n t r o l l i n g p a r a m e t e r s , r e s o -n a n t c o n d i t i o n s , k i n e t i c e n e r g y , e t c . T h i s i s f o l l o w e d b y a n e x t e n s i v e t e s t p r o g r a m t o a s s e s s v a l i d i t y o f t h e t h e o r y a s w e l l a s t o s u p p l y m o r e a c c u r a t e d a t a n e e d e d f o r p r a c t i c a l a p p l i c a t i o n s . E m -p h a s i s i s p l a c e d o n t h e c o n d i t i o n s f o r m a x i m u m d a m p i n g b y g e n e r a l l y o p e r a t i n g a t t h e n a t u r a l f r e q u e n c y o f t h e f i r s t a n t i s y m m e t r i c s l o s h i n g m o d e d u r i n g f r e e a n d f o r c e d o s c i l l a t i o n t e s t s , a s s u g g e s t e d b y p r e v i o u s i n v e s t i g a t i o n s 9 0 . P e r f o r m a n c e o f d a m p e r s f i t t e d w i t h a d d i t i o n a l d e v i c e s s u c h a s b a f f l e s i s r e a s s e s s e d i n t h i s p r o c e s s . T h e m a i n o b j e c t i v e i s t o a r r i v e a t a n o p t i m u m c o m b i n a t i o n o f s y s t e m p a r a m e t e r s s u c h a s d a m p e r g e o m e t r y (D, d, h), l i q u i d p r o p e r t i e s (p, Vf), a n d e x t e r n a l v a r i a b l e s o f e x c i t a t i o n a m p l i t u d e (eo) a n d f r e q u e n c y (ue) l e a d i n g t o m a x i m u m d i s s i p a t i o n o f e n e r g y t h r o u g h l i q u i d s l o s h i n g . S o m e o f t h e v a r i a b l e s a r e i n d i c a t e d i n t h e s k e t c h b e l o w . 11 Application of the concept to control vortex resonance and galloping types of wind-induced oscillations is subsequently investigated during wind tunnel tests. Al-though a successful model to approximate response of circular cross-section geome-tries is not yet available, considerable experimental data have led to well estab-lished empirical procedures 9 1 - 9 2. Furthermore, the galloping theory has shown to accurately predict oscillations of a square prism with viscous damping 9 3 - 9 4 . Ex-periments were therefore designed to permit analysis of the response based on this information. Elastically mounted circular and square cylinders fitted with various types of nutation dampers were tested in simulated conditions of smooth and tur-bulent winds rising the closed circuit laminar flow and the boundary layer wind tunnels of the Department. The models underwent either two-dimensional plung-ing or three-dimensional rotational motion. Quantitative assessment of the damper performance under these highly nonlinear excitation conditions was carried out, and effect of the controlling parameters such as damper geometry, liquid height, internal configuration, etc., compared with the results obtained during the liquid sloshing study to arrive at final recommendations. 2. A N A P P R O X I M A T E A N A L Y T I C A L A P P R O A C H T O P R E D I C T E N E R G Y D I S S I P A T I O N 12 2.1 P r e l i m i n a r y R e m a r k s T h e v e l o c i t y field w i t h i n a simple r i g i d torus d a m p e r 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 t r a n s l a t i o n c a n be a p p r o x i m a t e d b y a p o t e n t i a l flow s o l u t i o n w i t h the assumptions t h a t viscous effects are restricted to a s m a l l b o u n d a r y layer region a n d the flow is l a m i n a r . A n a d d i t i o n a l t e r m a c c o u n t i n g for the v e l o c i t y profile at the walls is i n t r o d u c e d to assess energy d i s s i p a t i o n t h r o u g h the a c t i o n of the viscous forces. T h e procedure is s i m i l a r t o the one adopted by Case a n d P a r k i n s o n 8 1 . A l t h o u g h the v a r i a t i o n a l f o r m u l a t i o n has lately been quite p o p u l a r to solve for the p o t e n t i a l f u n c t i o n , a conventional E u l e r i a n a p p r o a c h is used here to exploit some of the results f o u n d b y previous investigators. It s h o u l d be noted t h a t the s t u d y is restricted to s t r a i g h t w a l l d a m p e r s t o f a c i l i t a t e u n d e r s t a n d i n g of the p r o b l e m , a n d the pressure forces are c a l c u l a t e d a p p l y i n g B e r n o u i l l i ' s equation at the boundaries. 2.2 P o t e n t i a l F l o w S o l u t i o n 2.2.1 B a s i c E q u a t i o n s T h e p o t e n t i a l f u n c t i o n $ ( r , 0, z,t) represents a s o l u t i o n of the differential equa-t i o n , V 2 $ = 0, (1) w i t h t h e b o u n d a r y c o n d i t i o n s : dn = Vn a t t h e w a l l ; d 2 $ a $ a $ a 2 $ 2 a $ a 2 $ + ff^- + 2 — — — + — — - — — + ... = 0; 13 ( 2 a ) (26) dt2 * dz dr drdt r 2 dO d6dt r e p r e s e n t i n g c o m b i n e d k i n e t i c a n d k i n e m a t i c c o n d i t i o n s a t t h e f r e e s u r f a c e . H e r e $ = $ + $/; $ = p o t e n t i a l f u n c t i o n r e l a t i v e t o m o v i n g c o o r d i n a t e s r , 0,z\ $ / = p o t e n t i a l f u n c t i o n f o r t h e d a m p e r s o l i d b o d y m o t i o n . O t h e r g e o m e t r i c v a r i a b l e s a r e i l l u s t r a t e d i n F i g . 3 . 0 x 1 n e r t i a I R e f e r e n c e € 0 s i n c u e t F i g . 3 G e o m e t r y o f t h e s q u a r e s e c t i o n d a m p e r s e l e c t e d f o r a n a l y t i c a l s t u d y A t t h i s s t a g e , i t i s c o n v e n i e n t t o d e f i n e t h e d i m e n s i o n l e s s p a r a m e t e r s : 14 • potential function : $ = Rfan' • moving coordinates : f = z = Ro Ro • excitation amplitude and frequency : co = 4°-, CJ = • time : r = u>n t. 2.2.2 Linear Solution As the free surface occupies different orientations during the damper motion, a standard Taylor series expansion of equation (2b) around z = 0 is used and a linear solution obtained by neglecting second and higher order terms (Appendix I). Applying the procedure of separation of variables, the linearized system yields a solution in terms of the Fourier-Bessel expansion, as found by Bauer53, and is presented here in a dimensionless form: 5 » V ^ r r< i \ McoshAit(2 + fc) <P = e 2_j yitW (Ai,r) — s — - cos 0 cos UT; (3a) . coshAi,7i where: e = dimensionless amplitude of the disturbance, c0w; fu = amplitude coefficient of mode (l,i), i.e. 1st circumferential, ith transverse mode, ^}^U\ ^y.^1^.1*^ ', (36) [ ( w i i / w ) - I j A i , -with: a = g , and Alt- = j[C12(Ali)(A?1. - 1) - C2(AM«)(A?f.«a - 1)]; (jju = liquid natural angular frequency in mode (l,i), i.e., -^-ctu] Ro 15 Ait = e i g e n v a l u e f o r m o d e ( l , i ) , r e p r e s e n t i n g s o l u t i o n o f : C ( ( A l t ) = 0 , w i t h C1(Xlif) = J^Xuf)-^ n ( A i . T ) ; a n d a i , - = t a n h ( A i , 7 i ) , h = ——. RQ N o t e J i a n d Y\ a r e B e s s e l f u n c t i o n s o f t h e f i r s t a n d s e c o n d k i n d , o r d e r o n e , r e s p e c -t i v e l y , a n d p r i m e d e n o t e s d i f f e r e n t i a t i o n w i t h r e s p e c t t o f . O f c o u r s e , t h e l i n e a r s o l u t i o n c a n n o t b e e x p e c t e d t o b e a c c u r a t e f o r h i g h e r d i s t u r b i n g a m p l i t u d e s e, a n d i s n o t v a l i d n e a r r e s o n a n c e a s t h e e x p r e s s i o n f o r $ b e c o m e s v e r y l a r g e a n d g o e s t o i n f i n i t y f o r u>e = uu. T w o c a s e s a r e t h e r e f o r e c o n s i d e r e d t o e x t e n d t h e a n a l y s i s t o t h e n o n l i n e a r r a n g e . 2 . 2 . 3 N o n l i n e a r , N o n r e s o n a n t S o l u t i o n ( w e 96 w i i ) A p e r t u r b a t i o n m e t h o d i s a p p l i e d u s i n g t h e p r o c e s s o f i t e r a t i o n v a l i d f o r s m a l l p a r a m e t e r s c w h e r e a t h i r d o r d e r e x p a n s i o n i s a s s u m e d , i . e . , $ = 6$(D + ? 2 $ ( 2 ) + g » $ ( S ) + _ (4) H e r e i s t h e l i n e a r t e r m o f s e c t i o n 2 . 2 . 2 , a n d c a n b e d e r i v e d b y s u b s t i t u t i n g f o r i n t h e s e c o n d o r d e r f r e e s u r f a c e c o n d i t i o n ( A p p e n d i x 1 .2 ) . I t i s f o u n d ( A p p e n d i x I I . 1) t h a t , + / 2 n C 2 ( A 2 n f ) c o s h A 2 n ( z + h) c o s h A 2 n / i c o s 26 s i n 2 d r , w h e r e : 16 font fin — amplitude coefficients of (0,n), (2,n) mode, respectively; Aon>A2n = eigenvalues of (0,n), (2,n) mode, respectively, i.e., solution of: C 0(Ao na) = 0 and C'2(\2nO) — °. $^ ' can similarly be obtained although it was not considered here. The stability condition is presented in Appendix II.2. The solution is not always valid due to the resonance of the nonlinear higher modes at certain values of CJ, as discussed in section 2.2.5 later. The occurrence of such singularities is, however, localized to a small range of excitation frequencies and is generally not dealt with in this study. One case of interest involves the first transverse, second circumferential mode responsible for the resonant interactions around CJ = 1 , and is treated in the next section. 2.2.4 Nonlinear, Resonant Solution (we « w l t) A different expansion is required for the solution around the first axisymmetric circumferential mode (CJ « 1) by taking the excitation to be of the order of the nonlinear terms in equation (2b), i.e., where q < 1, determined through the iterative process. A detuning equality required to eliminate secular terms is defined as (5) (6) 17 The first order equation reduces to the free vibration linear case, i.e., d*$W i for the first transverse mode shapes oscillating at the natural frequencies u>u, and by neglecting the higher modes leads to the solution of the form $V > = Cx(Anr) — - fn cos[0 + (pii) coswr coshAn/i L + fn sin(0 + £n) sinwr , (8) where fu, fn are the amplitude coefficients for the solution, and <pu, £u are the phase angles in 6. However, for the case where higher circumferential natural frequencies are multiple of the first mode (o;nl « nwn), additional significant terms appear in equation (8). This is the condition for resonant interactions treated separately in the subsequent analysis. 2.2.4.1 No Interactions (wnl 96 n«„i) The analysis is similar to Hutton's theory for circular cylinders63 with a major difference in the type of Bessel functions describing the transverse modes. Only a short outline of the procedure and results, useful to introduce the next case, is therefore pesented. Second and third order expressions are found in terms of by substituting relation (8) into (2b). The exponent of the perturbation parameter is required to be q = 1/3, and the detuning parameters U\ and i / 2 of (6) as well as the coefficients fu, f l x and phase angles c p n , £ n of (8) are found by setting the secular terms containing cos(0 + <pu) cos CJT or sin(0 + £ 1 1 ) sin CJT to zero in the 18 2nd and 3rd order free surface conditions while using a multiple time-scale analysis. T h e expansion T = T0 + Ti + T2 + (9) where ro = T, T\ = e ^ r , and r 2 = e 2 / 3 r , yields two limit cycles: planar solution: f n = <Pu = tu = 0; and / u satisfying nonplanar solution: hi{K1f^1 + u2) + F1 = 0; (10a) stable for : - ^ - ( ^ - - 2^/^) < 0, (106) fii In (10c) hi hi V i i = £ u = 0; with fn satisfying fu[-{KKx - 2KJ& + v2\ + JZ±-FX = 0, (11a) and ch = fii + Fi 1 (116) stable f o r : ^ ^ ^ < 0,real. (11c) Here i / 2 is a function of the excitation. Fi, K\ and KK\ are damper geometry dependent parameters while B and C vary with fu, Kx, KKX (Appendix III.l). T h e second order terms are $( 2 ) = V 2 n s i n 2 0 c o s 2 w r + (V>on + V » 2 n c o s 2 0 ) sin2wr, (12a) where: V>0n = f W / u - f l l ) C o ( A o n ^ ) V'2n = n 2 n ( / 1 2 1 + c 1 2 1 ) c 2 ( A 2 n f ) c o s h A 0 n ( « + h) cosh Aon^ c o s h A 2 n ( £ + h) cosh A 2 n / i (126) (12c) 19 2 . 2 . 4 . 2 Resonant Interactions (oJni n w n ) It is found that, for dampers with a going to 1 and relatively small h, higher natural frequencies of the first transverse mode tend to be near multiples of w u , A For instance, w 2 i = 1 . 9 9 w n , u>31 = 2 . 9 6 u ; i i , etc., for a = 0 . 9 and h = 0 . 1 0 . This particular situation leads to relatively large terms in the nonlinear solution described in the previous section which would make the expansion invalid. T h e following development recognizes that the super harmonic modes multiple of w u are now solution of the 1 s t order free surface condition. T h e generating solution is then * ( D r> t\ ^ c o s h A n ( z + h) r = C i ( A u r ) v / n c o s ( g + ¥ ? i i ) c o s b ; r c o s h A n / i L + f i i s i n ( 0 + ( fn)sinwrj _ c o s h A 2 i ( z + h) r , , „ + C M A 2 i f ) — L / 2 1 c o s ( 2 0 + v ? 2 i ) c o s 2 w r c o s h A 2 i / i «• + f 2 i s in (20 + £2i) sin 2 w r J „ . , ^. cosh A 3 i ( z + h) r . . „ + C 3 ( A 3 i f ) u « ' / 3 i c o s ( 3 0 + v ? 3 i ) c o s 3 w r c o s h A 3 i / i >• + c r 3 1 s in (30 + £ 3 1 ) sin 3 w r ] + etc., ( 1 3 ) where the number of interactions strictly depend on the damper geometry parame-ters a and h. T h e problem, however, becomes quite complex with additional terms i n and it is assumed that / 3 1 , f 3 1 , / 4 1 , etc., are small compared to the coeffi-cients of the first two modes and can be neglected. This was found to be always true for a particular class of dampers (typically 0 . 5 < a < 0 . 6 ) where only two interac-20 tions occurred. This study is also restricted to the planar mode, and contributions from nonplanar f u and / 2 i coefficients to stability requirements are ignored. $ ^ thus reduces to jk(D t ra, f \ *x cosh A n (z + h) $l > = fu c o s ( d + v ? i i ) C i ( A n r ) - — s — - c o s w r cosh A n ^ ^ . , . . . cosh A 2 i ( z + h) . . . + f 2 i C 2 ( A 2 i f ) sin(20 + 6 1 ) - ;sin2£>r. (14) cosh A 2 i / i A second detuning equality as in (6) is now introduced, = 4 - foe* - 02e2* - (15) r.2 11 2 " 2 1 u> and a procedure similar to that of the previous case is used to solve for ux, i / 2 , Pi, P2, / i i i f 2 i , ¥>n and £n, with secular terms in cos(0 + <p\\) coswr as well as sin(20 + £21) sin2<2>r set to zero in both 2nd and 3rd order free surface conditions. This approach is similar to the one employed by Bajkowski et a l . 9 5 . The coefficients fiii f 2i> and phase angles <pu and £ n are now function of the slow time scales T\ and r 2 as defined in (9). Details of the analysis are given in Appendix III.2. The exponent of the perturbation is taken to be q = 1/3 as it is desirable to obtain a limiting solution that tends towards the previous case when the interactions become small. The results are as follows: <Pu = 0, £ 2 1 = - j ; with fu and f 2 i solution of the 3rd order nonlinear system of equations: / i i ( # i / i 2 i + # i f 2 2 i + V2) + Fx = 0; (16) f 2 l ( i f 2 f | i + ^ 2 / n + ) 9 2 ) =0. 21 H e r e u2 = v — 0 ^ 2 1 , a n d / ? 2 = P — b\ -^-> w i t h v a n d /? f u n c t i o n s o f t h e e x c i t a t i o n £ 2 1 a n d d a m p e r g e o m e t r y . T h e s t a b i l i t y c o n d i t i o n i s r e p r e s e n t e d b y ( / » ) • > Z ^ A , ( 1 7 ) a s s h o w n i n A p p e n d i x I I I . 2 . 3 . T h e s e c o n d o r d e r t e r m s a r e o f t h e f o r m ( A p p e n d i x I I I . 2 . 1 ) $ ( 2 ) = ( ^ l n c o s 0 + A i n c o s 3 0 ) c o s w r + (fan c o s 2 0 + A 2 n ) s i n 2 u > T + (03n c o s 3 0 + A 3 n c o s 0) c o s 3 w t + (V>4n c o s 4 0 + A 4 n ) s i n ACJT, ( 1 8 a ) w h e r e : c o s h A m „ ft. A m n = Y. ^nCP(Xpnr) COShX[f + ( 1 8 c ) n c o s h A p n / i w i t h p = 4 — m , f o r m = 1 , 3 , 4 ; a n d p = 0 , f o r m = 2 . 2 . 2 . 5 P r o p e r t i e s o f t h e P o t e n t i a l F u n c t i o n 2 . 2 . 5 . 1 V a r i a t i o n w i t h D a m p e r G e o m e t r y T h e a m p l i t u d e c o e f f i c i e n t s / m n a n d f m n o f t h e v a r i o u s m o d e s d e p e n d o n t h e A e x c i t a t i o n a n d t h e d a m p e r f l o w b o u n d a r y c h a r a c t e r i z e d b y t h e v a r i a b l e s a a n d h. T h e e f f e c t o f g e o m e t r y o n t h e f i r s t m o d e i s c o n t a i n e d i n t h e t e r m s F\,K\, KK\, e t c . , a s g i v e n b y r e l a t i o n s 3 ( b ) , 1 0 ( a ) o r 1 6 . F i i s t h e c o e f f i c i e n t o f t h e l i n e a r s o l u t i o n . A ^ I t i s n o t a f u n c t i o n o f h, a n d i t s v a r i a t i o n w i t h a i s n o t p r o n o u n c e d , r a n g i n g f r o m 22 1 . 4 4 a t a = 0 ( l i m i t i n g c a s e o f t h e c i r c u l a r c y l i n d e r ) t o 1 . 1 7 a t a = 0 . 5 , a s s h o w n i n F i g . 4 . A l l d a m p e r s s h o u l d t h e r e f o r e e x h i b i t s i m i l a r p r o p e r t i e s i n t h e l i n e a r r a n g e , i . e . , a w a y f r o m r e s o n a n c e . T h e n o n l i n e a r t e r m s a r e i n c l u d e d i n K\ f o r t h e p l a n a r m o d e , o r a c o m b i n a t i o n o f K\, K2, Ex, a n d E2 f o r t h e c a s e o f r e s o n a n t i n t e r a c t i o n s , w i t h KK\ c o n t r i b u t i n g t o t h e s t a b i l i t y o f t h e m o t i o n ( r e l a t i o n 1 0 c ) . A U n l i k e Fx, t h e s e t e r m s a r e s t r o n g l y d e p e n d e n t o n a a n d h. T h i s i s m a i n l y d u e t o t h e r e s o n a n c e o f t h e n o n l i n e a r m o d e s i n t h e p r o x i m i t y o f CJ = 1 . I n g e n e r a l , Kx i n c r e a s e s a s i t a p p r o a c h e s t h e n a t u r a l f r e q u e n c y o f o n e o f t h e h i g h e r o r d e r t e r m s . F o r t h e p a r t i c u l a r c a s e o f i n t e r a c t i o n s w i t h m o d e ( 2 , 1 ) d e a l t w i t h i n t h i s s t u d y , t h e a n a l y s i s y i e l d s a f a i r l y l o w v a l u e f o r Kx, w i t h l a r g e Ex, E2 a n d K2. C o n t r i b u t i o n o f t h e l a t t e r p a r a m e t e r s t o t h e o v e r a l l n o n l i n e a r b e h a v i o r m a y h o w e v e r b e m o d e s t a s t h e a m p l i t u d e o f t h e i n t e r a c t i n g m o d e p r o v e d t o b e u s u a l l y m u c h s m a l l e r t h a n fu-F, 3.0-2.0-1.0-0 0 0.2 0.4 0.6 0.8 1.0 a F i g . 4 V a r i a t i o n o f t h e l i n e a r c o e f f i c i e n t Fx w i t h a / 23 Plots of the various coefficients versus a show these large fluctuations quite clearly, as illustrated by Fig. 5(a) and (b), for h = 0.1 and 1.0, respectively. At low h, resonance of the mode (2,1) is alrealdy felt at a = 0.4, with very large, neg-ative Ki beyond this point (Fig. 5a). The interactions, however, keep K\ to the order of 0.5 or less, while E\, Ei and Ki start to grow with increasing a. At high h, resonance of both modes (2,2) and (0,2) result in very large, positive K\ near a = 0.20 and 0.37, respectively (Fig. 5b). The latter becomes quite small beyond resonance, thus suggesting that the response in the first planar mode is very close to that given by the linear solution for a > 0.6. The motion is, however, quite unstable as KKi decreases as well. It should be noted that the potential flow solution was not derived for these resonant modes as they are confined to a narrow range, e.g., from o = 0.35 to 0.45 for h = 1.0 with mode (0,2). For a damper with a given a, the nonlinear terms can be quite dependent on the liquid height, as shown in Fig. 5(d). No interactions are present throughout the A range of h considered, but K\ varies significantly from negative to positive values crossing the x-axis at several points. Negative K\ implies "hardening" character-istics, i.e., the resonant frequency increases with the amplitude of the excitation, whereas "softening" results from a positive value, and of course the solution is linear for K\ — 0. This general behavior was noticed during earlier work on nonlinear sloshing66 and can have interesting implications for maximizing energy dissipation, as the absence of nonlinearities theoretically yields infinite response amplitudes and 24 A damping at CJ = 1.0. This condition is met here for h = 0.32, 0.62 and 0.79, al-though the motion is unstable for the last two points due to small KK\. A t higher a, resonant interactions with the mode (2,1) are present at low h and the trends are similar to those mentioned earlier, i.e., small K\ and growing E\, E2, and K2 in the interacting region in contrast to a decreasing K\ away from resonance (Fig. A 5c for a = 0.608). However, fluctuations across the range of h are smaller here, and this damper is expected to be less sensitive to liquid height. A brief examination of the terms controlling the nonplanar coefficient f n , i.e. K\jKK\ and [KK\ — 2K\), suggests that this mode is equally sensitive to liquid height at a = 0.308 (Fig. 6a), and to the resonance of the higher modes as shown in F i g . 6(b) for a = 0.608. 2.2.5.2 Variation with the Excitation T h e nonresonant solution is a function of the frequency CJ while the displacement velocity of the damper walls 6 affects the resonant response in two ways. Firstly, it generates higher nonlinear terms as it grows larger, usually resulting in lower amplitude coefficients / i i or f u as well as important shifts in the resonant frequency (softening or hardening characteristics), as governed by relations (10a), (11a) or (16). Secondly, the resonant region expands as it is required that \CJ - 1| < — c 1 / 3 + higher order, (19) 2 non-interacting solution - 1 0 0 -h r l . O resonance of mode (0,2) - 2 -- 1 0 0 -I 1^ resonance of mode (2,2) / / (b) \i 0 . 2 0 . 4 0 J B 0~!i iTfJ a non-interacting solution resonance of mode (2,1) \ a r 0 . 3 0 8 (d) ~o^2 b 7 4 0 ^ 6 o T i A 1 * 0 h Fig. 5 Planar mode coefficients Ku KKX, Eu E2 and Ki as affected by the damper geometry to 26 (KK,-2K, ) K, /KK, a=0.308 6-2-a = 0.608 0 1 1— I—— r 0 0.2 0.4 0.6 0.8 1.0 A h F i g . 6 N o n p l a n a r c o e f f i c i e n t s —{KK\ — 2K{) a n d K\jKK\ a s f u n c t i o n s o f h f o r : ( a ) a = 0 . 3 0 8 ; ( b ) a = 0 . 6 0 8 f r o m t h e d e t u n i n g e q u a l i t y ( 6 ) . T h i s p h y s i c a l l y m a k e s s e n s e a s a l i q u i d o r i g i n a l l y s l o s h i n g i n t h e n o n r e s o n a n t r e g i o n u n d e r a s m a l l e x c i t a t i o n m a y l i t e r a l l y r e s o n a t e a t h i g h e r a m p l i t u d e s , w i t h t h e o c c u r r e n c e o f t h e n o n p l a n a r m o t i o n . I t i s i n t e r e s t i n g t o n o t e t h a t t h e s a m e p r i n c i p l e h o l d s f o r t h e h i g h e r n o n l i n e a r m o d e s . F o r i n s t a n c e , t h e 2 n d o r d e r t e r m s o f t h e n o n r e s o n a n t s o l u t i o n s h o u l d r e s o n a t e w h e n 27 | ^ - 2 | < ^ , (20) u 4 where: / = 0,2; n = 1,2,...; and UQ is of order 1 (Appendix II.l) ; a condition gen-erally easier to meet with larger c. Experiments, however, suggest that UQ does not need to be as high as 1, as discussed in Chapter 3. Thus more resonant interactions are expected at higher excitations. T o illustrate how the various regions overlap, F ig . 7(a) and (b) show / n , nor-malized by c 2 / 3 for the resonant solution to account for different expansions, versus CJ. T w o amplitudes e0 and damper geometries are considered. T h e coefficient / o i (now normalized by e 4 / / 3 at resonance) for the first configuration is presented in F i g . 7(c). Noteworthy are the hardening characteristics with increasing e 0 displayed in F i g . 7(a), and the corresponding reduction in fu, as opposed to more linear be-havior of the damper with resonant interactions (Fig. 7b). A region with two equilibrium positions is usually present with fu exhibiting the jump phenomenon as one branch ceases to exist or becomes unstable. In the case of a = 0.608, and h = 0.196, both branches collapse near resonance (CJ = 0.98) while another loop de-velops at a higher frequency. T h e interacting coefficient is shown as a fraction of fu in F i g . 7(d) and stays small due to the stability requirements ( | f 2 i / / n | < 0-32 here, from relation 17). Variation of f u , / o i and jfei versus CJ for different damper geometries and amplitudes 29 2.3 Pressure Forces The liquid sloshing motion generates a nonuniform pressure on the container wall. The resulting force is F = J j p cos OdA, (21) where: A = vertical wall area in contact with the fluid; p = pressure exerted by the fluid. p can be found using Bernouilli's equation for unsteady flow, ^ ( V $ ) 2 + * * + ^ = 0. (22) F can be nondimensionalized and expressed in terms of an added mass ratio as Mi M/w^ co thus indicating the departure of the liquid from the behavior of a corresponding solid mass Mi. Results for the nonlinear potential flow solution, expanded to the 2nd order, are listed below with details of the derivations given in Appendix IV.1.1: ( i ) t = ^ ) { ? / , ' ^ I C l ( A " ) - o C , ( A " < , ) 1 — €QCJ{AI — Bi)\ ainuT — -x c0w(Ai + B\) sinwr; (24a) > h{\ — a2) — > sin UIT — — sin3wr; (24o) h{l - a2) Cf K 1 30 1 , >Ul . . 1 fr 1 (eo^)1/3 W J fc(l-o2) 1 (cow)1/3 C4 . . \ sin wr >. sin 3£>r (24c) Here cases (i), (ii) and (iii) represent solutions for the nonresonant, resonant with-out and with interactions, respectively, and the A's and B's as well as C4 terms contain the second order cross-products given in Appendix IV. 1.2. These expressions were evaluated and used for comparison against experiments. Most of the results are therefore presented in the next chapter, however, a typical curve useful to discuss the solution characteristics and showing the magnitude of the response at CJ is presented in Fig. 8. As the flow is dominated by the mode shape of Ma the first order term for low amplitudes, | ——| essentially follows the variations set Mi by fu in the planar motion, with a maximum for the lower branch accompanied by a sudden reversal of signs at a resonant point different from CJ = 1.0 due to nonlinear effects. Calculations show that the nonlinear contribution, in expression (24b) for this particular case, is less than 10%, although it may be larger according to the damper geometry, and was substantial for higher e. The ratio of 3rd over 1st harmonic was also found to be of the order of 10%. Meanwhile, the nonplanar response shows a positive added mass beyond resonance that continues to grow Ma until the motion no longer exists. The nonlinear component of \-rj-\ and the 3rd Mi harmonic are also larger and amount to 20% of the first order term. The picture is, of course, quite different at higher amplitudes, as the contribution of the higher 31 m o d e s i s o f t h e s a m e o r d e r a s t h a t o f t h e f i r s t . N o w fu i s g r e a t l y a f f e c t e d b y t h e n o n l i n e a r i t i e s a n d n o l o n g e r d i s p l a y s a w e l l d e f i n e d r e s o n a n t r e g i o n f o r £Q a s l o w a s 0 . 3 a n d CJ a s h i g h a s 2 . 0 i n t h e c a s e c o n s i d e r e d h e r e . Ma/M, -4 a=0.40 hr0.30 €o= 0.036 Planar Nonplanar F i g . 8 T y p i c a l a d d e d m a s s c h a r a c t e r i s t i c s a t l o w a m p l i t u d e 2 . 4 D a m p i n g F o r c e s 2 . 4 . 1 E f f e c t o f V i s c o s i t y F o r i n c o m p r e s s i b l e f l o w , t h e N a v i e r - S t o k e s e q u a t i o n r e d u c e s t o 8 1 — * ( u . V ) u + | ? = -V(gz + I) + ^ + vfV*u, ( 2 5 ) dV w h e r e u i s t h e fluid v e l o c i t y v e c t o r i n t h e m o v i n g f r a m e o f r e f e r e n c e a n d — — i s t h e dt 32 acceleration of the same frame. S u b s t i t u t i n g u = V $ + u 2 , where u 2 is a correction —• v e l o c i t y due to viscosity , a n d recognizing t h a t V — V $ / , [ ( V $ + u 2 ) • V ] ( V $ + u 2 ) + | - ( V $ + u 2 ) at = -V(gz + V- + ^f) + t / , V 2 ( V $ + u 2 ) , i . e . , dU2 ( V $ • V ) V $ + ( u 2 • V ) V $ + ( V $ • V ) u 2 + ( u 2 • V ) u 2 + = -V{gz + + —) + i / , V ( V 2 $ ) + ^ / V 2 u 2 . (26) U s i n g B e r n o u i l l i ' s e q u a t i o n , i . e . , - ( V $ ) 2 + gz H 1- —— = 0, as w e l l as V 2 $ = 0, 2 p at reduce the above equation t o ( u 2 • V ) V * + ( V $ • V ) u 2 + ( u 2 • V ) u 2 + ^ = i / j V 2 ^ . (27) T h e corresponding c o n t i n u i t y equation is V • ( V $ + u 2 ) = 0, i . e . , V • u 2 = 0. (28) A s u 2 = — V $ at the w a l l , the c o r r e c t i o n is of the order of the p o t e n t i a l flow s o l u t i o n , a n d a n expansion of the following f o r m is assumed tT2 = e*u2l) + c 2'4 2 ) + e 3 ? « 2 3 ) + •••> (29) where g — 1/3 a n d 1, at a n d away f r o m resonance, respectively. S u b s t i t u t i n g into (27) y i e l d s , u p t o the 2 n d order: ^ - v^u™ = 0; (30a) 33 a - ( 2 ) - vfV2u[2) + • VJtZ^ + ( V ^ 1 ) • V)4X) + (4X) • V)V$<x> = 0. (306) The linear differential equation (30a) leads to a relatively simple solution for under simplifying assumptions80, i s then derived by substituting for in (30b). Details of the analysis and expressions for the correction velocities are pre-sented in Appendix IV.2.1. 2.4.2 Energy Dissipation and Reduced Damping Ratio On integration of the work done by the viscous stresses, the expression for energy dissipation rate in a viscous liquid can be shown to be 9 6 , dEd dt /x / |V x u\2dv + f (n- V)\u\2dS-2 f n • u x (V x u)dS. (31) Setting u = V $ + u2, considering V x V $ = 0 (irrotational flow), and u — 0 at the boundaries gives d f(V x u2)2dv + f (n-V)|V$|2dsl. (32) dEd dt -j v j z = n Here rjf represents the instantaneous free surface elevation, i.e., deviation from the undisturbed horizontal height. It must be noted that the first term expresses the contribution from the shear forces at the damper walls, and the second term is the effect of the free surface, often neglected in this type of analysis. The integral over a cycle permits calculation of an equivalent reduced damping ratio defined as, 34 Ed where : Ce = equivalent absolute damping coefficient, ~; 7 T W e C 5 Ed = total energy dissipated per cycle. rirti is therefore the damping ratio 77 of a single degree of freedom system of rigid mass M, damping coefficient C e , and natural frequency we, divided by the mass ratio Mi ——. Recognizing that the total energy of a solid mass M oscillating harmonically M with a displacement x = tosmuet is -Me^oj2, it is also the ratio of dissipated to 2 total energy of a corresponding rigid mass Mi during a cycle, divided by Air. A n expression more representative of the dissipation as a fraction of the actual energy in the flow is defined in the next section. Contribution from the boundary layer at the damper walls, complete to the second order without including the effect of the streaming layer (Appendix IV.2.1.2), is given below. Here relations (i),(ii) and (iii) correspond to the cases of nonresonant and resonant cases, as explained before: + I2n(m,n) + J2 n (m,n) ] + e 0 £ 2 A A i } ; (34a) -i&hjimMw^[fh+A)I**u(1'1} + Z/n(l , 1) + 7 2 n ( l , 1) + J2U(1,1)] + }; (346) ( m ) ( i 3 v a | 4 * " ( i - " + " » ( m ) + J 2 n ( l , 1) + J 2 u ( l , 1)] + v/S&Ifcfc^l, 1) + 4 / / 2 2 ( l , 1) + J 2 2 1 ( l , 1) + 4J22 1(1,1)] + + (34c) kk's ,//'s, J2's and J2's are combinations of Bessel and hyperbolic function cross-products (Appendix V.2.3), while AA's and BB3 contain the higher order solutions 35 as shown in Appendix IV.2.2.1. The smaller contribution from the boundary layer at the free surface is obtained to the 1st order only with the following results for the 3 cases: 0) "r,l = y * a2)J^{j2 IC /WlnAinO=ln[/Aii (m, n) v / m n + JAii(m,n) + Am]}; (35a) + J i 4 1 1 ( l , l ) + An]; (356) ( i U ) h ( ^ l h ( i ^ + A n ] + $ 2 2 i A 2 1 a 2 1 [ M 2 2 ( l , 1) + 4 J A 2 2 ( l , 1) + A2 2]}. (35c) It should be noted that the corresponding damping force has to be 90° out of phase with the excitation in order to dissipate energy. Fig. 9 shows the reduced damping ratio versus CJ for the same damper and excitation used during the added mass discussion (Fig. 8). rjrti clearly reaches a maximum at resonance, with magnitude one or two orders higher than that at CJ < 0.9, or > 1.2 for the lower branch of the planar motion, as it is a function of the square of the amplitude coefficient fu. As expected, the free surface boundary layer contribution is small and is of the order of 1% throughout the range of CJ considered. Although the wetted area along the walls is the same as that of the damper bottom in this case (the ratio of the two 2h areas is — y for a torus), the analysis suggests that there is more dissipation at the walls, by a factor of 1.2 — 1.5 for the planar, and around a factor of 2.0 for the 36 Fig. 9 Typical variation of theoretical rjr>i versus CJ nonplanar mode. The higher velocities near the free surface are in fact responsible for such behavior. Very low contribution from bottom surface is usually found at larger h as the velocity gradients become weak near z = —h, suggesting that high liquid heights be avoided for optimizing energy dissipation. Nonlinear terms are small for the planar motion at low amplitudes, but are significant in the nonplanar mode (of the order of 20%). Their magnitude is comparable to that of the first order term at higher ZQ (> 0.30) as was the case for the added mass. Of interest is the fairly constant value of rjr>i with CJ beyond resonance for the nonplanar motion. 37 2.4.3 Energy Ratio Er,i A quantity reflecting energy dissipation efficiency is defined here as Er,i = ^ , (36) where Ed refers to the dissipated energy of relation (31), and Et is the average total energy stored in the liquid relative motion during a cycle, Et = T + U, (37) where: T = average kinetic energy, — / -p / (V$)2dv dt; 2n J0 L2 Jv J U = average potential energy, — / \p I gzdv dt. 2TT J0 L JV J Using the expression of n/ as a function of $ (Appendix 1.3), substituting for $, and integrating using a procedure similar to the one applied to the calculation of the added mass (Appendix IV. 1.1) yields Er>l = 4h(l - a2)^-, (38) where Et is as follows for the various cases of nonresonant, resonant without and with interactions corresponding to (i), (ii) and (iii): (i) Et = ^ { a „ A 1 1 w V i 2 i ^ + / i i^ / iy /? i i (» , i ) [ / i4 1 1 ( t , j ) + JAuihJ) + altlLu]}+(eoCj)2AAAi; (39a) (") Et= 7 r ^ ( / ? 1 + f f 1 ) { | l l « & a A 1 1 + /? 1 1[7A 1 1(l,l) + J A i i ( l , l ) + a^An]} + * /aAAA2; (396) 38 + JAXX{1, 1) + a^Au] + ? 2 2 1^ 2 2(l, 1)[/AM(1,1) + 4 J A 2 2 ( l , 1) + aliAai]} + JXTBBBS + * , AAAa; (39c) where the BBB's coefficients represent the effect of 1st and 2nd order term cross products, and the AAA's coefficients include nonlinear terms only (see Appendix IV.2.3). It can be shown that the first order term in the expression for Erj is inde-pendent of the amplitude of excitation for a given ijr>i. However, it is a function of frequency as the potential energy contains acceleration terms. For a given damper liquid and geometry, the Reynolds number increases with CJ, which further con-tributes to the general downward trend shown in Fig. 10. The higher orders seem to present similar energy dissipation efficiencies as no significant changes in the curves occur at larger eb. To assess the effect of the geometry only, a parameter accounting for the varia-tion of the Reynolds number is defined as E;A = ErAVRe~, (40) since the reduced damping ratio was shown to be essentially proportional to 1/y/Re. It is plotted versus h and a in Fig. 11(a) and (b), respectively. Results indicate that low liquid heights and larger a's are most effective at dissipating energy. How-39 0.04 0.03 0.02 0.01-a=0.40 hr0.30 Re/d>= 2.497x104 € o r0.036, planar € o r0 .036, nonplanar e o r 0.36, planar T -lh 0.7 0.8 0.9 1.0 1.1 1.2 1.4 A 1.6 Fig. 10 Typical curves showing variation of Er>i with frequency and amplitude ever, the potential flow approach is less reliable at smaller h as the boundary layer occupies a more significant part of the liquid volume. The rapid growth of E* t for h < 0.26 in Fig. 11(a) may therefore not occur in practice. It may also be noted (Fig. l ib ) that the downward trend with decreasing a is somewhat stalled below 0.40. The velocity gradients, however, become unrealisticly large near the damper inner wall as a tends to 0 since the curvature effects were neglected in the analysis. The results obtained for small values of this parameter cannot therefore be fully trusted. Overall, the curves present useful trends with relatively short, slender dampers (small h, and a closer to 1) particularly effective in optimizing energy dissipation. Finally, the nonplanar motion consistently exhibited higher Erj compared to the planar mode, as illustrated in Fig. 10. 40 8 6 4 4 « o = 0 . 0 3 2 a = 0.308 A h r ^ — - - - i r - - - -0.260 h = 0.692 (a) i i • i • •-^4^0^40 . € c - 0 . 0 3 6 h - 0 . 2 0 0 (b) 0i8 0.9 i!o i ' . i ^'.2 A CJ Fig. 11 E*i versus CJ as affected by: (a) h; (b) a All the calculations for the theoretical solution were carried out on the main frame computer (Michigan Terminal System) using a Fortran program. 41 3. E X P E R I M E N T A L DETERMINATION OF D A M P E R CHARACTERISTICS 3.1 Preliminary Remarks Experiments in steady-state forced excitation with the damper undergoing a translational motion were designed to assess the theoretical predictions and fur-ther evaluate performance. The controlling dimensionless variables discussed in the previous chapter were varied and the effect of internal devices such as baffles investi-gated. Initially, a flow visualization study was undertaken to confirm the qualitative nature of the mode shapes. This was followed by an extensive set of measurements of the sloshing horizontal force transmitted from the fluid to the damper walls. 3.2 Test Arrangement and Models A Scotch-Yoke mechanism connected to a horizontal frame free to slide over supporting bearings, available in the Department, was upgraded to provide a smooth sinusoidal excitation (Fig. 12). A high inertia fly wheel driven by a D . C . motor generates a steady harmonic motion at frequencies as low as 0.7 Hz and damper amplitudes as high as 4 cm. The system can be operated safely up to 5 Hz for average amplitudes of oscillation, or higher for very short strokes (< 1 cm). A V A R I A C rheostat along with adjustable eccentricity of the Scotch-Yoke provided the means to vary the frequency and amplitude of excitation. Oscilloscope 1 L~ Spectrum Analyser Filter BAM BAM Strain Gauge — V W H J L { I •Damper Strain Gauge Moving Frame L . ~ . . . J 7 7 T T T 7 7 7 7 T r Bearings Scotch-Yoke Mechanism 3 " D.C. Drive Fig. 12 Test Arrangement 43 Two strain gauge arrangements were mounted on the apparatus: the first one was installed at the base of the damper supporting beam to measure the response of the sloshing liquid, while the second arrangement, a part of the ring shaped bracket, was attached to the main frame by a short spring to record the displacement. Careful design of the damper support was necessary to obtain proper sensi-tivity for minimal beam deflection, required to be small here compared to the main frame amplitude of excitation. A n aluminum plate, (0.318 cm thick, 20.3 cm long and 3.81 cm wide), clamped to the moving base and fitted with a horizontal plat-form to hold the container, was used to provide a linear range of strain versus horizontal sloshing forces with minimal impact due to pitching moments. The nat-ural frequency was much greater than that of the excitation, from approximately 40 Hz without damper to 12 Hz under larger loads. The output signal was amplified through a Bridge Amplifier Meter ( B A M , Ellis Associates) before being directed to a Spectrum Analyser (Model SD335, Spectral Dynamics Corporation). A filter (Model 335, Krohn-Hite) and a dual channel storage oscilloscope (Tektronics 564, Vertical Amplifier Type 3A3, Time Base 2B67) were connected in parallel to record the excitation from the other source simultaneously. The analysis in the frequency domain showed the magnitude of the response at different harmonics while the time domain measurements yielded the phase angle between the response and the exci-tation needed to calculate pressure and damping forces. 44 Small scale transparent plexiglas torus shaped dampers with square or rectangu-lar cross-section, such as shown in Fig. 13(a), were constructed in the Department's machine shop. Various sizes were required to investigate important dimensionless parameters, and two models were fitted with baffles and inner tube (Fig. 13b,c) found to be effective under certain conditions of excitation9 0 (Table I). Limited experiments were also carried out with circular cross-section dampers. Table I Details of the damper models used in the test program Damper d D Capacity Internal Cross-# (cm) (cm) (ml) Configuration Section 1 2.86 5.40 140 plain square 2 2.86 5.40 140 baffles square 3 2.86 5.40 126 inner tube square 4 3.50 7.31 147 plain square 5 3.79 6.99 315 plain square 6 3.15 7.55 235 plain square 7 2.84 11.7 297 plain square 8 1.42 3.93 39 plain square 9 4.70 4.70 326 plain square 10 2.20 2.20 38 plain square 11 3.50 15.9 640 plain square 12 2.98 8.08 177 plain circular 13 1.55 23.6 140 plain circular 3.3 Flow Visualization A n inspection of the various mode shapes was first conducted with the dampers 45 No of baf f les : z8 No of holes = 34 h V d - 0 . 5 0 d | /d = 0.39 b/d = 0.50 q; / d = 0.11 (b) (c) Fig. 13 Sketch showing several damper model internal configurations: (a) plain; (b) baffles; (c) inner tube oscillating near their natural sloshing frequencies. The conventional dye injection procedure and photographic equipment were used to visualize the free surface el-evation and its qualitative agreement with the theoretical predictions as given by relation (1.10). A large amplitude antisymmetrical motion with a stationary node at 90° to the direction of the excitation, and zero node across the radius, characterized the first mode shape studied using dampers 5 and 11, as predicted from the harmonic 46 (cos0) and Bessel function Ci(Aiir ) dependence of the 1st order potential flow so-lution (relation 8) along 0 and r coordinates, respectively (Fig. 14a). The small variation of the surface elevation in the radial direction at a given angle 0 (Fig. 14b), and the general pattern along the outer wall (Fig. 14c) suggests reasonable qualitative agreement. At times, a discontinuous wave front, however, appeared to be present. The nonplanar mode was observed at slightly higher amplitudes or frequencies, with a large swirling action about the damper circumference, seemingly 90° out of phase in time and angular position with the excitation. The similarity between the free surface shape in the radial direction for this mode and that of the planar motion, for a given 0 (Fig. 15), is again consistent with relation (8). It should be noted that the occurrence of the nonplanar response more readily took place for higher liquid heights in the torus. Although less useful for the purpose of this study, higher transverse modes were excited at their corresponding natural frequencies. They all exhibited a single circumferencial node at 0 = 90° , with minimum and maximum free surface elevation at 0° and 180° angles, and a number of transverse nodes at times more difficult to identify (1 for the 2nd mode, 2 for the 3rd, etc.). They are in agreement with the Bessel function C i ( A i n f ) dependence of relation (3a), and are shown in Fig. 16, 17, and 18. Of course, such modes distort the free surface plane considerably, with nonlinear effects becoming more pronounced at higher frequencies. Furthermore, theoretical calculations show that the spacing between two eigenvalues becomes (a) F r e e s u r f a c e d i s c o n t i n u i t y T h e o r e t i c a l M o d e s Ci (A„ f ) cos 6 Fig. 14 1st planar mode exhibiting: (a) antisymmetric motion about circum-ference; (b) variation across f; (c) variation along 6 49 (b) Fig. 17 Mode (1,3) shown as: (a) in the plane of the excitation; (b) perpen-dicular to the same plane Fig. 18 Close-up view of mode (1,4) 50 increasingly narrower, thus making the identification of the modes less obvious. Noteworthy is the apparent absence of turbulence throughout most of the flow field for at least the first two resonant states, a condition that was assumed for the the-oretical development. No attempt was made to visualize the modes corresponding to 2nd and higher order nonlinear terms. 3.4 Added Mass and Reduced Damping Ratio 3.4.1 General Procedure A force Fa transmitted from the sloshing fluid to the rigid damper walls in the direction of excitation causes the supporting beam to deflect proportionally to its magnitude provided the system is elastic and operates far away from its resonant state. A n additional force Fo generated by the system's own inertia is proportional to the amplitude and frequency square of the excitation, Fo = Mow 2 Co cos wt, (41) where Mo is the equivalent mass of the support (general "moving base" problem in vibrations). It contributes to the overall strain recorded by the sensor. A dynamic calibration procedure, consisting of measuring the output voltage after loading the support with dead weights under various conditions of amplitudes and frequencies, was therefore adopted to estimate Mo as well as the slope of the response curve (Fig. 19a). A n initial static calibration, with the system at rest undergoing bend-ing under a known stress, was used to verify the results. This is shown in Fig. 19(b), 51 where the y-axis represents the response as recorded by the spectrum analyser at the exciting frequency we, for the case of dynamic testing, while the oscilloscope provided the results for the static procedure. The other channel recorded the dis-placement of the moving frame and a simple calibration curve was produced by direct measurements of the stroke versus output voltage (Fig. 19c). Furthermore, phase angles between the excitation and response of the beam at the driving fre-quency were derived from real time measurements on the dual channel oscilloscope. As the system damping is due to the aerodynamic drag along the damper support as well as the hysteresis damping within the beam, and is very low compared with the effect of liquid sloshing, the frequency dependent phase shift introduced by the instrumentation (mainly the filter) was found by simply running the experiment without the damping fluid (Fig. 19d). The experimental determination of the four variables: sloshing force Fa, am-plitude eo and frequency we of the excitation, and the corresponding phase angle tb, supplied the necessary information for calculation of the added mass and damping ratios 1 " <S -11 cos*: and = 5 & ^ (42) where and \FB\ refer to the magnitude of the component oscillating at the Mi exciting frequency. The higher harmonics of the added mass can similarly be found as , A f a . \F.\n Mi n M/cow?' (43) 2^,, cm Fig. 19 Calibration curves to determine: (a) M 0 ; (b) slope for the response using both static and dynamic procedures; (c) slope for the excitation; (d) phase angle between response and excitation 53 where n denotes the rank of the harmonic considered. No provision was made for the estimation of the total energy contained in the liquid motion. 3.4.2 Results and Discussion From the theoretical development, the added mass and reduced damping ratios are expected to be a function of the set of dimensionless parameters (CJ, €Q, a, h, Re), representative of the excitation and damper characteristics. For convenience, the equivalent variables eo/d, D/d and h/d defined as: t o C o D 1 + a h h ~j = i — ; ~~T = i — ; a n d 7 = ; — ; (44) a 1 — a a 1 — a a 1 — a were adopted, and tests conducted through their systematic variation. Additional information concerning the effect of internal configuration and corresponding pa-rameters was also included in the study. Results mainly describe the damper behav-ior under excitations near the first sloshing frequency uu, as it is the condition of maximum damping, with the accuracy of the phase angle measurements diminish-ing away from resonance. Tests in free vibrations using an apparatus similar to that of reference 90 were carried out to extend the data in the low damping region. The significant findings of the test program are summarized in the following subsections. (i) General Shape of the Output Signals The excitation was first established to be purely sinusoidal (Fig. 20a), and the response was, in general, dominated by the driving frequency (Fig. 20b). A number 54 of superharmonics, odd multiples of the fundamental frequency (i.e., 3uet, 5uet, etc.), were also present as indicated by the spectrum analyser record (Fig. 20c). This is in good agreement with the assumed expansion for the potential function and the form of the derived added mass, as nonlinear even harmonics were found to be symmetrical in 0 and therefore cancel out when integrated over the damper wall. The higher resonant frequency of the damper support was often visible (at around 12 Hz here), but its effect on the first harmonics of the sloshing response was assumed to be negligible. (ii) Effect of Frequency As predicted by the theoretical model, the response is very sensitive to the fre-quency CJ. At relatively low amplitudes of excitation, a dramatic rise in the reduced damping ratio accompanied by a reversal in sign for the added mass characteriz-ing resonance was generally observed. The maximum r)rj usually coincided with M — 0 at a frequency slightly different from 1, as shown in Fig. 21 for a half-full damper with D/d = 4.1. The nonplanar mode generally took place for dampers with higher h/d or lower D/d ratios, extending the high damping region until the motion ceased to exist. For the case of h/d = 0.5, D/d = 1.89, this corresponded to CJ « 1.15 (Fig. 22). The theory generally anticipated these trends, although the damping curves are often too narrow with peaks higher than those measured (Fig. 21). In other cases, no solution near resonance can be found, thus preventing the occurrence of a large peak in r}r<i (Fig. 22). The stability boundaries for the 55 F . mV 20-( b ) 4 0 -3 0-D/d =1 .84 h / d ; 0.500 <iu : 0.924 « o /d=0.0 7 0 1 o -r e s o n a n c e o f d a m p e r 3rd h a r m o n i c I 5th h a r m o n i c \ b e a m s u p p o r t 1 o ( c ) 1 5 f. Hz Fig. 20 Output signals showing: (a) moving frame displacement; (b) damper support beam deflection; (c) frequency spectrum of the response Fig. 21 Variation of damping and added mass ratios with frequency for half-full damper#7 Fig. 22 Variation of damping and added mass ratios with frequency for half-full damper#l 58 planar mode are realistic, but the nonplanar region extends beyond what is experi-mentally observed with underestimated nr>i. Predictions for the added mass are, in general, quite reasonable with, at times, very good agreement in the linear range (e.g., 1.1 > CJ > 1.4 in Fig. 22). However, experimental values for the nonplanar motion were usually much smaller than expected. The potential flow solution for the case of resonant interactions was used for the damper of Fig. 21 resulting in two stable positions beyond resonance. The upper branch yields a damping ratio that remains high past CJ = 1 and seems to follow the experimental damping curve until rjTii suddenly drops at CJ = 1.14. The accurate prediction of the resonant point quite close to CJ = 1.0, indicative of weak non-linearities, is also very encouraging. At higher amplitudes, the effect of frequency was less pronounced with usually smaller peaks recorded. This is not surprising as nonlinear terms are now expected to be quite large. (iii) Effect of Amplitude In the nonlinear range, e r j / d has similar effects on the damper behavior as CJ: the occurrence of a resonant peak often followed by a nonplanar motion for certain conditions of excitation and geometries. When CJ > 1.0, the liquid motion is originally planar and small at low amplitude (linear range). However, it becomes M unstable with a large jump in both nr / and |—— | as the nonplanar mode takes over ' Mi (Fig. 23). A gradual reduction in damping then accompanies a further increase in 59 60 eo/d whereas very large rjTii can be obtained with decreasing amplitudes. For CJ < 1.0, an optimal damping is reached as the liquid goes through resonance with |-777-1 = 0 (Fig. 24a), followed by a rather unsettled motion which fails to at-tain the fully nonplanar mode. However, such trends for CJ > 1.0 or CJ < 1.0 can be reversed through the damper geometry and liquid height, as shown in Fig. 24(b). This particular behavior is related to the "softening" or "hardening" characteristics of the damper (i.e., reduction or increase in the resonant frequency with ampli-tude) as discussed in Chapter 2, and is generally predicted by the potential flow model. Here again, the calculations for the nonplanar mode yield a lower damping and higher added mass than those of the experiments (Fig. 23), thus suggesting significant dissipative mechanisms in the main flow field. The theory properly in-dicates a large region of unstable flow for the case of Fig. 24(a), but fails to find a solution at resonance, a situation already encountered while studying the variation with CJ. The results for the unstable nonplanar mode are indicated in this case for comparison against experiments, and show reasonable trends for the added mass and damping ratios. Finally, the hardening characteristics of the low liquid height damper of Fig. 24(b) yield a predicted resonant point at eo/d « 0.06, as opposed to a measured value higher than 0.14, that was never reached as the planar mode then became unstable. A possible cause for such dicrepancies may rest with the relative size of the boundary layer thickness, not accounted for in this analysis, that changes the effective h/d, or makes the potential flow approach questionable when Fig. 24 Resonant behavior of damper#5 with: (a) h/d = 0.5 and C J < 1 Fig. 24 Resonant behavior of damper#5 with: (b) h/d = 0.19 and CJ > 1 63 h/d is small. This is further discussed in the following paragraph. In all the cases, the predicted resonant peaks are narrower than those measured (Fig. 25). The right trends are however discernable at lower eo/d. Of course, the theory cannot be expected to be realistic for higher amplitudes as the assumed expansion for the perturbation method becomes invalid and turbulence dominates the flow. 7.0 4.0 3.0-2.0 1.0 0.0 € 0 / d = 0.046 * o / d = 0.091 €0/d =0.475 —' ' D/dr4.10 \ . h/d r 0.500 R e / c u r 3.35 x104 0.5 0.7 0.9 1.1 Fig. 25 Peak damping ratios as affected by amplitude i / > . • i i 1.4 1.6 A CU (iv) Effect of Liquid Height This geometric parameter significantly affects the position of the resonant re-gion, as discussed during the theoretical development. For the damper of Fig. 26, a value of h/d = 0.48 is expected to result in a purely linear response (i.e., K\ = 0 64 D / d = 1 . 8 9 h / d = 0 . 5 0 0 * 0 / d = 0 . 0 4 6 P l a n a r U n s t e a d y o r N o n p l a n a r 0 . 7 0 . 9 1.1 h / d * 0 . 2 5 0 * 0 . 3 7 5 * 0 . 5 0 0 • 0 . 6 2 5 • 0 . 7 5 0 1 .4 1 . 6 A CU Maximum damping and added mass ratios for various liquid heights at e 0 /d = 0.046 65 in relation 10a of Chapter 2) with resonance at CJ = 1.0, and infinite damping and added mass ratios. This is supported by the experiments, as h/d = 0.5 generates a pronounced peak near CJ — 1.0. However, the peak value of rjrj is around 3.0 and is essentially somewhat insensitive to liquid height in the range h/d < 0.625. In gen-eral, a shift in the resonant frequency with liquid height was not as severe as that predicted by the theory. The potential flow solution still continues to provide the right softening or hardening trends for the damper with various h/d as illustrated in Fig. 27 (higher amplitudes). Resonance for h/d > 0.75 is no longer possible as the planar mode becomes unstable thus suggesting, once more, that high liquid heights be avoided in designing a damper. At higher D/d ratios, both theory and experiments indicate a more linear behavior as it is the case of resonant interactions discussed earlier, with smaller shifts in the resonant frequency and well defined peak responses (Fig. 28). The experimental results proved to be quite useful in assessing the theoretical model. For instance, the energy dissipation in the potential flow regime is obviously not negligible as the sloshing action is well contained at resonance in spite of the relatively small nonlinear effects at certain liquid heights, as discussed earlier. This may be responsible for the weaker than predicted nonlinear effects at other h/d. Moreover, the theory is again shown to be less reliable at low liquid heights, possibly due to a relatively large boundary layer thickness. 66 2A-1 . 6 0 . 8 I M 0 / M ( 4 . 0 -1 . 0 0 . 0 - 1 . 0 D / d = 1 . 8 9 h / d r 0 . 5 0 0 € o / d = 0 . 1 0 5 4— ).T P l a n a r U n s t e a d y o r N o n p l a n a r 0 . 9 i l l TT4 h / d r e s o n a n t f r e q E x p t s T h e o r y • 0 . 2 5 0 1 . 1 2 1 . 5 0 0 . 3 7 5 1 . 0 3 1 . 3 0 0 . 5 0 0 0 . 9 3 • 0 . 6 2 5 -• 0 . 7 5 0 — — 1 .6 A r e s o n a n t f r e q u e n c y 1 .4 1 .6 A CU - 4 . 0 -Fig. 27 Variation of the peak response with liquid height for damper#l at e0/d - 0.105 6 7 3.2 1.6 D / d = 4.10 €0/d =0.091 — r - z ^ -o.r h / d 0 . 3 7 5 0 . 5 0 0 • 0 . 6 2 5 • 0 . 7 5 0 0.9 1.1 1.4 i ; e A Fig. 28 Variation of the peak damping ratio with liquid height for damper#7 at e0/d = 0.091 (v) Effect of Diameter Ratio Maximum damping ratios have been shown to be higher for D/d = 4.10 due to the smaller nonlinear effects (Fig. 21), a result supported by the theoretical development of the resonant interactions. In fact, experiments indicate a contin-uous improvement in the performance with increasing diameter ratio. The fre-quency spectrum curves of Fig. 29 exhibit smaller superharmonic response as D/d is changed from 1.84 to 2.40 (ratio of 3rd/lst harmonic from 0.57 to 0.37 despite a slightly higher eo/d), while the maximum rjrj in the frequency domain remains 5 0 -y. mV 40-3 0 H 2 0 H 1<H (a) 0 -y, mV 3 0 H 2 0 H 1(H CO h/d = 0 D/d = 1 . 8 4 CJ r 0 . 9 2 4 € 0 / d r 0 . 1 5 8 R e = 3 . 2 2 x 1 0 4 h/d = 0 D/d = 2 . 4 0 Uj = 0 . 9 2 4 * o / d = 0 . 1 9 8 Re= 2 . 9 4 x 1 0 ' Cb) h/dz0.500 Ay 3 /Ay, - 0.57 (d) h/dz0.500 Ay 3 /Ay, - 0.37 1 5 10 15 0 5 10 15 f, H z f, H z Fig. 29 Frequency spectrum of the response for damper#5 and #6 showing the effect of D/d OO 69 higher with amplitude as the diameter ratio varies from 1.0 to 4.1 (Fig. 30). It should be mentioned that the curves for the response without damping liquids are also presented in Fig. 29 as their peak amplitudes have to be substracted from the total response to obtain the net sloshing force. Results for very slender dampers (D/d > 10.0) suggest that the trends persist 9 0, promising a very efficient design. Theoretical predictions are not always straightforward as the transition point from the no interaction to interacting solution has to be arbitrarily chosen. Both formu-lations should converge, however, the latter becomes unstable as the interactions weaken (increasing & i coefficient) while the other is not yet representative of the situation. Unaccounted viscous effects may again be responsible for generating the experimentally observed stable transition range. Furthermore, the estimated boundary given by relation (20), where c is substituted by e 1 / 3 at resonance, also suggests that VQ should be smaller than 1.0 for better agreement with experiments. (vi) Reynolds Number Effect The equations of Chapter 2 along with previous investigations dealing with the sloshing motion linear r a n g e 8 7 - 9 0 established the reduced damping to be propor-tional to i E e - 1 / 2 . The present tests however indicate this may not be the case near resonance, as shown for two different geometries in Fig. 31. Several liquids such as water, alcohol, kerosene, and oil of different viscosities, as well as two damper sizes with otherwise identical geometric parameters h/d and D/d, were used to vary the 70 4.0 3.0 2.0 1.0 D / d • 1.00 • 1.89 A 2.40 • 4.10 0.4 0.2 0.8 1.2 €D/d Fig. 30 Maximum damping ratios versus amplitude for various D/d Ma Reynolds number. Results show that r)rj and Ij^ rl remain generally unaffected by a change in Re in the nonplanar mode (for Re as low as 1.73 x 104 in Fig. 31a), with a slight downward trend in the planar mode (Fig. 31b). At very low Reynolds number (580 or 700 in Fig. 31), the curves drop significantly with the planar mo-tion becoming stable over the entire range of excitation amplitude. Hence, it can be speculated that the higher dissipative effects at lower Re are offset by a reduction in the sloshing motion through the combined action of dissipation in the main flow field and the larger boundary layer thickness. Although dominant, the nonlineari-ties alone are not the only mechanisms restricting the response at resonance. The importance of small damping terms at CJ = 1 can be well illustrated by the analogy 71 3.0-D/d =1.89 h/d = 0.500 cu r1.15 Re(x1 04) A 4.01 A 2.82 1.6- \ • — . Non-planar • • Planar 2.66 D 1.73 o o 0.070 0.8- X A» A & A a 0.0 4 » ..•_'.J° 4 0.0 o T - * H — 0.4 0.8 lMa/M,l 4.0; 1.2 € 0 / d 1.0 0.0 -1.0-3 -4.0-Fig. 31 Damping and added mass ratios as affected by Re for dampers with: (a) D/d = 1.89 72 D / d =4.10 h / d z 0 . 5 0 0 to =1.02 2.8-2.0-1.2-0.8-0.4-A» A -Unsteady Planar Re(x1 0A) A 3.42 A 2.41 1.47 o o 0.058 \ . A A 0.2 i Ll i 0.4 0.8 1.2 € Q /d I M a / M | 4.0-1.0-0.0 -1.0^ • » - • • -A —i—r 1 /4 i — 9v2? 0.4. A A _ _ • ^ » j A A 2 € D /d A • -4.0-Fig 31 Damping and added mass ratios as affected by Re for dampers with: (b) D/d = 4.10 73 of a simple mass-spring-dashpot linear system (Appendix VI), for which inertia and stiffness are otherwise the controlling parameters in the nonresonant region. The liquid sloshing response also exibits a gradual sign reversal in the added mass near resonance, as pointed out earlier. This is in contrast to a sudden jump characteristic of an undamped model. As the energy dissipation is quite motion-dependent here, the addition of even small viscous effects in the flow might be sufficient to improve the present formulation. A further point of interest is the absence of variation in the resonant frequency with changing Reynolds number, for all the dampers con-sidered in this study. The smaller free stream (potential region) at lower Re due to an increased boundary layer thickness is also accompanied by a larger D/d ratio, eventually resulting in a cancellation effect and the observed trend. (vii) Effect of Configuration Baffles or inner tube positioned inside the damper have shown some success at promoting energy dissipation for a certain range of frequencies, as reported by previous investigations90. The study is extended here into the optimal region of resonance by using 3 dampers with D/d = 1.89. Typical results are presented in Fig. 32 which suggests that the baffle configuration generally suppresses the nonplanar mode (Fig. 32a, Co > 0.03). Furthermore, the added mass was found to be lower than that for the plain damper, thus indicating a reduction in the amount of sloshing motion. The effects are somewhat different with the inner tube where the interference with the free surface now generates a negative added mass 74 at low amplitude. Although more dissipative mechanisms are present, such as the formation of a wake behind the baffles, or increased friction against the tube, the additional restrictions imposed on the flow result in a net loss in rjr,h A similar picture is obtained by varying the frequency (Fig. 32b), with lower maximums for the added mass and damping ratios. The peak response for ijr>i appears to be wider prior to resonance, as expected for such systems, however, the absence of nonplanar motion for the baffle arrangement and the interference with the inner tube yield an overall lower efficiency for CJ > 1.0. As in the case for low Reynolds number flows, any configuration preventing the large motion of the free surface appear to affect the damper performance. Note the change in the resonant frequency with the introduction of the baffles (|-r^-| = 0 at CJ = 0.92). The inner tube-liquid contact \ was also found to make the planar mode more stable well into the region where\the rotating motion would have otherwise started with a plain configuration. (viii) Note on Damper Cross- Section The cross-sectional shape allowing for larger sloshing motion is likely to max-imize the damper efficiency. Straight wall containers (i.e., square or rectangular cross-section) were studied here as they are easier to construct and simpler to anal-yse theoretically. Some tests on circular cross-section models were also conducted with the performance comparable or lower than that of an otherwise similar straight wall damper. For instance, a geometry with D/d = 3.00 and h/d = 0.5 yielded a peak rirti = 2.2 at resonance, for an amplitude eo/d = 0.06. The same trends were 75 3.0 1.6 o.e-\ 0.0 D / d =1.89 h/d = 0.500 U) r1 .15 Non-planar Planar • Plain a B a f f l e s * Tube Re=2.66x10 E I - - - J 0.0 0.2 ~t^~i— 0.4 0.8 1.2 €0/d 1.2 €0/d M Fig. 32 Effect of internal configuration on nr>i and | - r - ^ - | versus: (a) e r j / d 76 3.0-^  1.6H 0.8 IMQ/M,I 4.0 1.0H 0.0 -1.0H -4.0 D/d = 1.89 h/d=0.500 € o /d = 0.049 Non-planar Planar / P fi : • Plain ° B a f f l e s A Tube Re/oj=:2.31x104 — r - z ^ -0.7 0.9 i i 1.1 -r—tA r 1.4 1.6 A A •a » A lit-0.7 0.9, — i — 1.1 —I I—— 1.4 1A.6 CO ,-' r • \ - * Fig. 32 Effect of internal configuration on rjr i and versus: (b) £> Mi 77 observed during free oscillation experiments, although here resonance was more dif-ficult to establish and no quantitative results could be obtained in this region. A n interesting concept consisting of a sloping cross-section (Fig. 33), allowing for the breaking of the liquid sloshing waves, was similarly tried. The logarithmic decre-ment method showed some improvements are possible for particular geometric ratios of d i / d , 0 * 2 / 0 * and 0 * 3 / 0 * over the square geometry. More systematic tests in forced oscillations would have to be conducted to validate the idea. The introduction of flexible walls is another area to be examined. d -I I * — d i — H Fig. 33 Proposed sloping cross-section 3.4.3 Comparison with Free Oscillation Tests The distinct character of the free oscillation tests and associated apparatus9 0 (Fig. 34) provided a means to verify the steady-state excitation results. The ad-ditional parameters representing the amplitude decay deo/dt, the small rotational motion induced by the pivoting arm, and the variation in the natural frequency due to fluid-structure interactions, are variables likely to influence the response. Fig. 35 78 NUTATION DAMPER V SPRING MOVEABLE COLLAR BEARING RECORDER OR OSCILLOSCOPE n B A M STRAIN GAUGE ADJUSTABLE SUPPORT AND HEIGHT TO CONTROL NATURAL FREQUENCY Fig. 34 Apparatus used for free oscillation tests compares some results as given by the two methods. The amplitude decay approach clearly tends to smooth the curves due to the transient effects (Fig. 35a) . In most cases, nr>i is lower than its corresponding value for the forced vibration tests (Fig. 35b). This is not surprising as the flow is likely to need some time to respond to the increase in rjTii with dimishing amplitude. Furthermore, the boundary for the transition from planar to nonplanar motion is delayed. The trends are, however, the same and the shift in magnitude could be attributed to the system rotation, as discussed in the next chapter. It should be pointed out that deo/dt was minimized by using a large dead weight (i.e., small fluid to total mass ratio). The reduced damping ratio was taken as 79 B a f f l e s D/d =1.89 h/d = 0.50 cu =1.15 Re=: 2.66x10' 0.8-•—•* Non-planar • • Planar Steady-state Free oscillations 0.0 0.0 3.0-1.6-Plain D/d 3 1.89 h/d r 0.50 cur 1.39 Rer3 .21 x104 0.8 Fig. 35 Damping characteristics versus eo/d as obtained by steady-state and free oscillation experiments for: (a) damper with baffles; (b) plain damper 80 ^ - - 2 ^ r i ( 4 5 ) according to the logarithmic decrement method normalized for the relative inertia of the fluid. Here: xm = system amplitude for the mth cycle; y = fluid to total inertia ratio for the pivoting system (equivalent to Mi jM for translation). The relation is valid for discrete values of amplitude corresponding to m=l , 2, etc., and can be made continuous by taking the limit as m tends to 0, i.e., i \ v l n x i / x m J j * ' I ( X ) = ^ o - 2 ^ 7 ' ( 4 6 ) or * M — - - L — j , (47) where x is the amplitude function, x{m) = xm. (48) A polynomial fit for the envelope of amplitude decay was then applied to facilitate the analysis, x(m) = A0 + Aim + A2m2 + (49) as shown in Fig. 36, which yields 1 A x + 2mA2 + ... + pm^-^A^Ii ^ 1 ( X ) = ~2^[ A0 + Aim + ... + ApmP ]T ( 5 0 ) 81 x =2.50 - 0.075m + 0.0019m2 0 40 80 m , c y c l e s Fig. 36 Amplitude decay for the half-full damper#l oscillating at CJ = 0.924 with an initial displacement of eo/d = 1.28 (chart recorder Type TR322, Gulton Industries) 3.5 Concluding Comments This experimental program combined with the theoretical development have resulted in an in-depth understanding of the nutation damper behavior. The major findings are summarized below: • The damping characteristics are entirely governed by the liquid motion. The condition of resonance with the damper operating at its first sloshing natural frequency results in a substantial gain in rjr>i. Any configuration restricting the action of the free surface, such as baffle arrangements, inner tubes, or even high viscosity fluids, further contributes to a drop in efficiency. 82 Nonlinearities play a major role at resonance. They should be minimized as they generally limit the liquid motion, reflected by a reduction in damping and added mass ratios. They are also responsible for the softening or hardening characteristics governing the response versus the amplitude of excitation. The appearance of the nonplanar mode has often beneficial effects as it extends the high efficiency region beyond resonance. Whenever possible, long and slender dampers with relatively low liquid heights (high D/d and h/d < 0.5) should be used as they exhibit weaker nonlinear effects. When resonance can only be met at smaller diameter ratios, particular h/d can also be found to provide similar characteristics. The theory serves as a useful tool in understanding the damper behavior. The resonant frequency at low amplitudes, as well as the hardening or softening trends, are often properly predicted. The peak response and the nonlinear effects are, however, too pronounced and suggest that significant dissipation takes place in the main flow field. The analysis is quite demanding, in terms of time and efforts, since many cases of resonant interactions need to be considered, at times leading to unstable solutions. However, the procedure provides considerable insight into the effect of the various controlling parameters. Damping forces outside the boundary layer are likely to restrict the motion of M the main stream at resonance (lower peaks for r/r>j and |-r-p|) while promoting 83 dissipation elsewhere (wider region). A numerical approach for solving the full Navier-Stokes equation would therefore be more accurate. However, the three dimensionality of the flow and its time dependence, combined with the highly nonlinear free surface boundary condition, would make this process fairly costly. Furthermore, the presence of such phenomena as discontinuous, turbulent wave fronts mentioned in section 3.3 would still be unaccounted for. Improvements to the potential flow solution could also be implemented to correct for the boundary layer thickness. The variational method, allowing for the introduction of an empirical dissipative term in the equation of the main flow field67 is another possible avenue of research. • Finally, turbulence was never found to be beneficial as rjr>i did not change its trends at higher amplitudes or frequencies of excitation (with the transition from laminar to turbulent flow). 4. W I N D I N D U C E D I N S T A B I L I T Y S T U D Y 84 4.1 General Description Effectiveness of the dampers in controlling vortex resonance and galloping insta-bilities was assessed in both laminar (u'/V < 0.1%) and turbulent flows for two and three-dimensional bluff bodies undergoing translation and rotation, respectively. The closed circuit laminar flow wind tunnel with a test section of 0.69 x 0.91 x 2.44 m, and the large boundary layer tunnel (24.4 m long, with an initial cross-section of 1.58 x 2.44 m) fitted with 20.74 m of roughness board upstream of the model to pro-duce desired boundary layer thickness and turbulence intensity, were used to simu-late the external environment. Dampers were mounted on a variety of aerodynamic models with square or circular cross-sections. The two-dimensional arrangement, useful for predicting the response of tall structures such as smokestacks, buildings, etc., spanned the height of the laminar flow wind tunnel, while a horizontal set-up simulated a transmission line configuration. The rotational motion was studied with three-dimensional models of finite aspect ratio. 4.2 Two-Dimensional Tests in Laminar Flow 4.2.1 Preliminary Remarks Although the natural wind is essentially turbulent, the vortex resonance and galloping response of two-dimensional, square and circular cylinders in laminar flow 85 has been well documented. Consistent empirical results91 combined with the de-velopment of a successful galloping theory 9 3 permit an approximation of the cross-flow oscillations, provided the aerodynamic reduced damping 77,. > a , also called mass-damping 6 or stability parameter8, is known. These tests are therefore well suited for the evaluation of the nutation damper characteristics under conditions of nonlinear, wind-induced forcing excitations. As the amplitude growth of the model response is usually assumed to be slow until a limit cycle is reached, due to relatively high rjTta in wind engineering problems, the steady-state results of Chapter 3 should apply. 4.2.2 Test Arrangement and Model Description A rigid frame located outside the wind tunnel and supporting four air bearings, in turn carrying a sliding shaft at top and bottom on which aerodynamic models were mounted in a vertical position, was used to conduct the two-dimensional tests (Fig. 37). Four springs provided the structural stiffness and an inductance coil type displacement transducer recorded the amplitude response. This already available set-up, specifically designed for the study of aeroelastic problems, was also equipped with eddy current magnetic dampers. More information on the test facility is given in reference 97. Relatively large (10.2 cm « 4") yet light models were constructed in the Department's machine shop to produce the desired instability region, with rjr>a as low as 2.0 to 3.0 to allow for some flexibilty in the choice of nutation damper size. A smaller 5.1 cm (2") square cross-section cylinder was also used to evaluate the performance for a different value of aerodynamic reduced damping ratio. 0.64 86 Displacement Damper Transducer \ i V v End Plate Air Bearing Shaft Air Bearing Block •V\A- I Function Conditioner i—r t i i—i Dead Weight Generator Amplifier Spectrum Analyzer Chart Recorder or Oscilloscope Fig. 37 Wind tunnel set-up for two-dimensional tests cm (0.25") thick balsa wood provided reasonable bending and torsional stiffness while two thin aluminum plates bonded to the balsa wood defined sharp edges of the square configuration. The 10.2 cm diameter circular cylinder was made of 0.64 cm thick P V C pipe section. The models were provided with medium size end plates, following a careful design procedure, as explained in section 4.2.4. The details are given in Table II. Although the model weight ranged from 349 to 786 g, the moving shafts, clamping mechanisms, and damper supports contributed more to the inertia of the oscillating system (1094g ± 3 7 g, according to springs used). It could also 87 be changed with the addition of metallic plates inside the model, or dead weight outside the tunnel for finer adjustments. Table II Physical description of the two-dimensional aerodynamic models MODEL# 1 2 3 CROSS-SECTION * 102 m m • 1 IX51 mm C"~^ 10|2m LENGTH(mm) 673 673 673 MATERIAL BALSA & ALUMINUM BALSA & ALUMINUM PCV MASS (g) 786 349 745 END PLATES 102 m m 2 0 4 m m ! 3 0 5 1 m m m " 1 t 2 5 1 m m 1 • 2 0 4 m m t 3 0 5 | m m i Static force measurements on the square cross-section models were carried out with the six-component pyramidal strain gauge balance (Aerolab). Drag Dr and forces perpendicular to the flow Sp were recorded over a range of angle of attack a giving the side force, Fy = Sp cos a — Dr sin a, (51) useful in prediction of the galloping response. 88 4.2.3 Calibration Procedure The displacement transducer was connected to a chart recorder and a spectrum analyser (same instruments as in Chapter 3). A static calibration procedure with the chart recorder responding to a given displacement of the model resulted in the curve of Fig. 38(a). The peak amplitude y of a purely sinusoidal signal such as that obtained during the vortex resonance or galloping excitation was measured by the spectrum analyser, and in turn calibrated against the recorder. It gave a constant value of 31.78 m V / c m (Fig 38b). 4.2.4 Model Characteristics Although large end plates, or the more recent double plate configuration97, are desirable to reproduce conditions of two-dimensionality98, the drag and weight penalties can weaken the excitation and response during the dynamic tests. This was assessed during a preliminary determination of the static side force coefficient on the 5.1 cm square cross-section (Fig. 39a). Results clearly indicate a relatively lower initial slope of C / y versus a, for both the no end plate and the large end plate configurations, known to delay the galloping instability. Less pronounced trends were observed for the 10.2 cm size square model (Fig. 39b). Medium size end plates, sufficient to minimize the effect of suction caused by slots in the tunnel walls were therefore adopted. Of course, the slots were necessary to permit vibrational motion of the system. More results dealing with this particular aspect are discussed in the next section. The position of the plates also played a significant role in earlier 89 Fig. 38 Calibration constants used during the tests for: (a) chart recorder; (b) spectrum analyser 90 0.6 0.4 0.2 -0.2-R = 1 . 2 3 x i o 4 Model#2 dm bm/dm • • 0 0 (no plates) • • 5 2 (med.) A- -A 5 3.5 (large) Model# 1 0.6 0.4 0.2 R r 1.17x10" a m / d m bm/d m . > o 0 • o 3 2 £r -A 5 3.5 -0 .2 a, o Fig. 39 Effect of end plate dimension on C / y for: (a) model#2; (b) model#l 91 tests on an aluminum, 7.6 cm (3 in) diameter circular cylinder. Presence of a 1.27 cm plate-wall gap resulted in a vortex resonance response much lower than expected. A smaller gap of 0.63 cm, necessary to install the models, was used in all subsequent experiments. The high blockage ratio (11%) of the 10.2 cm section model (model #3) was responsible for larger peak displacements, compared to the standard vortex res-onance response for a circular cylinder 9 1, as shown in Fig. 40. With a shift in dimensionless amplitude Y but otherwise similar shape, this curve can be used as the reference for the nutation damper performance. The response at different levels of Vr,a was obtained by activating the electromagnetic dampers at different voltage settings. As expected, the damping ratios were found to be essentially constant with amplitude, over the range considered (Fig. 41). The logarithmic decrement method in conjunction with the amplitude decay polynomial fit approximation, used in the analysis of data, was described earlier towards the end of Chapter 3. The inherent system damping ne (i.e., no damper, 0 mA) was also found to be constant at low Y but quickly increased beyond a certain threshold. This is in agreement with earlier studies using such test facilities9 9, and is beleived to be caused by the higher bending stresses at larger amplitudes. The deformation during the initial displacement may however be different from that applied by the distributed load-ing of the air flow pressure field and such a rise in r}8 may not occur during the dynamic tests. The galloping response can be predicted by the quasi-steady theory Fig. 40 Maximum displacements of model#3 undergoing vortex resonance 0.3 0.5 0.6 y Fig. 41 System damping for different electromagnetic damper settings 93 using the measured side force coefficients of model#l and 2 shown in Fig. 42. The results for model#2 (5.1 cm side) compared quite well with those by Brooks 1 0 0 . The higher blockage ratio in Fig. 42(b) causes the Cfy values to be larger at small angles of attack. The dC/y/da slopes at a = 0° are approximately 2.6 and 4.6 for the two models, respectively. The galloping response prediction, obtained using a 13th order polynomial approximation for C / y , agreed with the experimental results for the electromagnetic damper set at higher nria (i.e., galloping onset velocity L70 far from vortex resonance Ur), as illustrated in Fig. 43(a) and (b). For a lower value of the aerodynamic damping, both vortex resonance and galloping regions overlap, and the response is rather insensitive to rjr>a until the latter is large enough for a distinct peak near Ur (Fig. 43c). The data were taken for a model natural frequency of 2.00 Hz, and agree with the low turbulence data of reference 97. The validity of the curve at several other frequencies was also established by changing the stiffness of the test arrangement, as shown in Fig. 44. A well defined response to the vortex shedding excitation was often visible on the frequency spectrum outside resonance, particularly for the 10.2 cm models (Fig. 45a). With the record of the correponding wind velocity, a Strouhal number of 0.196 was found for the circular cross-section (Fig. 45b), whereas the square configuration yielded St = 0.127 and 0.139, for the 5.1 and 10.2 cm side, respectively (Fig. 45c). The accuracy of the data was lower for the smaller model due to the reduced aerodynamic forces. 94 0.6-2.60a - 0.0075a3 - 5.93 x 1 0 3 Q 5 + 2.34 x 10 sa 7 - 2.25 x 10 6a 9 + 1.23 x 1 0 9 a n 0.4-0.2-' A A s A 9 Model#2 A 0/A t=5.5% -0.2-(a) 0.6-0.4-0.2-• R (x104) 1.23 1.74 2.74 3.88 q. i f • A A A 4.60a - 0.018a3 - 1.96 x 10 3a 6 + 8.56 x 10 5a 7 - 7.99 x 106 a 9 - 9.63 x 10V 1 + 1.23 x 10 9 a 1 3 Model#1 • Ae/At=11.1% • -0.2 (b) i — 8 A T 1 6 24 32 a, o Fig. 42 Cfy versus a for: (a) model#2; (b) model#l 95 Y/IL 0.8- uc/ur • 0.44 4.63 47.4 a 1.04 10.7 112. 0.6- * 1.50 15.7 162. 0.4 0.2 (a) from ref. 93 from polyn. fit in F ig . 41 1?>(%) U0/Ur ^ , 0 • 1.20 1.98 32.7 • 1.50 2.47 40.9 * 1.72 2.83 46.9 ( b ) 0.5 1.0 1.5 2.0 0.5 U/U0 Y 0.8 0.6-I 0.4 0.2 Model u0/ur • 2.86 1 0.17 • 5.73 1 0.35 o 1 1.7 2 1.14 A 12.8 1 0.77 • 21.5 1 1.30 (c) 0.6 1.4 2.0 2.8 U Fig. 43 Galloping response for: fa) model#2 with high damping; (b) model#l with high damping; (c) low damping 96 Y U Fig. 44 Response of a two-dimensional square cylinder without damper for: (a) model#l; (b) model#2 97 Fig. 45 Vortex shedding excitation on two-dimensional models showing: (a) frequency spectrum of the response; (b) Strouhal number for the cir-cular cylinder; (c) Strouhal number for square cross-sections 98 4.2.5 Results and Discussion 4.2.5.1 Vortex Resonance Response of a Circular Cylinder The tests were conducted at a frequency of 2.50 Hz with the response of the model without dampers exceeding the physical limits imposed by the slot size in the wind tunnel walls allowing for the motion. The addition of damper#l with various liquid heights resulted in a significant reduction in amplitude even for h/d = 0.125, as shown in Fig. 46(a). The oscillations were almost completely eliminated at h/d > 0.5 with the damper operating near its first natural sloshing frequency, established to generate high damping ratios at low amplitude with the occurrence of the nonplanar mode. Quantitative information can be obtained by recognizing that the aerodynamic reduced ratio r)r>a is related to t)r,i (liquid reduced damping ratio) through , 1 1 , 1 V 4 7 T M/paLmdm Mi/M where Lm and dm are the model length and diameter, respectively; pa is the air density; M is the total system mass; and rj8 is the inherent damping determined to be 0.105% for small oscillations during the experiments. Using the model character-istic curve of Fig. 40, the peak amplitudes Y = 0.094, 0.05, and near 0, associated with h/d = 1/8, 1/4 and 1/2, correspond to rjT>a « 12.5, 20.0, and > 40.0, re-spectively. This in turn requires rjrji to be of the order of 0.36, 0.31 and > 0.35, respectively. This is indeed the case with h/d = 1/2 as observed in Chapter 3 (p. 59) where r}r>i was found to be greater than 2.0 in the nonplanar mode at low exci-99 tation amplitude. Similarly, the half-full baffle and tube configurations suppressed the oscillations (Fig. 46b), while for the more slender damper#13 operating far from its resonant frequency a large response persisted (Fig. 46c). Although no data for damper#l with h/d = 1/8 and 1/4 were obtained for these conditions during the steady-state experiments, a record of the amplitude decay was taken prior to switching on the wind to assess the damping (Fig. 47). Despite CJ being much smaller than 1.0, the nonlinear effects at higher amplitudes produce a resonant peak at e0/d = 0.25 and 0.45, for h/d = 1/4 and 1/8, respectively, due to the hardening characteristics at low liquid height. Larger values of r)rj = 0.50 and 0.59 are also obtained for the two h/d ratios at an amplitude of eo/d = 0.18 (i.e., Y = 0.05) and 0.334 (Y = 0.094), respectively, as compared to T)rj « 0.31 and 0.36 estimated from the response during the wind tunnel tests. Possible innac-curacies may enter due to the fairly flat slope of Fig. 40 in the range considered here, with small errors in Y leading to a large variation in r / r ) a . Furthermore, it was shown that the logarithmic decrement method does not follow the fluctuations in damping with eo/d precisely (section 3.4.3), therefore rjrj is likely to be lower prior to attaining the resonant peak . The order of magnitude and the relative amount of damping for the two liquid heights are, however, quite comparable in both cases. Several approaches are available to verify the value of T]rj. One way would be to use available data on the excitation due to vortex shedding. A fairly compre-100 Damper Parameters h/d = 0.500 M| /M= 0.038 0.8 1.0 — no damper • plain (#1) • baffles (#2) A tube (#3) D/d - 1.89 CJ =1.15 Re=2.66x104 U Fig. 46 Vortex resonance response on model#3 showing: (a) effect of liquid height and w; (b) effect of internal configuration 101 Y 0.4 0.2 0.0 ^ , 0 = 2.92 D a m p e r # 1 & 13 l i m i t Damper P a r a m e t e r s h / d = 0.500 M i / M =0.038 (c) 0.6 D / d \ u > R e (x104) • 1.89 2.66 A — - A 15.2 6.70 \56.8 0.8 1.0 Fig. 46 Vortex resonance response on model#3 showing: (c) effect of diameter ratio and CJ 0.6 0.4 0.2 h / d = 0.125 0.4 0.8 € 0 / d Fig. 47 Variation of rirj with amplitude during free oscillations for damper#l with h/d= 1/8 and 1/4 102 hensive study by Diana and Falco 1 0 1 permits the prediction of the response over a wider range of U. With the knowledge of the work done on the model by the wind, at various amplitudes and frequencies , an iterative procedure based on the input to dissipated energy balance results in the comparison of Fig. 48. Notice-able discrepancies are apparent although the general trends are well reproduced. A n alternate approach is the partly successful Hartlen-Currie oscillator model 1 0 2 requiring the empirical determination of the variables and 6 ,^ using the lightly damped response here (Figs. 49a, b), and applying it to the model fitted with nu-tation dampers (Figs. 49c, d). The static lift coefficient CJO when taken to be the same as that determined by Feng 1 0 3 underestimated the response for h/d = 1/8 and 1/4 (with no solution generated beyond the resonant peak), as shown in Fig. 49(c). The input damping ratios were based on rjr>i = 0.36 and 0.31 for h/d = 1/8 and 1/4, respectively, as found earlier from Fig. 40. Furthermore, the half-full damper corresponding to rjrii > 0.35 was found to suppress the oscillations (not shown). A higher C j 0 of 0.5 to account for the larger blockage ratio was then considered and peak amplitudes closer to the experimental results obtained (Fig. 49d). A minimum t]rti of 0.62 was then needed for h/d = 1/2 to bring Y down to near 0. Overall, the method further verifies the level of the input damping ratios, as the predicted left-hand side of the response curve matches the data. In general, vortex resonance can be controlled using nutation dampers with a mass less than 1% of the total weight of the system (model#l with h/d= 1/8). Even 103 Y Fig. 48 Comparison between experiments and predictions based on the energy balance method smaller sizes with CJ closer to 1 are expected to perform equally well. The higher blockage ratio of the 10.2 cm diameter model, responsible for a larger excitation than that in free air, makes this estimate conservative. 4.2.5.2 Vortex Resonance and Galloping Response of a Square Cylinder The dampers were first mounted under the more unstable conditions of model#l with a low initial aerodynamic damping ratio of 2.92. Qualitatively, they perform according to the characteristics determined in Chapter 3, with the frequency param-eter CJ closer to 1.0 more successful at delaying the onset of galloping and virtually suppressing the vortex resonance peak (Fig. 50a). The tube and baffle configura-1. 0. 0. 8 i b h =1.0 C | o = 0.3 0.03 0.05 / 0.10 ' 0.20 a h =0 .05 C | o = 0.3 0.7 ' 0.6 l i m i t 0.2 (a) 0.6 l i m i t (b) 0.8 1.0 1.2 0.6 U 0.8 0 . 2 H 0 . 0 a h = 0.05 b h = 0.6 C| o =0.3 A h/d 0-125/ 0.250X, (d) 0.5 1.0 h/d 0 * 0.125 Expts • 0.250 Hartlen-Currie a h - 0 . 1 5 bh=:0.5 C| o =0.5 U 0 . 8 1.0 Predictions of the Hartlen-Currie model showing: (a) determination of ah with empty damper; (b) determination of bh', (c) response with partially-filled damper and C / 0 = 0.3; (d) response with C/o — 0.5 u o 105 tions were found to be relatively less efficient with the galloping onset velocity UQ of 3.9 and 2.65, respectively, as compared to 4.0 for the plain damper (Fig. 50b). Of course, the problem is made more complicated by the highly amplitude dependent damping ratio. For instance, it has been established that rjrj exhibits a jump at CJ = 1.15 and 1.39, with low amplitude excitation, followed by a rapid decrease in efficiency with CQ (section 3.4.2). This does not appear to prevent the build-up of oscillations under vortex resonance, however, Y subsequently stays close to zero until UQ is reached. On the other hand, a gradual increase in response began around Uo = 2.0 for CJ = 0.92, with rjTti reaching a maximum at higher amplitude (Chapter 3, p. 61) and hence further delaying the onset of instability (Fig. 50a). Experiments with the smaller model#2 continue to show CJ to be the govern-ing parameter. The half-full damper essentially suppressed the galloping instability over the entire range of U, while h/d = 7/8 allowed for a high response at U « 14 in spite of the larger mass (Fig. 51a). It may be pointed out that the wind speed was limited to U < 18 to prevent possible damage to the model and the loss of air bearing low friction characteristics under a large static load. Oscillations quickly appear for h/d — 1/8 with an initial low damping ratio but the hardening charac-teristics, leading to sloshing resonance at higher amplitudes, resulted in the stalling of the response until the excitation becomes strong enough at U « 4.8. Meanwhile, the damper with D/d= 15.2 (CJ much larger than 1.0) is quite uneffective with Y still following the low damping, combined vortex resonance-galloping curve (Fig. 106 ^ , 0 = 2.92 D a m p e r P a r . D / d =1.89 h / d =0.500 M,/M = 0.038 D a m p e r #1 R e (x104) 2.13 2.66 3.22 6.41 — No D a m p e r U 0.92 R e = 2.13x104 0.0 — No D a m p e r A — - A P l a i n (#1) — -• B a f f l e s (#2) Q T u b e (#3) Fig. 50 Galloping response of model#l showing: (a) effect of frequency; (b) effect of internal configuration 107 Damper Par. l imi t U l i m i t Fig. 51 Galloping response of model#2 showing: (a) effect of liquid height and CJ; (b) effect of diameter ratio and CJ 108 51b). This corresponds to rfr>a < 21.5, according to Fig. 43(c) showing the data for the electromagnetic damper tests, a condition certainly met here. The nutation damper performance can be further assessed using the gallop-ing theory. The latter proved to be reasonably accurate at predicting the system response with viscous damping (section 4.2.4). In principle, it should be applicable to any energy dissipation function through the definition of an equivalent viscous damping ratio, provided the time derivatives are small, and the vortex resonance velocity Ur is much lower than U0" (galloping onset velocity). With the knowledge of r)rj versus Y and the necessary modification to the stability analysis (Appendix VII.l), an attempt was made to compute the response of the half-full damper#l mounted on the 10.2 cm square-section model. Analytical results are compared with the experimental data in Fig. 52(a). The theory predicts that the onset of galloping should be considerably delayed as T]rj is high at low amplitudes. Moreover, there exists an upper stable branch, fairly constant with increasing wind speed, due to the diminishing damping ratio with Y. However, experiments conducted at several values of CJ showed that the system starts to gallop before Uo is reached, without stabilizing at the higher limit cycle. This behavior suggests that the transient effects are important. The energy dissipation is of course generated by the liquid motion, and some lapse of time for the system initially at rest is likely to be needed before the damping level reaches the steady-state conditions of Chapter 3. Meanwhile, the structure may gain momentum with r/r>/ further dropping at higher amplitude, thus 109 leading to an even larger response. Results for the 5.1 cm diameter model show better agreement with the the-ory (Fig. 52b). The half-full damper successfully postponed galloping to U > 18, although no upper branch for Y in the range 0.2 to 0.5 was found. Oscillations beyond the physical limits of the test facility (Y > 1.2) could however be excited by imparting a large disturbance at U « 0.9. The configuration with h/d = 1/4 essentially followed the lower branch of the predicted response. The general trend for h/d = 1/8 is also fairly representative of the experimental data with a shift along the x-axis characteristic of an overestimated damper efficiency at low amplitude. This is probably due to inaccurracies in the input damping coefficient based on the free vibration tests of Fig. 47. From the part of the curve where Y slowly increases from 0.1 to 0.15 with U changing from 1.5 to 4.3, it can be inferred that rjTii varies from a very low value (< 0.1 at Y = 0.1) to about 0.21 at e 0 /d = 0.26 (Y = 0.15) before the system starts to gallop. This is compatible with the upward trend for r/r>/ until e 0 /d = 0.5 (Fig. 47) combined with its value of 0.36 for eo/d = 0.35 estimated in section 4.2.5.1. Evaluation of the performance during vortex resonance is more difficult as there is no well established universal response curve characterizing the effect of the system parameters. A comparison between Fig. 50(a) and the data obtained with the elec-tromagnetic dampers (Fig. 43) however shows the half-full configuration oscillating 110 -Theory Expts Y 0.6-0.4-0.2-model 1 h/d=0.500 w=1.15 O J - 1 . 3 9 T ) r | : F i g . 2 3 \ \ : F i g - 3 5 Ir.l Y 1.2 0.8-I 0.4 model 2 h/d = 0 .125^ T l r , : F i g . 4 7 ^ (b) h/d = 0 . 2 5 0 T V - F ig . 4 7 • h/d = 0 . 5 0 0 ^ ^r,! : F i g . 2 3 20 30 U Fig. 52 Predicted galloping response of square prisms fitted with nutation rreaictea galloping res o se ol s are dampers for: (a) model#l; (b) model#2 at CJ — 1.15 should contribute to an overall ij >1.70% at Y = 0.13, since a maximum response of 0.16 was recorded at U = 1.58 on model#l with the eddy current damp-ing. Similar conclusions can be drawn at frequencies of CJ = 0.92 and 1.39. The upper return loop also indicates that TJ is less than 1% at higher amplitudes. This generally agrees with the results of Chapter 3, although the response for CJ = 1.15 and 1.39 should have stabilized at a lower Y where T)rj is much larger. This again I l l suggests that the acceleration of the structure, initially at rest, is an important factor because of the time needed for the damper to grow to its full potential at low eo/d. The predictions of the Hartlen-Currie lift oscillator model, originally based on the empirical parameters of reference 104 (i.e., = 0.13, = 2.50), but sub-sequently modified to fit the data for the electromagnetic damper tests, are shown in Fig. 53. Assuming they give some indication about the system parameters, the damping ratio for CJ = 1.15 and 1.39 is of the order of 1.85% (i.e., rjr<i = 0.46), and 6.94% (r)rti = 1.8) for CJ = 0.92, which are close to the values of the steady-state ex-periments, for the maximum amplitudes obtained here. Thus at a lower frequency, the structure appears to capitalize on the initial high damper efficiency. A comment concerning the aerodynamic model design would be appropriate. Without end plates, damper#l with h/d = 1/2 and CJ = 0.92 successfully delays galloping (Fig. 54a), while allowing for large vibrations starting at U = 4.0 in the presence of end plates (10.2 cm square cylinder). It should be mentioned that the response was identical for both cases in absence of nutation dampers. The effect of model size is illustrated in Fig. 54(b) with damper#l now postponing the onset of instability beyond U = 16 on the 5.1 cm square section. 4.2.6 Concluding Comments A comparison between the two-dimensional wind-induced oscillation tests and the steady-state forced excitation experiments of Chapter 3, along with the free 112 Y 0.3 0.2 0.1 0 Y 0.3 0.2-0.1-0 0 C,0=1-4 — a h =0.13, b h=2.50 ah=0.30 , bh=11.0 Hartlen-Currie • Expts 77 = 1.70% • i I 1.85 C ) o = 1.4 — 6.94 a h - 0 . 3 0 , b h =1 1.0 A GO D (b) A 0.92 • 1.39 9 1.3 1.7 U Fig. 53 Hartlen-Currie model predictions for model#l with: (a) electromag-netic damping; (b) nutation damping vibrations data using the wind tunnel set-up, has led to the important conclusions summarized as follows: • The optimal damper parameters obtained through the steady-state analysis help minimize the response during wind-induced oscillations. For instance, liq-uid sloshing resonance with CJ close to 1 resulted in maximum efficiency, while baffle and inner tube arrangements were less effective at reducing vortex res-onance and galloping instabilities. Low liquid heights with CJ > 1 are better suited for restricting the response at high amplitudes, due to their hardening 113 Y 0 .4 0.2 0.0 l imit %,a = 2.92 Damper Parameters D/d = 1.89 h/d = 0.50 M|/M =0.038 UJ-0. 92 Re=2.13x10 4 Damper# 1 0.5 1 . 5 •—• End P l a t e s A — - A N o E n d p | a t e s 2.5 3.5 U Y 0 .4 l imit 0.2 0 0 •—• 2 .92 A A •( ! . 7 4 w 8 16 U Fig. 54 Response of a square prism with nutation damper as affected by: (a) end plates; (b) model size 114 characteristics, whereas the opposite is true for larger h/d ratios. • The semi-empirical galloping theory proved to be useful in studying the energy dissipation characteristics. The Hartlen-Currie model of vortex resonance is promising as it helped establish a good correlation between the left hand side of the response curve and the damping ratios af Chapter 3, for both circular and square cylinders. • Time dependent parameters involving the acceleration of the model appear to affect the damper performance. Transient effects during liquid sloshing are then significant and a steady-state approximation is no longer sufficient to predict performance on the larger square cross-section. For the weaker excitation of the 5.1 cm square cylinder or the circular model, better agreement with the results of Chapter 3 was observed. In general, dampers whose rjrti versus amplitude curves do not drop too quickly are preferred to avoid premature onset of instability. • Only a small amount of liquid is needed to control the vibrations. The vortex resonance of circular cylinders is limited to Y < 0.1 with a liquid to total mass ratio less than 1% (damper#l, h/d — 1/8). The same arrangement also postponed the onset of galloping by a factor of 4 for the 5.1 cm square model (U~o = 4.8). In the presence of larger excitation of the 10.2 cm square cylinder, a mass ratio of about 4% proved to be more effective (UQ = 4.0 for model#l at h/d = 1/2, CJ = 0.92). It can be reduced significantly at lower frequencies 115 where resonant sloshing conditions can be met with larger, more efficient D/d ratios. 4.3 Three-Dimensional Tests 4.3.1 Preliminary Remarks The effectiveness of nutation dampers was next assessed for finite aspect ratio models free to oscillate about a fixed axis. Both the wind-structure interactions as well as the liquid sloshing motion are now more difficult to analyse than those of the two-dimensional case. The experiments conducted in the boundary layer wind tunnel provided valuable information about the energy dissipation under this type of dynamic excitation. A series of tests in both laminar and turbulent flows was conducted to permit a comparison between the different wind environments. 4.3.2 Test Arrangement and Model Description A 67.6 cm long aluminum rod fastened to a freely rotating shaft, supported by two air bearings, held the aerodynamic model at the upper end and the damper at the bottom (Fig. 55). The arrangement, originally designed by Sullivan 9 4 , was modified to position the damper outside the wind tunnel thus avoiding interfer-ence with the flow, as explained in the next section. Furthermore, the original, longitudinally pressure compensated air bearings were found to permit significant oscillations in the in-flow direction at higher wind speeds. Therefore, the arrange-116 D a m p e r R e c o r d e r o r O s c i l l o s c o p e Fig. 55 Wind tunnel set-up for the three-dimensional tests ment was modified to include two adjustable, hardened steel pins acting on the shaft center of rotation. They fully secured the model without adding any signif-icant inherent damping. Light, 50.8 cm (20") long, square and circular cylinders, similar in design to their two-dimensional counterparts described in section 4.2.2, were used in the test program. Particular attention was directed towards the upper connecting end where an aluminum rod extended half way inside the model. Rein-117 forcements were used to maximize structural rigidity. A physical description of the models is presented in Table III. Two springs provided the desired stiffness to the system, while a strain gauge mounted in series with a Bridge Amplifier Meter and a spectrum analyser (same as the instrumentation used in Chapter 3) recorded the displacement. Table III Physical description of the three-dimensional aerodynamic models M O D E L * 1 2 3 4 C R O S S -SECTION ^ ^ 102 m m I • 102 m m 1 * | | J_5I m m | 11 5I m m LENGTH(mm) 508 508 508 483 MATERIAL BALSA & ALUMINUM BALSA & ALUMINUM PCV BALSA MASS (g) 644 244 515 253 4.3.3 Model Characteristics The static side forces were first measured on the square cylinders in laminar flow. The resulting aerodynamic coefficient was found to be much lower than that of the two-dimensional models. The slope at zero degree angle of attack is small (dCfy/da « 0.60 for model#2), probably due to suction across the opening in the bottom wall (Fig. 56a). The end effect at the top is also likely to contribute to generally lower Cfy. The higher blockage continues to show a larger excitation with 118 a slope of 1.4 at a = 0° and an improved curve for Cfy until a = 10° (model#l, Fig. 56b). Combined with a nonuniform horizontal displacement along the length of the model due to rotation, this should generate weaker galloping instabilities than those of the two-dimensional case (for the same tip displacement). The integration of the side force is presented in terms of a moment coefficient in Fig. 56(c). A small gap of 0.48 cm between the wind tunnel bottom wall and the models was used throughout the tests. Measurements of the natural frequency in free oscillations and the spring stiff-ness were used to arrive at a system inertia of 0.2021 Kg-m 2 for a total mass of 1.044 Kg. More significant contributions came from the aerodynamic model and the damper rod support, due to their relative height extending away from the pivot point. With an inherent damping of 0.1%, or less, derived from the amplitude decay curve of the system without damper (Fig. 57), the aerodynamic reduced damping r}r>a based on the cylinder tip deflection was found to be in the range 0.85-1.13. It was again noticed that r]B generally increased with amplitude in free vibration. Earlier work by Sullivan 9 4 used a constant viscous damping and the agreement with theory was found to be reasonable. The value at low amplitude was thus assumed to be valid throughout the range as explained earlier in section 4.2.4 (different load-ing). The Strouhal number for the 10.2 cm square model was found to be 0.12 (Fig. 58). Its lower value than that of the two-dimensional cylinder is compatible with the results of other investigations1 0 5. The vortex shedding excitation away from 119 0.6-0.4-0.2^ • • dCfv/da = 0.60 • A« •a \., L»-BA Model#2 A D /A t =5.5% L n / d m Z l O . I -0.2-(a) 0.6-R (x104) • 1.23 o 1.74 * 2.74 • 3.88 0.4" 0.2 dCfv/da = 1.40 • • ° ° • U A • 6B# A • OA" 4»i • A O A " Model* 1 A 0 /A t =1 1.1% U / d m = 5.03 -0.2 ( b ) — i — 8 16 —i— 24 32 a, Fig. 56 Static side force for three-dimensional square prisms as: (a) measured on model#2; (b) on model#l; 120 0.12 0.08 0.04 -0.04 • • • • 2 D • R r 1.23x10 4 • • • • • model 1 • • • • model 2 8 * • i 1 i 1 i 1 i (c) 8 16 24 32 a, o Fig. 56 Static side force for three-dimensional square prisms as: (c) expressed as a moment coefficient resonance was too difficult to monitor for the other models. The calibration proce-dure was repeated for each set of springs affecting the force per unit displacement transmitted to the strain gauge. It was similar to that described in section 4.2.3. A last point addresses the position of the damper with respect to the struc-ture. Dampers were first installed at the top of the cylinder, as would be the case in a real life situation. However, a significant weakening of the galloping instabilities was usually observed, with, at times, complete suppression of oscillations even be-fore the liquid was inserted. This can be expected as the axisymmetric shape of the damper contributes to the drag without generating any static side force. Its signif-121 V, % 0 0.4 0.8 Fig. 57 Inherent damping ratio for the 3-D set-up and two frequencies of excitation f, Hz Fig. 58 Strouhal number for the large square cylinder icant size thus resulted in a drop in C / y . Fig. 59 illustrates the effect in laminar flow. Without damper, the aerodynamic model#4 exhibits a well defined vortex resonance peak followed by the onset of galloping at U = 6.0. With the empty 122 damper, the interference is such that galloping never occurs. Hence the damper was supported outside the wind tunnel such that its displacement was equal to the cylinder tip deflection. Fig. 59 Effect of damper position on the response of a square prism 4.3.4 Results and Discussion 4.3.4.1 Vortex Resonance Response of a Circular Cylinder The tests conducted at a frequency of 2.50 Hz showed that relatively low liquid heights can suppress the oscillations. The damper with h/d = 0.046 limited the response to Y « 0.15 in both laminar and turbulent flows, while Y remained lower than 0.05 for h/d = 1/8 (Fig. 60). No response was noticeable with the half-full 123 0.4n o.2i 0.0 (b) 0.6 U Fig. 60 Vortex resonance response of model#3 as affected by h/d and CJ in: (a) laminar flow; (b) turbulent flow 124 damper, even with the baffle or inner tube configurations, as illustrated in Fig. 61. This is consistent with the weaker excitation of the three-dimensional models reported in the literature 1 0 6. It is interesting to note that the response is slightly higher in turbulent flow. This is somewhat unexpected as the vortex formation is thought to be less organized here as compared to the laminar condition. Perhaps the characteristic velocity profiles (Fig. 62), together with a different blockage ratio, are responsible for such behavior. The large scale turbulence induced resonant interactions reported in reference 106 is another possibility. Damper Parameters 1,0 = 1.13 D/d =1.89 h/d = 0.500 CUr1.15 Re=2.66*10 4 M./M z 0.067 Y 0 . 4 -— No Damper 0 .2 -0.0 0.7 0.9 1 . 1 1.3 1 .5 U Fig. 61 Effect of internal configuration on the 3-D model response 125 In general, the damper performance was found to be quite comparable in both laminar and turbulent flows, with the vortex resonance shifting from U « 1.0 (lam-inar) to 1.6 (turbulent case). However, amplitude decay plots for h/d = 1/8 and 1/2 indicate the damping to be generally lower for the three-dimensional models compared to their two-dimensional counterpart (Fig. 63a and b), and gradually de-creases with the amount of rotation as defined in Fig. 63(d). rjrj is fairly constant €D/d Fig. 63 Damping characteristics as affected by liquid height for nutation dam-pers undergoing rotation 127 at high eo/d (Fig. 63c). The dependence on CJ is quite pronounced: the same type of damper but of different diameter ratios shows reversed performance according to the exciting frequency. At f=1.00 Hz, damper#13 (D/d= 15.2) has a larger liquid motion which suppresses the oscillations, while for D/d = 1.89 (damper#l) the model response reaches an amplitude Y = 0.37 (Fig. 64a). However, at f=2.50 Hz, damper#13 allows Y to grow to 0.1 while no response is observed for the smaller diameter ratio (Fig 64b). It should be mentioned that the results for the system without dampers do not collapse onto the same curves for all frequencies (Fig. 65). This is likely due to the variation in inherent damping as affected by the use of different springs. 4.3.4.2 Vortex Resonance and Galloping Response of a Square Cylinder With a low value of reduced aerodynamic damping ratio, the model in lami-nar flow exhibited vortex resonance merging with the onset of galloping instability (similar to the two-dimensional case). The forces are, of course, weaker on the 5.1 cm cross-section (system frequency of 2.5 Hz) and damper#l with h/d = 1/8 essentially suppressed the vibrations over the entire range of U (Fig. 66a), while allowing for an amplitude build-up of Y = 0.10 during vortex resonance on the 10.2cm model (Fig. 67a). The same arrangement remained effective in turbulent flow with Y slowly increasing towards 0.1 until the galloping onset velocity was reached at U = 5.0 (model#l, Fig. 67b). No resonant peak was visible here, thus suggesting that the vortex shedding excitation is small in this type of flow. Similar 128 ^ = 1 • 1 3 Damper#1 & 13 Damper Par. h / d : 0.500 M , / M r 0 . 0 6 7 D/d 6J Re (*104) * 1.89 0.46 1.06 — - - 15.2 3.10 22.7 N o D a m p e r D/d to Re (x10 4) »—* 1.89 1.15 2.66 — -• 15.2 6.70 56.8 — N o D a m p e r Fig. 64 Effect of D/d and CJ on the vortex resonance response of model#3 129 Y Fig. 65 Vortex resonance response for model#3 without dampers at various frequencies trends can be observed at smaller liquid heights, with a large response at resonance for the laminar case for h/d = 0.064 (Fig. 67a), as opposed to the significant gal-loping oscillations with the boundary layer pofile (Fig.67b). This agrees with other studies 1 0 7 indicating turbulence increases the static side force coefficient. With the addition of liquid &th/d= 1/4, the amplitude is further reduced to Y « 0.04 in lam-inar flow (vortex resonance) and 0.05 with turbulence (U = 5.0), while being totally eliminated for h/d > 3/8. Similar trends were observed on the 5.1 cm model with smaller liquid heights, as shown in Fig. 68. CJ remains the controlling parameter, as demonstrated in Fig. 69(a) with the half-full damper#l excited at the various frequencies, or in Fig. 69(b) with two different diameter ratios. Damper#8 with 130 Fig. 66 Galloping response in 3-D for model#2 with nutation dampers in: (a) laminar flow; (b) turbulent flow 131 Damper Par. D / d r 1 . 8 9 Re=2.66x104 M|/M=:0.134xh/d 0.5 1.5 2.5 3.5 U Fig. 67 Galloping response in 3-D for model#l with nutation dampers in: (a) laminar flow; (b) turbulent flow 132 Vr n - 3 . 1 7 D a m p e r P a r a m e t e r s Fig. 68 Effect of low liquid heights on the 3-D galloping response of model#2 h/d = 1/2 and CJ very close to 1.0 is more effective than damper#l at h/d = 1/8 and CJ = 2.21 at low amplitudes in turbulent flow (Fig. 66b), as expected from the sloshing resonance characteristics. However, an upper branch was found with damper#8 in the presence of an initial disturbance suggesting a region of lower damping at higher amplitudes. This was not observed with damper#l as expected from the hardening characteristics at low liquid height resulting in an increase in rjr<i with displacement. The galloping theory can be used here to predict some of the results obtained in laminar flow. Although a local force coefficient is preferable to the average values 133 0 . 5 1 . 5 2 . 5 3 . 5 Fig. 69 Effect of D/d and CJ on the 3-D galloping response of model#2 134 of Fig. 56 9 4 , the first term A \ of the polynomial fit, found to be 0.316 and 0.251 for model#l and 2 , respectively, suggests the onset velocity to exceed the investigated range of U (up to 4.5 and 18 for the two models, respectively) provided the damping ratio for the system is more than 0.45%. This condition is met for damper#l with h/d = 1/8 and e0/d > 0.080 (Fig. 63a) or a corresponding Y > 0.02 for the larger square prism. The configuration with h/d = 0.064, found to exhibit a more uniform damping versus amplitude characteristic and a maximum r)rj « 0.031 in free oscilllations, is also expected to delay galloping beyond the imposed boundaries for U. This, of course, is also true for larger amounts of liquid (h/d > 1/4). The absence of oscillations observed at higher velocities for all cases in laminar flow thus conforms with the predictions. No static forces were measured for the turbulent case and therefore no such analysis can be carried out here. It is, however, interesting to notice that small liquid heights, i.e., h/d — 0.064 or 0.043, postponed the instabilities in a way similar to the two-dimensional flow case with h/d = 1/8 (Fig. 51), where a stalled progression for Y (Figs. 67b, 68) corresponds to the region where r}r>i improves with amplitude (Fig. 63c). 4.3.5 Concluding Comments With the weaker excitation on a three-dimensional bluff body and the higher inertia ratio achieved by positioning the damper at a distance from the center of rotation equivalent to the tip of the structure, relatively smaller amounts of liquid were needed to control the oscillations. The important findings are listed below: 135 The governing damping parameters in rotation are the same as those determined for translation. The response is quite sensitive to CJ and low liquid heights show improved performance at higher amplitudes. A relatively low liquid to system mass ratio of 1.5% (damper#l with h/d= 1/8) was sufficient to keep Y < 0.1 in all cases. rjrji was, however, estimated to be lower than its two-dimensional counterpart in otherwise similar conditions of amplitude and frequency. Vortex resonance dominated the response of the lightly damped 10.2 cm square section cylinder in laminar flow while galloping was the governing mechanism in turbulent conditions. The oscillations on the circular model were also found to be higher in the boundary layer tunnel. Overall, the results justify the need to conduct tests in the simulated natural wind, the smooth air stream results being not conservative. Both experiments and the galloping theory predict the speed for onset of insta-bility to be beyond the investigated range (for the dampers considered here). The transient effects are expected to be small with the weaker aerodynamic forces generating slow accelerations on the models. A steady-state approxima-tion of the damping characteristics should therefore apply reasonably well. 136 4.4 Application to Transmission Lines 4.4.1 Preliminary Remarks This series of tests was designed to demonstrate the applicability of the concept to wind-induced oscillations of transmission line. A two-dimensional cylinder with an arbitrarily chosen square section, mounted horizontally in the laminar flow wind tunnel, was used to generate both vortex resonance and galloping instabilities. A l -though this particular shape is not likely to be representative of a cable under icing conditions, it has a well documented response and permits a comparison with the results of the two-dimensional tests (section 4.2.5.2). The main objective is to assess performance of the nutation damper when a bluff body executes oscillations in the vertical direction as is the case with the transmission lines. The torus container is now part of a more complicated device, similar to the commonly used Stock-bridge damper, so that the vertical motion of the aerodynamic model can generate a significant liquid sloshing to dissipate energy. 4.4.2 Test Arrangement and Model Description A simple support consisting of 8 springs, two of which held by sensitive beam-like strain gauges to record the displacement, was positioned inside the wind tunnel (Figs. 70, 71a). A 86.4 cm (34") long, 10.2 cm (4") side square cross-section cylin-der, with a mass of 1.279 K g and otherwise similar in design to the two-dimensional models of section 4.2, was used in the test program.The 2.5 cm ( « 1") gap between 137 each end of the model and the wind tunnel walls allowed for possible rolling motion and prevented the tunnel corners from interfering with large translational displace-ments. End plates (same dimensions as in section 4.2) were installed to promote flow two-dimensionality. Relatively thick aluminum reinforcements were used inside the structure for increased rigidity as well as to provide a firm base for mounting the springs and damper support. The nutation damper was fixed to a horizontal platform, connected by a torsional spring to a light metallic arm, in turn attached to the aerodynamic model (Fig. 71b). The arrangement allowed for the rotational degree of freedom needed to impart significant sloshing motion. The device was designed to minimize drag . Its width was facing the flow and spring arrangement located in the wake of the container. The center of gravity of the rotating part (damper and supporting platform) was kept under the axis of the cylinder to avoid inducing pitch motion of the model due to response of the damper. The previously described instrumentation of the three-dimensional tests was again used here. Damping Device Fig. 70 Sketch of the horizontally mounted wind tunnel set-up 138 Fig. 71 Horizontally mounted wind tunnel set-up showing: (a) front view of the oscillating system; (b) close-up view of the damping device 139 4.4.3 Model Characteristics The spring lengths were adjusted to ensure that the model at rest was centered at mid-height across the wind tunnel while providing a natural frequency u>\ « 2 Hz without damper (spring constant/unit length « 3.60 N / m 2 ) . With the installation of the rotating damping device, two distinct natural frequencies, characteristic of any two-degree-of-freedom system, were observed. They depend on the torsional spring stiffness and damper mass (Appendix VI.2). Although a number of pa-rameters can be optimized to minimize the response of the model at resonance, the frequency ratio W 2 / W 1 (wi=natural frequency of the model without damping device, o»2=natural frequency of the damping device alone) was kept relatively close to 1.0 with aerodynamic model to damping device mass ratio 1712/mi « 0.10 (Appendix VI, eqs. VI.4, VI.7). A hard torsional spring with a stiffness constant of 0.557 N-m was used to test heavier nutation dampers while smaller amounts of liquid required the installation of a soft spring with k2 — 0.285 N-m. The inherent damping ratio of the system (i.e., no damper) was estimated to be 0.04% corresponding to a very low rjr>a = 0.75, making the structure aerodynamically quite unstable. The energy dissipation in the rotating mechanism of the damper was investigated separately using the strain gauge arrangement of Chapter 3. A simple calibration procedure, with the beam positioned horizontally to support the damper (Fig. 72a), led to a free-oscillation damping ratio r}82 (rjB2 = , where C<j2 is the damping coeffi-2 r a 2 U > 2 cient of the sytem) of approximately 3.0 to 4.0% (Fig. 72b). The experiment was repeated for the partially filled containers to express performance in terms of ijrj. 140 x / d Fig. 72 Evaluation of the secondary system damping ratio showing: (a) cali-bration procedure; (b) r/ a 2 and r)rj versus amplitude 141 4.4.4 Results and Discussion In absence of the damping mechanism, a combined vortex shedding-galloping response, similar to that of the two-dimensional tests (section 4.2.4) was obtained for the square cylinder (Fig. 73a). The damping device was mounted but not activated, with the torsional spring replaced by a rigid bracket to account for the additional aerodynamic forces. Although the model was free to move in any direction, a well behaved one-degree-of-freedom vertical translation was observed throughout the test. This was however not the case with the action of the secondary system. Without liquid, the damper platform oscillated vigorously, and the inherent energy dissipation was sufficient to restrict the first vortex resonance amplitude Y to 0.265. The wind velocity is nondimensionalized with respect to u>i (natural frequency of the main system, « 2.00 Hz) to show the shift in the response. Behind the resonance peak, the lock-in phenomenon was suddenly interrupted with a change in frequency, from 1.68 Hz (wni) to 2.40 Hz ( 0 ^ 2 ) . A beat motion during the transition (Fig. 73b) was often observed and a significant build-up in amplitude did not occur. With increase in wind speed, the response settled at 2.40 Hz and galloping finally occurred near U = 7.0 coupled with a rolling motion. The latter is probably due to a lack of uniformity along the cylinder length. With a natural frequency of 2.12 Hz, it was relatively easy to excite roll given the proper conditions, as discussed later. The introduction of liquid reduced Y to 0.1 at resonance, for both h/d = 1/4 and 1/2 (Fig. 74). The arrangement proved to be effective at controlling both 142 0 . 5 / / m 2 /m 1 — O J 2 / C J l = 0.12 0.96 freq.(Hz) • • 2.08 no damper • • 1.68 j empty • • 2.40 j damper meas. calc. "2.5 1.68 2.40 1.71 2.43 1st Peak 2nd Peak 0.64 0.37 3 . 5 U 7 . 0 1.68 & 2.40 Hz , beat (b) time t Fig. 73 Response of the system without damping liquid showing: (a) effect of auxiliary device; (b) beating motion 143 0.4-0.2-" n l \ Trans. C U n 2 data points only rolling h/d • 0 * 0.25 o 0.50 Damper #1 D / d = 1.89 m 2 /m 1 =:0.12 c u 2 / ^ i = 0.96 0.5 Fig. 74 Effect of liquid height on the system response vortex resonance peaks, with the half-full damper performing better at un2, where CJ = 1.11. It should be mentioned here that the damping characteristics were found to be qualitatively quite similar to those of Chapter 3, with CJ close to 1.0 generating optimal rjrj, as illustrated in Fig. 72(b). The reduced damping ratios were generally lower, a result consistent with the earlier discussion (section 4.3.4.2) which showed that rotation reduces efficiency. This, however, is not a problem here as the liquid to secondary system mass ratio was quite large, with peak 772 > 10%. With the response of the damper, a strong rolling action often accompanied the plunging vortex resonance motion, as illustrated by the frequency spectrum of Fig. 75(a). It persisted at higher U (Fig. 75b). Under slightly different conditions, 144 a galloping type of instability even occurred in roll (Fig. 76a). The damping mech-anism was designed to respond to a translational motion only and therefore could not react to any rolling as it was positioned half-way along the model length. A different arrangement, with a damper fixed at each end of the cylinder and facing the axis perpendicular to the flow, would probably be more effective at controlling both modes and could be a subject of further studies (Fig. 77). y, mV 1 ' ro t In , i i i rot ' A I I I I I ' 1 2 3 4 5 f, Hz Fig. 75 Frequency spectrum of the response for: (a) fv w frot', (b) / „ >• frot 145 Fig. 76 also shows the effect of other parameters such as ui/ui and mi/mi. The empty damper configuration now allows the response to exceed Y = 0.35 with U2/U1 = 1.32 (Fig. 76a), which represents a significant change compared with y = 0.265 for w 2 /wi = 0.96 shown earlier (Fig. 73a). The quarter-full damper is quite ineffective, but more liquid and ct; closer to 1.0 reduces Y to 0.2 at h/d = 1/2. Somewhat different results were obtained with a reduction in m^/mi. The first resonant region is now confined to 0.15 and 0.075, for the empty and half-full damper#8, respectively. A typical low damping vortex-galloping curve dictates the response at the other natural frequency (w n2, Fig. 76b). This overall behavior seems to agree qualitatively with the vibration absorber relation (VI.6) that predicts the resonant amplitude under a constant excitation F: the larger wind-induced oscillations correspond to higher calculated Y\ (Figs. 73, 76). Although beyond the scope of this study, minimizing Y\ is likely to result in a design quite effective in controlling the vibrations. Of course, a more rigorous analysis should include interactions between the system parameters and the aerodynamic forces. 4.4.5 Result Summary The experiments showed that the partially filled torus containers are suitable for transmission line application as significant reduction in vibrations is possible. The following observations can be made: • The presence of two resonant frequencies appears to be beneficial as their mutual m2/m]zz 0.12 a^/cu,^ 1.32 Damper# 1 D/d =1.89 h/d • 0 o 0.25 A 0.50 A (Jj 1.33 0.94 <Z\ |  Trans-rolling ri(xJW*b) w n i ^n2 1st Peak 2nd Peak 2.04 0.877 1.98 1.76 2.98 0.5 4H 1.5 m 2 /m 1 - 0.078 OJ2/CU1 = 1 .00 Damper#8 D/d = 2.77 3.5 U 7.0 2.5 h/d 0 A 0.72 <*>ni wn2 1st Peak 2nd Peak meas. 1.72 2.28 calc. 1.74 2.30 0.73 0.32 (b) 0.5 1.5 — 1 1 *—1—XX 1 1 — 2.5 3T? 7.0 U Fig. 76 Response of the model as affected by: (a) w2/u>i; (b) m 2 / m i 147 Fig. 77 Sketch of the two damper arrangement useful to control roll interaction can disrupt the first vortex shedding lock-in region. On the other hand, the nutation dampers are then required to be effective for both excitations. This condition can be met by certain damper configurations. Alternatively, two separate containers designed for the individual frequency may be used. • A combined vortex resonance-galloping curve can develop at either natural fre-quency for the lightly damped system, depending on the parameters u>2/u>i and mi/mi. With an increase in damping, the onset of galloping is delayed and the model oscillates at un2-• The condition of liquid resonance still maximizes the energy dissipation. The reduced damping ratio continues to be lower in rotation compared to that in translation. A light support with the liquid positioned far away from the center of rotation can, however, give the desired energy dissipation. • More systematic tests to optimize the system parameters (i.e., u^/wi, mi/mi, 148 etc.), as well as a configuration that reduces rolling motion interfering with the main mode of vibration, would be necessary to properly assess and finalize the damper design. 5. C O N C L U S I O N S 149 This investigation has provided information useful in the design of nutation dampers for controlling wind-induced instabilities. With the objective of optimiz-ing the energy dissipation parameters, it has also contributed to the understanding of nonlinear liquid sloshing problems using both theoretical and experimental pro-cedures. Extensive tests with two and three dimensional models in laminar and turbulent flow wind tunnels suggest that the concept of nutation damping can ef-fectively suppress both vortex resonance and galloping instabilities. Based on the study, the following general conclusions can be made: (i) The damping characteristics have been established through a comprehensive test program evaluating influence of the damper's dimensionless parameters. The theoretical development proved useful in understanding the liquid motion and the corresponding role of nonlinearities leading to a consistent variation of the damping ratio with frequency, amplitude, liquid height, etc. (ii) Reliance on the experimental results is still necessary as the potential flow ap-proach in conjunction with the boundary layer correction, although predicting the correct trends, does not account for several mechanisms for energy dissi-pation. Discrepancies between calculations and measurements, in both viscous stresses and pressure fields, indicate that additional damping terms should be included in the equations governing the flow. 150 (iii) Whenever possible, dampers should be designed to operate at their liquid slosh-ing resonance, as shown by the theory, sloshing table experiments, and further verified by the wind tunnel tests. Conditions of low liquid heights and large diameter ratios are more efficient, resulting in higher peaks in damping ratios and a smaller variation with amplitude of excitation. Low Reynolds numbers and internal devices such as baffles or inner tubes should be avoided as they restrict the action of the free surface. (iv) The damper behavior in rotation is similar to that in pure translation with optimal efficiency at the condition of sloshing resonance. However, free oscilla-tion tests show the damping ratio to reduce with an increase in angular motion about the horizontal plane. (v) The wind tunnel tests were useful in assessing the effect of external forces. In general, the better damping characteristics obtained during a steady-state excitation resulted in improved control of wind-induced oscillations. Time de-pendent parameters related to the acceleration of the structure proved to be significant for the case where the aerodynamic model is unstable and the damp-ing ratio is strongly dependent on amplitude. (vi) Relatively small nutation dampers were usually adequate to suppress the vibra-tions. The two-dimensional circular cylinder, with a low r}rta of 2.9, required a damping liquid to structure mass ratio lower than 1% under vortex resonance. 151 Somewhat larger ratios from 1% to 5% (depending on T7r,o) were necessary to significantly delay galloping of the square cross-section. (vii) The weaker aerodynamic excitation associated with the three-dimensional mod-els required even smaller dampers to be used, less than 1% in all cases. For square cylinders, vortex resonance is the main mechanism of instability in lam-inar flow whereas galloping governs the response in the turbulent wind, with maximum displacements approximately the same in either case for the range of wind speed investigated . The motion of the circular cylinder under vortex shedding was found to be of similar magnitude for both flow conditions, with a larger response for the model without dampers under turbulent excitation. (viii) The nutation dampers can easily be applied to transmission lines with the design of a support allowing for rotational motion. They provided significant energy dissipation with an effective control of the instabilities. Results are promising and optimization of the system parameters can lead to further improvements. (ix) Nutation dampers are particularly suited for structures with low natural fre-quencies. For example, at 0.3 Hz or less, it is estimated that a liquid to total mass ratio of 0.75% (Mi/M = 3%) is capable of restricting the response of steel chimneys, with initial aerodynamic reduced damping of 1.9, to Y < 0.1 accord-ing to the available data (Appendix VII). Thus, in addition to being simpler in design, nutation dampers promise to be lighter compared to the conventional 152 tuned mass devices. Some Thoughts on Future Work • The thesis has provided some insight into a class of nutation dampers' behavior. However, an accurate analytical prediction of the damping ratio still remains a challenging task. The proposed formulation, resulting in a 3rd order character-istic equation for the liquid's amplitude response, was found to be incomplete in spite of the special consideration given to the the important phenomenon of resonant interactions. A more sophisticated analytical or numerical scheme accounting for additional sources of dissipation would therefore be worth inves-tigating. This can be combined with a more elaborate experimental procedure to detect and study the nonlinear component of the response through the use of large-scale models, surface sensors, etc. • The time-dependent sloshing response was shown to be significant during the wind tunnel tests, as the liquid is initially at rest and damping is generated by the motion (as for any type of tuned mass damper). A systematic study providing the damping characteristics versus rate of amplitude change (i.e., de/dt, d2ejdt2, etc.) would therefore be quite useful in practical applications. This could be included into a broader evaluation of the performance under different types of excitation encountered in other fields (e.g., earthquake and ocean engineering problems) where such dampers could be used. 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Part II. Development of a Mathematical Model for Full-Scale Application", Journal of Wind Engineering and Industrial Aerodynamics, Vol. 12, 1983, pp. 75-97. [114] Vickery, B.J., Isyumov, N., and Davenport, A.G. , "The Role of Damping, Mass and Stiffness in the Reduction of Wind Effects on Structures", Journal of Wind Engineering and Industrial Aerodynamics, Vol. 11, 1983, pp. 285-294. 162 A P P E N D I X I: N O N L I N E A R F R E E S U R F A C E C O N D I T I O N 1. Basic Equation In polar coordinates, the kinematic boundary condition: drjf_ d^drjj_ 1 dr\j_ _ d§_ dt dr dr r 2 dO dQ dz' { ' and the Bernouilli's equation applied to the free surface (i.e., z = nj) , 2 + _ + _ ( V $ ) 2 = - — rcos*; (7.2) can be combined by eliminating rj/ explicitly. This yields the following expression at z = nf dt2 + 9 d z + dr drdt + r 2 dQ dOdt + dz dldi + (1fr> "cV2" + ( — ) 2 1 ( — ) 2 [ — ) 2 J ydz) dz* r*yd9' dQ2 rzdrKdO} r 2 dr dQ dQdr a * a$ t92$ 2 t9$ d$ a 2 $ + 2 — — — — + dr dzdr r 2 50 dz dfldz d3x n d2x.\d§ . n dh t r . _ r c o s , + _ _ [ _ _ s m , _ _ c o s , ] . ( J.3) 2. Perturbation Series Expansion Introducing: „ , = + £ 2 * r , ( 2 ) + e 3^} 3 ) + (7.5) and substituting into (1.2) gives a n d in terms of and $ ( 3 \ By using a Taylor series expansion around z = rjf as * * t9$. 1 ,d 2$. , r . and replacing for rjf (section 1.3) gives an expression for $z=rif in terms of $*=o, needed to get the full nonlinear free surface conditions. Relation (1.3) then reduces t 0 e«AW + e2*AW + e3*A<3) = 0 +... 163 at z = 0, or expressed in dimensionless form: * (,.7) m a 2$< 2) i a$<2) a & ^ a 2 ^ 1 ) 2 a $ ( 1 ' 3 2 $ ( 1 ' A w = 1 f- 2 1 dr2 <*iiAn dz df dfdr f 2 50 d6dr 1 2 5 $ ( i)a 2 $ ( i ) A a ^ 1 ) a ^ c 1 ) | i a 2 ^ 1 ) az 55C9T dr dzdr2 o n A n dz2 M a 2$< 3) i a$( 3) r a $ ( x ) a 2 $ ( 2 ) a * ( 2 ) a a * w , ' = 1 ; 1- 2 1 + dr2 Q!iiAn dz df dfdr df dfdr 2.d&1>da$W d&2'd2$W d$Wd2$W d&2>d2$W f2 ^~dl d$dr + ~dl dddr * + ^ dz dzdr + ~dl dldT r d * w a 3$( 2) i a 2$( 2> a$<2) a 3$(*) a i l A l l t dr 1 dzdr2 + a i l A „ dz2 J + dr [ dzdr2 """onAn a s 2 J/ + l af ' a f 2 + f 4 ^ a* J ae 2  a ^ 1 ) aa 2^^) i a i w a a $ ( I ) a $ ( 1 ) a 2 $ ( 1 ) + ( as ' a s 2 f 3 af * de ' + 2 af as afas 2 a $ ( 1 ) a $ ( 1 ) a 2 $ ( 1 ) 2 a s ^ a s ^ a 2 ^ 1 ) _ ttllA„ a 3$(*) + f 2 df de dfde + f 2 as a* asa# 2 * a s a r 2 1 a 2 ** 1 ) .add) i , a * ( 1 \ 2 / ^ ( 1 \ 2 i + — 1 — ^ 2 - H r H + 2^ ( - 3 5 - ) + H r H a i i A n az 2 ar r 2 dQ dz r a$( x) a 2 $ ( 1 ) a 2 $ ( 1 > a & w a 3*^) 1 a 2 * ( 1 ) a 2 $ ( J ) 1 1 1 1 1 dr dfdz dfdr df dzdfdr f2 dzdd dBdr i # ) a 3 $ w a 2 $ ( 1 ) a 2 $ ( 1 > a & ^ a 3 ^ 1 ) ^  + f2 de dzdedr*~dz~2 a l a T + as a s 2 a r U r 1 x 2 / r ^ ( 1 ) 1 a 2 $ ( l ) 1 a $ ( 1 > a 2 $ ( 1 ) . a 4 ^ 1 ) + ( a i l A l l J \ l a I 5 5 : 5 " + o „ A 1 1 a s 2 J ar asar + l a l 2 a 7 2 " H r T T O — — \ + ru* cos 0 cos ur. (1.9) ttuAn dzA dr i ' 3 Free Surface Equation Using expressions (1.2), (1.4), (1.5) and (1.6), and using nondimensional param-eter r)f = results in a$( 2) fit =&a.\\\\i\siCjfcos0cosu/r —] — e ? a n A n [ — - — ar ar + anAnK-Wcosflcoswr - _ ^ _ ) - ^ - + ^ V * * 1 * ) 2 ] + (1.10) where st- = 1 for q = 1 , and zero otherwise, and 2 = 0. A P P E N D I X II: N O N L I N E A R , N O N R E S O N A N T P O T E N T I A L F L O W S O L U T I O N 164 1. Second Order Terms Substituting for - E / i ^ i ( A H r ) C ° S h A l t ( " t h ) cosflcosfrr • coshAi,7i into the second order free surface boundary condition (relation 1.8) and rearranging yields, a 2 $( 2 ) 1 a$( 2) + = E E ^ ri { ^ ( A i * f ) c i ( A l i f ) dr2 o:iiAn dz ^ ^ 2 * i + [Ci (A l t f )Ci (A i y f ) - ^ C 1 ( A l t f ) C 1 ( A i y f ) + AX,yCi(A l tf)C1(A i yf)]cos2f;}sin2u;r. (II.l) Assuming $(2) = E E fnmCn{Xnmf) cos^^nm(* + )^ c o s n# sinpc2;7-, and inte-rn n coshA n m/i grating (77.1) as / {II.l)Cn{Xnmf)rdf J a to use Bessel function orthogonality condition (Appendix V.l) , it is found that ^ ( 2 ) = E { / 0 ^ 0 ( A 0 n f ) C ° S h A ° f t K ) n K coshAon/i » ~ /, «coshA 2 n (z + )^ , + / 2 n C 2 ( A 2 n f ) 2 n K T ; cos 29 \ sin 2wr, (77.2) coshA 2 n/i •> where: / 2 n = ^ ^ / l i / i y ^ O n i f n [LLuo(t,j ,n) + JJiio(»,j ' t n) + AKjjKK\ 10 (i, j , n) ] ° n 2u;[ (a ;on/^ 2 -4]Ao„ 5 and / 2 n = ^2 E / i * / i j ' n 2 n , * n = i L L n 2 ( ^ ^ n ) ~ ^ i i 2 ( » ' » i » w ) + AKjjKKii2{i,j,n)]  2 n 2Cj[(Cj2n/Cj)2-4}A2n/X2n 165 Here LVs, JJ's and KK's are Bessel function triple product integrals, A's are coefficients related to J C2(Xnrnf)fdf as defined in Appendix V.1.2, and AKij = auaijXuXij + ^ - -Afy. (JI.3) It can be shown quite readily that the solution is singular for (&/„/£;) 2 — 4 = 0 (/ = 0,2), as fon and f2n become unbounded. Furthermore, it is of order e - 1 for (uin/Cj)2 — 4 = UQE, where i/o is of order 1, a condition that makes the original perturbation series expansion invalid as some of the terms are now of the 1st order. Expression (II.2) thus only applies for: ( ^ ) 2 - 4 > u0v, (IIA) or, \-F--2\>—. (JJ.5) 2. Stability and 3rd Order Equation A standard stability analysis assumes a general 1st order solution of the form 1 0 8 , = E [ ( e i * c o s ^ + c3t sin0) coswr + (e2t cos 0 i cosh A u(z + h) + en sin0) sinwr]Ci(Aitf)-cosh \\ih where en, e2{, e3,- and e4t- are function of the slow time scale T2 = er, thus fol-lowing a procedure similar to Hutton's theory of resonant oscillations in circular cylinders63. Subsequent substitutions into (1.8) and (1.9) lead to the 3rd order system of equations: {-^ •jr1 + P2n + Di^2 e 2 « E ^2(«ijeik + e2je2k + e3je3k + e4je4k)] 2 i j k + DDX e3i[^2 ^2(e2je3k - txitAk]} cosflsinwr = 0; t y k 2 i j k + DDi ^ e4i(T] yZ(e2jesk - eiye4fc)]} cosflcoswr = 0; t y k {-^- + Pin ~ e 4 , ( E E ( e i J e i f c + e2j'c2Jfc + e3ye3jfc + e4ye4fc] 2 i j k + DDi eu(y] ^ T{e2je3k - eiye4*)]} sintfcoswr = 0; 166 de^x + P3n + Vl Z ^ e 3 » l i ' 3 * 72 i j k + n D i ^ e 2 , [ ^ ^ ( e 2 j e 3 t - eiyc4fc)]} sin 0 sin CJT = 0. 3 k Here D\ and DD\ are complicated, frequency dependent expressions otherwise similar to K\ and KK\ of Appendix III. 1, while p i n , . . . , P4n are the third order terms: pin = Di E E E e i » e i y f i i * ; » j * (77.4) P2n = PZn = P4n = 0. The stability of the solution previously derived is studied by considering a distur-bance in the ith mode such as: eu = hi + AieXr2; e2i = A 2 e A r 2 ; e3i = A 3 e A r 2 ; and e4t-+ A i e A r 2 . Substituting into (II.4) leads to the following conditions: A 2 = - 3 ( D i / 2 t ) 2 ; (77.5) A 2 = - ( / i 2 t ) 2 [ ^ i ( i ? i - i ? i i ) ] . (77.6) The solution is stable for nonpositive real part of A and requires 7?i(£>i — £>u) > 0 in (II.6). A P P E N D I X H I : N O N L I N E A R , R E S O N A N T P O T E N T I A L F L O W S O L U T I O N 167 1. No Interactions Using Hutton's theory for a circular cylinder 6 3 and substituting for the Bessel function solution of the torus problem, it is found that: • Detuning parameter: "1 = 0; CJ2 - 1 UJJ.1) = " = a 5 P 7 r • Coefficients of the 3rd order equation for fu'. „ C i ( A n ) - q C i ( A „ a ) * i = 7 ; A n Kx = -1L[SUM1 + GI); {111.2) An KKy = -±[SUM2 + G2], An where: SUM1 = ^ { - 1 8 / i ( l , 1,1,1) + A 2!(3 - 7a 2 1 ) I 2 ( l , 1,1,1) + S A ^ a 2 ^ - a 2 t ) J 3 ( l , 1,1,1) + 6J4(1,1,1,1) + 3A 2 X(3 - 7a 2 1 ) I 5 ( l , 1,1,1) -12/ 6(1,1,1,1) +6 / 7 ( l , 1,1,1)}; SUM2 = - L { - 6 J i ( l , 1,1,1) + A ^ l + 19A2 X) J 2 ( l , 1,1,1) + A ^ a 2 ^ - a 2 x ) J 3 ( l , 1,1,1) + 2J4(1,1,1,1) + A 2 !(3 - 70^)75(1,1,1,1) - 4 / 6 ( l , l , l , l ) + 2 / 7 ( l , 1,1,1)}; GI =Y,{[CK1KKl0i{l,n,l) - / / i 0 i ( l , n , l ) ] f l 0 n + \-^KKl2l{l,n,l) n - i / J i 2 i ( l , n , l ) - J J i 2 i ( l , n , l ) ] n 2 n } ; G2 = Y^{[CKiKK10i{l,n, 1) - 27/ioi(l ,n, l ) ] 2 n 0 „ n + [-CK2KK121 (1, n, 1) + 7/1,2,1 (1, n, 1) + 2 J J121 (1, n, l ) ] n 2 n }; Here I\, I2,...,Ij are Bessel function multiple product integrals defined in Appendix V.1.2 with: o - f°n. o - * 2 n -**0n — f2 ' **2n — » 2 j / l l Hi 168 and CKi = a 0 n a i i A 0 n A i i - - A ^ n + A n ( l - a u ) ; CK2 = a 2 „ a i i A 2 n A i i - ^ A 2 n + A n ( l - a n ) . • Coefficients for stability relations: 5 = AKKlMflx + ^ r 2 F l ] ] (7/7.3) (777.4) 2. Resonant Interactions 2.1 Second Order Terms and Detuning Parameters Substituting equalities (6) and (15) into (1.7), and (14) into (1.8) yield, after neglecting the amplitude time derivatives and phase angles, the following second order relation : —V\f\\C\{X\\r) cos0cos&r — /?if 2iC 2(A 2if) cos20sin2wr +AK0C12(X11f)} + [C[2(Xnf) - C l ( A l l f ) + AK0Cl2(Xlxf)) cos 20} sin2£r + / n f 2 1 { ^ K A i i ^ C U A a i f ) + C 1 (A 1 1 f)C 2 (A 2 1 f) , ^ g i ( A n f ) C 2 ( A 2 1 f ) ] c Q s 6 + ^ K A n f j C ^ A ^ f ) _ C 1 (A 1 1 f )C 2 (A 2 1 f ) , A V Ci(Auf)C 2 (A 2 1 f ) 1 l A , r f3Cj(Aiif)C 2(A 2 1f) +AKi — ——^ -J cos 39j cos WT + [—— ' -, 3C 1 (A 1 1 f)C 2 (A 2 1 f) , A t r C 1 (A 1 1 f )C 2 (A 2 1 f ) 1 a , f 3C; (A„f)C^(A a i f ) + ^ + AK2 - J cos0+ [ -_ 3 C l ( A n r ) C , ( A 2 1 f ) + A K ^ M X 2 l r ) , , r* 2 J J + f e a i { [C 2 a (A a i f ) + 4 ^ ^ 1 + AK3C2{X2lf)] + [C^2(A2 1f) - 4 C * ( A » f ) +A7f3C|(A2 1f)] cos40} sin4u>r = 0. (777.5) Here: 0 " 2" ' AK"i = ( a n - 1)A*^  - anQ:2iA n A 2 i + ^ A ^ ; 169 AK2 - (alx - 1)A^ + 5a i ia 2 iAi iA 2 i -AKZ = ( ~ 2 q : 2 1 + 1 ) A 2 i + 2 a 2 1 a 1 1 A 2 1 A 1 1 . Assuming = ( ) C n ( A r e m f ) C ° s h t k ) cosnO - » \emnj COShAnm/l and integrating (III.5) as l (III.5)Cn(Xnmf)fdf to use the Bessel function orthonogality condition as before, it is found that, $(2) = $(2) + $(2), (7/7.6) where: $ i = no interaction solution, i.e. relation 12, with n > 2 in fan', *(2) V ^ / r j n (\ ^coshA l n(z + A) $2 = 2_.\ [dinCi(\lnr) v cosfl n 1 coshA l n/i , ^ / \ M coshA 3 n (z -I- h) . + ein ^ 3 ( A 3 n r ) 1—s—-cos30j coswr cosh A 3 n / i r . ^, /» «\ coshA3n(z + /i) + [<*3nC3(A3nf) * n l ' C O S 30 cosh A 3 n / i „ . cosh Ai„(5 + )^ „, + e 3 n w ( A i n r j s cos0j cos3u>r cosh\\nh , rj f\ ^coshA 4 n(z + &) . + [d4nC,4(A4nr) i — x — - cos 40 cosh A 4 n / i , /<w\ ^coshA 0 N (2 + ^ ) , . cosh Aon^ with: din = ftinfutii', [7/ 1 2i(l,l ,n) + J J m ( l , l , n ) + ^KK12l(l,l,n)}  1 H ~ ~ 1)A../Af „ 5 n ^ 2 ; C l n = 7 l n / l l f 2 i ; [|77i2 3(l,l,n) - J J 1 2 3 ( l , l , n ) + ^LRK123{1, l > n )] 7 l R (*!» " l)A3n/AL d3n — 0 3 r e / n f 2 i ; f t J J W l . l . n ) - 3 J j W l . l . n ) + ^ K i i W l . l . n l l 170 C3n = 7 3 n / l l f t i ; [ | / / i 2 i ( l , l , n ) + 3 J J m ( l , l , n ) + ^KK121{l,l,n)} 1 S T l 2 ("in ~ 9)Aln/A 2 n 5 ^4n = n 4 n f 2 i ; _ [ / / 2 2 4 ( 1 , 1 , n ) - 4 J J 2 2 4 ( l , 1 , n ) + AK3KK224{1,1, n ) ] (*42n " 16)A 4 »/Aj n [ / / 2 2 0 ( l , l , n ) + 4 J J 2 2 0 ( l , l , n ) + A X 3 ^ K 2 2 4 ( l , l , n ) ]  l 4 n Hn ~ 16)A 0 „/A 2 n Here d\\ = d2i = 0 for the resonant interaction with mode ( 2 , 1 ) , and ^31 = <f4i = ... = 0 with the higher modes, to eliminate secular terms. Regrouping cos 0 cos d>r terms in (111.5), it follows that Integrating as leads to f (JJ / . 8 ) C i ( A i i f)fdf, J a ui — a i f 2 i , 1 TT fl 1 l \ 1 T T ft 1 l \ 1 -Afl _ [|JJ m ( i , 1 , 1 ) + J J i 2 i ( i , 1 , 1 ) + *qiKK131{i, 1 , 1 ) ] Using (6), this results in W H E R E ' 0 1 A777*T7 = & 2 - 1 aifti or i / 2 = v — ajfcij with oj = gl/S* Similarly, /?iftiC 2(A 2if) - f-f\C'*{\ur) - £i!^llQ. + AX 0 Ci 2 (Aiif ) ] = 0, (7/7.9) which, after integrating as f\lII.9)C2{X21f)fdf J a f2 gives: /?i = &i— ?21 171 where, bx - 1 V1"^1' *' *) ~ JJ^(h 1, 1) + AK0KK112{1,1,1)] 2 A 2 i / A 2 21 Furthermore, from (15): 02 = P-bl&, with, fi=%*L; and 61= 6 1 f a i ' ' & 2 e 2 / 3 ' W 1 gi/3' 2.2 Third Order Equation It should be recognized that equality (6) is equivalent to » . X - g ^ - ( * + • * / « > , . / . _ . . . . (//7.10) u) 2 2 y Applying (6) to relation (1.5), (111.10) to (1.8), and substituting for $ W and $(2> into (1.9) yields { - , 2 / u C l ( A u f ) + ^ / l l f t l [ C " A » f f ^ f ) + + ( A g 1 + 2 a j 1 A ; i - 4 « » « a i A ^ + { - / ? 2 i T 2 i C 2 ( A 2 1 f ) + ^ - A M c i ^ A n f ) - + (AlTo cos wr 2 - . ~ 3 2 $ ( 3 ) 1 a m -r-a^Af 1 ) C 1 2 ( A n f ) ] cos 20 sin2wr + + d r 2 a i i A n d £ —Pn cos 0 cos wr — Q 2 2 cos 20 sin2<2>r — P 3 1 cos 30 cos CJT —... — Pnm cos n0 cos mur — Qnm cos n0 sin mwr = 0. (777.11) Here P n m ' s and Q R m ' s are complicated expressions representative of the various mode shapes of $ ^ and $^ 2 \ and Pn = -Pii + T H -Note: Pi\ = term of the no interaction case, = - E { / » ^ n [ C 7 f 0 C 1 ( A n f ) C o ( A 0 r e f ) - CKAnfJCofAon*)] + ^ [ C 7 i T 1 C 1 ( A 1 1 f ) C 2 ( A 2 n f ) - 2 C l ( A l i r ] f 2 ( A 2 - f ) - C { (A n f )C 2 (A 2 n f ) ] } + ^ { - W l C ^ C A n f j C ^ A n f ) ] + A2!(3 - 7a 2 + 3 A f ^ ( 3 - afxJCfCAnf) + 6 C ? ( A l l f ) + SAJ^S - 7 a 2 1 )C 1 (A 1 1 f )C i 2 (A 1 1 f ) _ 1 2 W ^ ^ ( / / j i 2 ) CKi a n d CK2 are defined by (III.3); a n d P n = expression due to the interaction w i t h second mode, n - | c 2 ( A 2 i f ) C 3 ( A 3 „ f ) ] + f 2 i e 3 n [ C 2 ( A 2 1 f ) C i ( A l n f ) C / f 4 - 5 C 2 ( A 2 l f l f l ( A l w f ) - ^ ( A ^ f j C x C A ^ f ) ] + ^ [ 0 ^ ) 0 ^ ) 0 K 5 - C l ( A u ^ l ( A l - f ) - ^ C { ( A n f ) C l ( A l r i f ) ] + c21d3n[C2{\21f)C3{\3nf)CK6 - 3 C 2 ( A 2 1 ^ f 3 ( A 3 - f ) - i c 2 ( A 2 1 f ) C 3 ( A 3 n f ) ] } _,_ , > 2 / C i ( A i i f ) C 2 ( A 2 1 f ) C j ( A n f ) C 2 ( A 2 1 f ) C 2 ' ( A 2 1 f ) + 1 ~2  C i ( A u f ) C 2 2 ( A 2 1 f ) C ( ( A n f ) C 2 2 ( A 2 1 f ) + f4 + f 3 + M i d C A x x f j C K A ^ f ) + ^ 2 C 2 ( A 2 i f ) C i ( A n f ) + ^ ^ ( A n ^ C K A a i f ) + M 3 C 2 ( A 2 1 f ) C 2 ( A 2 1 f ) C 1 ( A 1 1 f ) o C 2 ( A 2 1 f ) C 2 ( A 2 1 f ) C U A 1 1 f ) 1 " 2 *2 where: CA- 3 = 1A 2 - iA2 -2 A 3 n 1 3 « n « 2 i A n A 2 i + - a n a 3 n A n A 3 n — - a 2 i a 3 n A 2 i A 3 n ; CK4 = ?-A2 -- 1 A 2 -2 l n 9 5 3 a n a 2 1 A n A 2 i + - a i i a i n A i i A i n - - a 2 i a i n A 2 i A l n ; CK5 = 1A 2 - 1A 2 -2A l n a n a 2 i A n A 2 i + ^ a i i a i n A i i A i r e - ^ a 2 i a l n A 2 i A x n ; CK6 = £ \ 2 _ 1A 2 -2A 3 n 9 1 3 a n a 2 i A n A 2 i + - a i i a 3 n A i i A 3 n - - a 2 i o ; 3 n A 2 i A 3 n ; DKi = 1 2 " g « 2 1 A 2 A 2 -A 2 1 A 1 1 - T a n ^ i A u A f ! + (1 + a i i ) o ; 2 i Q : i i A f i A 2 1 4 + « i i ( 2 - ^ « L ) A 2 i A i i ; DK2 = - a\x) - 2 a n a 2 i A n A 2 i ; DK3 = ^ ( 1 - 2a2,!) + 2a\x\\x + i a u a 2 1 A n A 2 1 . 4 I Similarly, Q 2 2 contains interacting terms only as, 173 Q22 = -^{hie^diX^CsiXs^CK, - 3 C l ( A l l ^ f 3 ( A 3 " f ) n - C K A u f j C ^ A a n f ) ] + fuesniCiiXiiVCiiXm^CKe + C l ( A l l^f l ( A l n f ) - CKAnfjCftAmf)] CxCAnfJdCAmf) where: ?2 + / i i d i n [ C i ( A i i f ) C 1 ( A l n f ) C t f 9 + - C i ( A „ f ) C i ( A l f t f ) ] + / n d 3 n [ C 1 ( A 1 1 f ) C 3 ( A 3 n f ) C K 1 o - 3 C l ( A l i r ] f 3 ( A 3 - f ) - C i ( A n f ) C 3 ( A 3 r i f ) ] + f2ie 4 n[C 2 (A 2 1 f)Co(Aonf)Cii ' 1 1 - 2C 2 (A 2 1 f )C 0 (Ao n r ) ] + f 2 1rf4n [ C 2 ( A 2 1 f ) C 4 ( A 4 n f ) C J r T 1 2 - S ^ 2 1 ^ 4 ^ - C 2 ( A 2 1 f ) C 4 ( A 4 n f ) ] } + / 2 ^ 2 1 { - C 2 ( A 2 l f ) f 2 ( A l l f ) C ; ( A n f ) C [ ' ( A l x f ) C 2 ( A 2 1 f ) d ^ A n Q C a C A a i f ) 2 f4 ^ l C ^ C A n ^ C ^ A a i f ) , _ „ C i 2 ( A n f ) C 2 ( A 2 i f ) + 4 ^ + DKA + D K C ^ u ^ ^ Q _ + +DKBc'1*{x11,)Ca{Xait) + ^ 6 C 1 ( A 1 1 f ) C i ( A 1 1 f ) C 2 ( A 2 1 f ) - | g i ( A i ^ ) g i ( A i i ^ 2 ( ^ ) } 4- , 2 / V ' U « r ' * f i 1 ? C 3 ( A 2 i r ) 3 C 2 ( A 2 1 f ) C 2 ( A 2 1 f ) + £ > X 7 C 3 ( A 2 1 f ) + DK%~2V;*1'' + M 9 C ^ ( A 2 1 f ) C 2 ( A 2 1 f ) r 3 C 2 2 ( A 2 i f ) C 2 ( A 2 1 f ) f 2 / ' 2 L 2 / ! ~ 2 \ , 1 \ 2 ^ C K " 7 = - A j ^ l - aj x ) + - A f ^ - - a i i a : 3 n A i i A 3 n ; 4 4 4 ^ \2 _2 \ 1\2 - 5 Ci^s = - T A I J I - a n ) - - A l n + - a i i a i n A n A i n ; 4 4 4 1 1 5 CK9 = - A n ( l - a 2 i ) + -X\n - -ctuctlnXnXln; 3 1 5 CK10 = - - A i ^ l - a 2 x ) - - A | n + - a i i a 3 n A i i A 3 n ; 4 4 4 C K u = 16anaonAnAon — 2A2 1Aona2iaon — 8 o : 2 i a : i i A 2 i A u — A 0 n + 2A^; CKX2 — 8 a n a 4 n A n A 4 n - a 2 iQ:4nA 2 iA 4 n - 4 a 2 1 a n A 2 i A n - - A 2 n + A 2 X ; 174 DK4 = z-aiiAnAju + Tttiia2iAf 1A 2i - -ai iaaiAiiAfj + - a f i ^ i A f x A 2 i ; o 4 o 4 DK5 = -A 2 - ! - a n a 2 1 A n A 2 i ; M 6 = i A 2 1 ( l + a 2 1); DK7 = ^ i A ^ A ^ l + 4a2x) + ^ a i i o ^ A n A 3 ^ - 3a2!) - ^cc221X421; 3 5 DK8 = gA^Cl - 4a|J + - a 1 i a 2 1 A 1 i A 2 1 ; 9 15 DK9 = r ; 2 -A 2 1 (l - 4a2 1) + —aua2iAi iA 2 i . Setting the terms with cosflcoswr and cos20sin2u>r in (111.11) to zero, integrating as ' / ( /J/ .llJdCAiifJfdf and I {III.ll)C2{\21f)rdf; J a J a and replacing vx by aif 2 i leads to: / i i ( tf i / i 2 i + ^ i f l i + v2) + 7i = 0; M#2<r22i + £ 2 / u + /?2) =o, (777.13) where JFfi and F i are defined by (III.2), and A 2 £?i = -^-[SUMZ -G3- 773]; A n A 2 E2 = -™-[SUMA -GA- HA}; {111.14) A 2 i A 2 K2 = -^-[SUM5 - GS]. A 2 i Note: SUM3 = \h (1,2,2,1) - i / i (2,1,2,1) + M 2 7 2 (2,2,1,1) + DKJs(2,2,1,1,) 7? 7^  + 74(2,2,1,1) + -^/ 5 (2 ,2 ,1 ,1) + £>7r3/5(2,1,2,1) - 276(2,1,2,1) + 77(1,2,2,1); ST/M4 = ~\h{2,1,1,2) - ^(1,1,2,2) + 7?7T572(1,1,2,2) + Z?7C,73(1,1,2,2) 4 2 + 74(1,1,2,2)+7J>7C575(1,1,2,2)+ £>i f 6 7 5 ( l , 2 ,1 ,2 ) - ^76(1,2,1,2) + I/7(2,1,1,2); 4 5(7 Af 5 = -^ -7 ! (2,2,2,2) + DK*I2 (2,2,2,2) + DK7I3{2,2,2,2) 16 + 374(2,2,2,2) + M 97 5(2,2,2,2) - ^76(2,2,2,2) 175 + § J 7 ( 2 , 2 , 2 , 2 ) ; 4 G3 = J2{[CK3KK231{l,n,l) - 9JJ231{l,n,l) - ^II23i{l,n,l)]lln n + [CK4KK211{l,n,l) - 5 J J 2 1 1 ( l , n , l ) - ^II211{l,n,l)]l3n + [CK5KK211{l,n,l) - J J 2 1 1 ( l , n , l ) - ^ / J 2 1 1 ( l , n , l ) ] u l n + [CK6KK231{l,n,l) - 3 J J 2 3i ( l , n , l ) - i / / 2 3 1 ( l , n , l ) ] n 3 r i } ; GA = J2{[CK7KK132(l,n,l) - 3 J J 1 3 2 ( l , n , l ) - J / 1 3 2 ( l , n , l ) ] * y l B n + [ C K 8 X K 1 1 2 ( l , n , l ) + J J u 2 ( l , n , l ) - / J 1 1 2 ( l , n , l ) ] 7 3 n + [CK9KK112{1, n, 1) + JJ112 - I J n 2 ( l , n, l ) ]n l n + [CK10KKl32{l,n,l) - 3 J J 1 3 2 ( l , n , l ) - II132{l,n, l)]n 3 n}; G 5 = ^ { [ ^ 1 ^ 2 0 2 ( 1 ^ , 1 ) - 2 7 7 2 0 2 ( l , n , l)] 74n n + [ C X 1 2 7 r X 24 2 ( l , n , 1) - 8 J J 2 4 2 (1 ,n , 1) - JJ 2 4 2(1,n, l)]n 4 n }; with, and lin = 7 — — , " i n = 7— - — J n = 1,2,3; /11?21 / l l f t l l4n £4n_. f) — ^ 4 r a . 2 ' l'4n — o 5 f 2 l ?21 <*1 r A n , 2 2 # 3 = T r k i T r " + ( a i i A u - 2 a i i a a i A n A 2 1 ) i i r / r 2 1 2 ( l , l , l ) ] ; H4 = ±[bx ^ - + «2n\\XKK112{1,1,1)]. L A 2 1 2.3 Solution Stability A general solution with nonzero time derivatives and phase angles, combined with a slow time scale TJ = c 1/ 3 , leads to a set of equations similar to (III.8) and (III.9) previously developed. On integration using the orthonogality conditions, as before, and introducing definition of the variables: fix = fu cosv?u; eJi = - / i i s i n y ? u ; f 2 1 = & i cos £ 2 1 ; e21 = f 2 i sin £ 2 1 , to simplify cross product expressions of the form, cos(&r + £>u) sin(wr + ipu) = s^in2(<2>r + <pu); it cos(wr + ¥ ? u)cos (2u ; r - ( - c ; 2 i ) = \ cos(wr + £ 2 i - <pu) + ^ cos(3wr + £ 2 i + <£>n); 176 etc., gives: -"1/1*1 + oi(/uf2i- € i i e 2 i ) -"1«11 - a l ( / l l e 2 1 + ell£*l) -Pitii + biiffi-ell) = -2 = 2 = 4 <*e2i. Ml (II1.15) -/3 1e* 1-6 1(2/r ie n) = - 4 - ^ - . Stability is studied in the neighbourhood of the steady-state solution derived earlier by introducing a disturbance such as: / i i = / n + A 1 e A r i ; e n = A 2 e A r i ; & = C 2 i + A 3 e A r i ; and = A 4 e A r i ; and recognizing that V\ — aif 2 1 , /?i = Now (111.15) reduces to ftl / o 2A aif: i i 0 A M i A o —2A — 2aif2i 26i/n 0 V 0 - 2 & X / U 4A This then yields (determinant=0) ?21 - « l / l l -4A L / l l / S21 A 2 V A 4 ; = 0. (7/7.16) f 4 16A2 + b\J-4±- + Soifc!/2! = 0, or (4A)a = - ( 6 j ^ + 8o 16 1/f 1). i21 The solution is stable for negative (4A)2, i.e., * i $ - + 8ai*i/ii >0, m or ( ^ ) 2 > - 8 ^ . f21 Ol (7/7.17) (777.18) 177 A P P E N D I X TV: A D D E D MASS A N D D A M P I N G RATIOS: DETADLS OF T H E ANALYSIS 1. Added Mass 1.1 General Procedure The domain of integration in relation (21) is subdivided into two regions as /0 r2ir rrij r2w / pcos9rd9dz + / / pcosOrdQdz. {IV.l) -h Jo Jo Jo The total force is the expression evaluated at r = RQ minus F at r = R{. The procedure is similar to the one described in reference 65. It is based on a Taylor series expansion of p around z = 0 to eliminate the dependence of the second term on rjf, i „ \ / „ dp(r,9,z). 1 2d2p(r,9,z). ,rTT~\ p(r,9,z)=p(r,9,0) + z V K ^ l\z=o+-z2 U=o + - • {TV.2) Setting z = rjf in the above relation, integrating over z from 0 to rjf, and solving dp for rjf in terms of p, ——, etc., gives, after simplifications, dz /0 ,2* ,2* 2 / pcos9rd9dz+ / -^-cos9rd9, (IV.Z) -h Jo Jo *P9 where po = p(r,9,0). Pressures are found from Bernouilli's equation (22) using $ derived earlier. (IV.3) can then be integrated analytically, with only the third or lower order being retained in the final expression for simplicity. It should be mentioned that the following equalities were used for integration over dz: J. L 0 cosh A n (z + h) coshXm(z + h) Xnan - Xmam _ dz = - r - r tor M f m ; -fi coshA n/i coshA m/i A n — A m = I( ^ + ^ i ) 2 cosh2 Xnh A n for n = m; 0 sinhA n (z + h) s inhA m (z + h) Xnam - Xman dz= r-5 r-= for n^m; (IVA) -h coshA^/i coshAmh A 2 — A 2 , U — + ? ) * cosh Xnh A n for n = m. Well known trigonometric relations such as: •2»r cos n9 cos mflcW = 0 for n ^ m; = TT for n = m, 178 were employed when integrating over 0. 1.2 Added Mass Higher Order Terms The A's, 2?'s and C4 expressions of relation (24) are presented below: » p E E E W i * [ 3 ( a i ^ i i ) 2 ' " i i ( » , i ) ' : ) + U i i ( i , i , ' : ) ] i « J k Bi = a n A u ^ ^ ^ / x . f / o p i i o ^ p ) + ^t12{i,p)}; »' j (») A > = / n E { ( / n " A ) n o p [ ^ ^ - A n a n ^ i o l L p ) ] + ^ [ ( / n + 3 £ ) p ^ y 1 ^ + ( / " + 4 f » ) ' » ( l , P ) " (A 2 i + ? I 2 I ) " H A I I * 2 * I 2 ( 1 , P ) ] + ^ 3 Y ^ [ ( / i 2 i + l l f x 2 i ) « n ( l , l , l ) + ( a n A n ^ S / ? ! + B 2 = / u X : { ( / i 2 i - f ? i ) n o p [ ^ f i l ^ + a n A n ^ M l , * ) ] + ^ [ ( / I ' I p " f n ) ^ ^ + " 2 f » ) M l . P ) + ( / « + f i a i ) «nA i i f i » a * i a ( l , p ) ]+ £ i i3A J ^ [ ( / i 2 i " 7 f i 2 i ) « n ( l , l , l ) + ( a i i A u ) 2 ( 3 / u + 5 f l a > 1 1 ( l , l , l ) ] } ; (Hi) A 3 = / i i f t i f E n i p X ^ l C i ^ i p ) - a C x ( A l p a ) ] - + / 2 1 ( i , 1) p l p - «nAn f i»*2 i ( i | i ) ] } ; Bz = / i i f c i { £ 3 7 l p ^ [ C . ( A 8 p ) - a C . ( A 8 | , a ) ] - + J 2 1 ( l , l ) 1 p A 3 p 2 2 + « u A n w t 2 1 ( l , l ) ] } ; P + 3/ 2 3( l ,p)] + « . . A 1 i 6 ' [ ( " " ' ' ; 3 7 " ) « » . ( l , r t + ( ' " " " / " ' " I M l . r i ] B 2 + / „ & £ { ^ [ ^ + M M ! + P = 179 + 3/ 2 3(l,p)] + ^ £ f c i o ( l , p ) + a n A n w 2 [ ^ 2 1 ( l ) P ) + ^t23(l,p) + 2 7 4 p r 1 0 ( l , p)) + £ i l l ^ [ t t 2 1 ( i , l , l) + ( d l 3 l 4 4 + a2 iA2ia i iAi i ) iy 2 i ( l , l , l ) ] } ; P + ^ * i o ( l , p ) + « n A n ^ [ ^ 2 i ( l , p ) + ^ 2 3 ( 1 , P ) +2 7 4 p*io(l ,p)] _ ^ [ u 2 i ( 1 ) M ) + _ ^ A ^ a n A u ) ^ ! , ! , ! ) ] } , where A;'s and /'s are combinations of the Bessel and hyperbolic function, and Vs, u's and tv's are the Bessel function products, as defined in Appendices V.2.3 and V.1.3. f*i is taken to be zero when using A2 and B2 in the expressions for A4 and B4. 2. Damping Ratio 2.1 Correction Velocity ui 2.1.1 First Order Taking u 2 to be harmonic, i.e., u 2 = U2etnWet, equation (30a) becomes 1 d dU^ d2U^ Neglecting curvature effects, i.e., ——(r—-2-—) « ——|—, and assuming the gradi-r dr dr dr2 ents perpendicular to the boundaries to be large compared to the change in other directions, i.e., near the vertical and bottom walls, reduces (IV.5) to: ijW =  vs  d*u2 } near r = Ro or r = R.. frV(5) tnu)e dz£ rjW^JOJ^L near z =-h. (77.7) tnwe dzl Setting = Ae*r near r = RQ and r = i?,-, and substituting into (IV.6) gives 180 or 172(1) = oTe A i r + a^ex* r, where Ai = lmuje and A 2 = — \ \ . Introducing the boundary conditions: u\ = — at r = Ro, u 2 = 0 far away from the solid walls, taking the real part and defining the dimensionless velocity, 4*=sib <™> yields a nonzero solution near the rigid wall for the various cases considered here, (i) Nonresonant case = —c n ai cos(u>r + fl), (TV. 10) where ai = / 0 Ci(XuR) R gu sin 9 V Y f u C l ( x u R ) g u cose , near f = R, R = 1 or a, J as expressed in the (r,9,z) cylindrical coordinates, and fl = ^ / - ^ » f ° r -R = 1, and V 2 (f — a), „ „ , / — A 1 coshAi t(S + h) ,k I, for R = a, with / = vite . Also $71, = —J— ?—- (Appendix \/2 coshAi,7i V) , and yi,- = dg/dz. The velocity near 2 = — h is similarly derived from (IV.7), and has the same form as (IV. 10) where C{(A l t f) cosh A i,-Ci(A l f-f) cosh Ai t-0 COS0 \ s i n 0 , a n d n = - i i ± i l / . \/2 (ii)Resonant, No Interaction Case (Planar and Nonplanar) u2x) = -e n [oTcos(a>T + fl) + M i n ( w r + 0)], (7V.11) where fl and a\ are the same as for case (i), with t = 1 only, and 61 accounts for the nonplanar terms as \ / c n C ' l i X n f l sing ^ cosh A11 h h = fi 0 Ci (A n J? ) R gn cos 9 criiCifAuJE^usinfl J . C i ( A n f ) r n . f n cos 0 f coshAn/i 0 near f = R, and z = —h, respectively, (iii) Resonant, Interacting Case fiW = -[enaTcos(wr + fl) + e^fi^sin^wr + fl\/2)], where a~[ and fl are identical to those of case (ii), and 62 is given by, / 0 \ C2{\nR) bo = -021 cos 20 feig8(A21f) s . n 2 g ^ V f2iC2(A2ii2)02isin20 J near f = R, and z = —h, respectively. 2<r2: cosh A21/1 C 2 (A 2 1 f) V f coshA2i/i 0 ~cos 20 J 181 (TV. 12) 2.1.2 Second Order u£} Equation (30b) becomes ^2JT(2) near r = R0, Ri, or near z = — h by replacing + ( u 2 1 ) - V ) V * ] (7 .^13) 2 r7(2) d2u. 2r7(2) with d2 u. Considering the dr2 dz2 simpler nonresonant case where $ ^ = ^^/iiCi(A l tf)c7i,cos0cosit;r and d 1 d d -e ai cos(ur + fl), and using V = (^-,--^r, ^—), it is found that near r = R or r ad oz (72 = 1 or a) • V ) 4 X ) + ( V $ ( 1 ) • VJfiW + ( u ^ • V)V*] = 7E0u;21{e1T^2-[l+ cos2(u;r + fl)] + ej—[cos fl + COS(2U>T + fl)] f, (7F.14) where: / 0 \ e l l = - ^ E hihiCx{\xiR)Cx{\xiR) sin 20,01,-ffiy , , . - g " ^ - \—J2 SU9W sin29 , 2», , \ -jjj-gugij + cos Ogxigxj J 182 62 = €21 + C22; eir= - ^^ / i .7 iyC , i (Ai , - f )Ci(Aiyi2) » 3 \ sm2dfngugij , , h R„ ~2~R~^2~f2 susui1 + y)\ sin2fl , f -75—I0it<7i/ + JJitfiy^] V +2cos20t7lt(j1y J * 3 sin20 -^ltAly + cos Ogugij sin 20 / - 0 i t 0 i y V 2iE * " " 1 J v ^ 2/1 / ^ cos Ogugij-^ J Upon substitution of (IV.14) into (IV.13), an analytical solution for ?72 ^ can be found by introducing the simplifications of the Taylor series expansion near the boundary, i.e., Ci(Ai.-f) = CxiXuR) + (f - R)C[(XUR) + Uf - i?)2C('(AltiE) + 1 (^-15) Ci (A l t f ) = Ci(A l tiE) + (f - J2)c;'(Alt-J2) + -(f - i2)2C{"(Ai<i2) + ... for i2 = 1 or a. Furthermore, the correction velocity is significant only fo fl of order 1, which implies that the boundary layer thickness is of order since y/2 -I from previous development. Recognizing that C'^XuR) = 0 and / is a large number leads to: C I ( A I . T ) «Ci(Ai,-J2); C i ( A i , r ) « ( f - i E ) C l ' ( A l t i 2 ) . {iv.ie) Taking f « J? and substituting (IV.16) into expressions for en and e2 yields a closed form solution for u 2 . Neglecting terms of order / , applying the boundary condition u 2 2^ = — V$^2^ at f = R, and nondimensionalizing acccording to (IV.9) gives fi(2) = I{ flL (1 - e 2 n) - S i Sin fl + '-f [l + [(1 - 0) sin fl - cos fl]] - sin2(u;r + f l ) ^ 2 " + sin(2u;r + fi)[% + % f l + 2 _ 2_ -cos(2wr + n ) - 2 ^ e n + sin(2a;7- + nv / 2)[-| - l a l e " ^ + cos(2u;r + fiv^^-e0^2}, 2 ^ 21 e22 (iv.n) 183 where e n , e21 and e 2 2 are equivalent to en , t2\ and e22 when Ci(Xuf) is replaced by Ci(AiiJ*), C{(Altf)//v^ by C['{\UR), and f by Rt with - 2 / : 2p" 0 C2(A2p.R) sin 26g2p \fopCo{XoPR)9op + hPC2{X2pR)92p cos 20 J u2 thus includes exponentially decaying, 2u>r terms, and asymptotically growing time independent terms across the boundary layer. The steady component must, however, decay in a region extending further into the flow, and is responsible for the presence of the so-called "streaming layer" 1 0 9 . Solving for the corresponding velocity profile however adds another degree of complexity and is not considered in this analysis. A similar development can be carried out near z = —h. Although an exact solution for (IV.13) can be found, it is easier to make simplifications similar to (IV.15): cosh Ai,(z + h) « 1; ' , ( / K 1 8 ) sinhA l t (z + h) « (z + h); near z = —h. This leads to an expression for identical to (IV.17), with now different vectors e n , e2l, e22, and E3. Applying the procedure to the resonant cases also gives results of the same form, where additional steady and harmonic, 2GJT terms now account for the nonplanar mode, and expressions in CJT, 3CJT and 4CJT originate from the resonant interactions. 2.2 Reduced Damping Ratio 2.2.1 Contribution from Rigid Boundary By neglecting terms of order 1, small compared to /, the dissipated energy per cycle from the first term of relation (32) becomes: Ed^^I0u{h{^)2+{^)2]dv}dti near r=iE°' *; (/y-i9) Ed*»J0^Uj{^)2+{^2]dv}dt> near (/y-2o) where u 2 2 ^ = («2r»«2c7> « 2 « ) in the (r,0, z) coordinates. By nondimensionalizing the above relations, taking the derivatives with respect to f and z, squaring and integrating over v, the reduced damping ratio nrj is obtained as per relation (34), where the nonlinear terms are listed below for the various cases of resonant and nonresonant conditions: 184 , (382y/2-567) T , (18y/2-40) r H — Li\\L>\2 H Lll&Z + 9 1 f 9 \ ^ w 2 H \ ^ , / 2 1 3 w 3 V ^ w 2 ( U ) = 4 7 ! i " l O ~ M l 1 + ~ T ~ 2 2 ~ 2 5 M l l M 2 2 + — M 3 1 17^2 (155 - 43V2) ( 1 7 5 - 1 1 2 ^ ) 8 3 2 30 1 2 6 1 6 (51U/2-657) 18 4 2 18 6 2 (382\/2 - 450) , , , , ,272 2782>/2. „ , + '-M12M42 + {— - — ) M 1 3 M 5 2 + ( 1 8 ^ - 4 0 ) M i 2 E 3 + ( 7 6 - 3 2 V 2 ) M i 3 ^ + G L ^ ^ * + ( 2 5 2 - f 4 ^ M 5 2 ^ + 8 « + *42)}; frnl AA, - AA* + 1 f(872-225y/2) 2 (1647 + 66y/2) 2 (m) AAS-AA1 + ^ 2 4 g JVn + 2 g g JV22 (178>/2- 105) 9 w » r (199- 1 3 5 v ^ ) w „ + - ^ '-MnNn - —M22Nn + * - — ^ M u ^ , (400-253v/2) , (5608 + 425y/2) A T + — '-M22N22 + - z^rz -NUN22 54 l o U U + < S ^ S . N u E r + + « + b 9 I zo o 103 , 1071 1231 597 + 1 2 5 ^ " H o " " " " ~ l 2 5 N l l N 8 2 + T o " 7 2 " 8 2 + 2 \ / 2 £ ? + 4v /2^f + 4 V 6 " £ | } ; BB3 = 2oi£ l5, where A A J = A A i when considering the first mode only, and L 2 , , L 2 2 , L11L22,—, aiEs are complicated expressions involving Bessel and hyperbolic function cross-products as defined in Appendix V , such as, I 2 ! = D ! ( t , j,Jk,0i[G1 1 1(.-,i>fc f 0 + G 1 1 ( , , i , f c , 0 ( l + 3 A 2 y A 2 , + 2A2,) + G»(i,j,k,l)} + D2(iJ,kJ)j[Clll{l'J6'k'l) + G?(i,jtk,l)(± *% a CL + SA^-A 2 , + ^-) + C " ( ' ' ' + ^ y H ^ / ^ " ( t , j,fc,0 J a a 4 + 2 J £ > n ( t , y , A : , / ) + / ^ ( t . i , * , / ) + 2 J D 1 1 ( » - , y , f c , / ) - 2W\l (t, y, * , /) + 2iD\l (.-, y, * , /) + I D } 1 (t, j, K 01; £ l 2 = Bs(»\ J , *> 0 j lG} 1 (•*. i . f c . 0 + s ^ 1 (», J. M R ^4(», 3, k, I) 4 1 G J ^ M + sG^f.-.j,*,/)] + F m ^ [ 3 I D ? { i J , k , t ) 4 a' 4 + /i?JS(«,i, *,/)]; £ n £ 2 2 = Ds{iJ,k,t)±[G\%j,k,t) - (3X2xj + 2)G\1(i,j,k,l)) + D 6 ( t , y , f c , / ) i [ ^ M M _ ( 3 A ? y + ^ ) G5 1 ( . - , i ,* ,0] + J i i w ^ l s / u J 1 ^ , i,*,0 - «>ii(«\i,M - iD\\{i,j,k,i) + 2IDH(i,j,k,l)}; L X X E 3 = £>7(t,i,2)[GGl2(t,i,n) - GG5 2 (t,j,n) - £ - L ^ G G j a ( . \ j , n ) ] + I>8(t ,J,2)[ ^ - ( 1/ a 2 + A 0 ) G G i 2 ( t < J > ) ] _ U 7(,-,y,0)[GGl°(i, j,n)(l + Ajy)] - D8(i,j,0)[GG\°(iJtn){\ + A2,)] - Fi3o[IDD\0(i,j,n) + IDDl°(i,j,n)} - ^[WD\2(i,j,n) - IDD\\i,j,n) - 2IDD\2{i,j,n) - 2IDD\2{i,j,n) + 2IDD12{i,j\n)]; L L 2 2 E 3 = D9{iJ,2)[GG\2{i,j,n) + \GG\2 (», j, n)] + £ > 1 0 ( t , j, 2) [ ° C " J ' n ) + iGGi 2(t , j ,n) ] + [D9(.-,j,0) + Z?10(t,y,0)]GG10(t,y,n) - FijoLLioi{i,n,j) - Fij2\^LLX2X{i,n,j) + JJi2i{i,n,j)}; M2X = {G\U\ + G 2 M(2, 2 - * 2 ) ( 1 + 2X2XX) + Xtx(2s2 + s2)} + G | * s + a C l 4 ( 4 A l i a ){ g a^ i + G i M M - *2)(^r + ^  + A n ( 2 5 l + *2)1 + G"4} + , ^ £ t o I ( 2 ^ 1 + •?) + 2IDl\2s2 - s2) <*• } 4 cosh Xxxh K + 2 s 2 J P * 1 + s2{IDll - 2ID\X + 2ID\l + ID]1)). Here: «i = fii ~ & a n d *2 = fh + fiiJ _ C 2 ( A 1 1 ) G j / 2 ( A n ) [ r l l , 2 , r i i f o , 2 , j 8 » | agi^AuttjCy'CAna) M 2 2 = - [Gx s 2 + G 2 ( 2 s 2 + sx)\ + -[G»4 + g . 1 ^ ! + 4)) + A X\\ Ai^l + A)™? + s2IDll}; a 4cosh Xxxh MxxM22 = ig i ' (A u )g i 3 (Ai i ) {GJ^ 2 - G\l\2s2{\ + X2XX) + s2X2xx}} 186 4 cosh An/i = 4fn&{c«(A„)(G» + 2G» + G*1) + + ^ + Gi 1)}; M | 2 = 2 / ? l f ? 1 { G i ' [ C ? ( A I l ) c f (A,,) + C } ( * . . « ) C f ( * » « ) ) a f J i y 1 0 f) + At: cosh4 Ax M213 = / l A { c f ( A n ) [ G » + G^CA 4 , - 2A h - 1) + G 3 X ] + aCf ( A „ « ) [ ^ + G ^ A f , - - 1 ) + % + \ AID? - 2ID? a o a cosh An/i + J D l 1 - 2ID\l + 2ID\l + ID}1]}; Mh = fhtii{lCl{Xii)CZa(\n) + Cf (AnajC^CAnoJajGj 1 cosh An/i } M 1 3 M 5 2 = / ? 1 f ? i { c J ( A „ ) C l w ( A 1 1 ) G | 1 ( l - A 2 X) + aCf (AnajCflAna) - Ah) + - ^ T T T ^ 1 " / £ > " " 7 D ^} ; a cosh Au/i > M 2 2 , M%2 and M12M42 are identical to M 2 X , M 2 2 and M11M22, respectively, after permuting 52 and si; M12E3 = X ; { /2nS2[C2 (A 2 n )C 2 (A n ) [GGl 2 ( l , l ,n) - GG* 2 ( l , l ,n ) n + g g P ( M . " ) ( 1 - A?,)] + a t 7 , ( A a . « ) C ? ( A l l 0 ) [ ° S ^ l i l = ) a 2 2 V G G j ° ( l , l , n ) ( l + A 2 J + aCo(A 0 n a)C 2 (A 1 1 a)GGi 0 ( l , l ,n ) (4 2 cosh An/i L2coshA2n'i - 2IDD\2(\, l,n) - 2mE>4-2(l, l,n) + 2JZ>£>|2(1, l,n)] + [ 2f°"Slt[IDD\0(l,l,n) + IDDl°(l,l,n)}} }; coshAon'i J 187 + 6 ? G i a ( l , l , n ) ( l - X2n)\ + aC 2 (A 2 n a)C 1 2 (A 1 1 a)[ 2 C g " ( 1 » 1 . w ) + 2 G G i ' ( l , l , . ) + G G , 2 ( l l n ) ( l _ A h ) 1 T [ J ^ 1 2 ( 1 , 1,n) - mD$ 2 (l ,1 ,n) 9 i cosh An/icosh A 2 r i / i + 2IDDl2{l, 1,n) - 27Z?£>] 2 (1 ,1 , n) - 2 / £ > £ > i 2 ( l , 1,n)}}; M 4 2 ^ 3 = ^ { ^ [ ^ ' ( A u j C ^ A x O d C A ^ t G G l 2 ^ , ! ^ ) + G G 4 2 ( M , " ) j n + a C r i A x x ^ C ^ A n a j d C A ^ a ) ! ^ 2 ^ 1 ' ^ + + [Ci'CAujC^AujCoCAon) + aCKAuaJdCAnajCoCAona)] G G 4 0 ( l , l , n ) / 0 r t 3 x - * " J / a » * 2 [ | L L 1 2 1 ( l , n , l ) cosh A n / i Lcosh A 2 i / i * + J J m (1, n, 1)] + / ° w S l L L 1 0 i (1, n, 1)1; COSliAon" M 5 2 ^ 4 = / l l ? l l X ) f 2 l { [ C l ( A l l ) C ' i , ( A l l ) C 2 ( A 2 n ) n + a C 1 ( A 1 1 a ) C i ' ( A n o ) C 2 ( A 2 n a ) ] G G l 2 ( l , l , n ) ^2 , A t" J £ W l , n , l ) + 2 J J 1 2 1 ( l ,n , l ) ] } ; cosh A n n cosh A 2 n a ' ^11 = ?2 4i{c 2 4(A 2 1)[76G 2 2 + G22(h421 + 2A 2 X - 32) + 4G 2 2 ] + aC 4 (A 2 1 a) /~<22 q \2 on /~<22 i o [ 7 6 % + G 2 2 ^ 4 , + 2 - ™) + 4%) + l—^\llD22 a 6 4 a 2 a4' a 2 c o s h 4 A 2 1 / i 4 + 2 / P | 2 + IDf + SID22 - 21D\2 + 8IDl2 + 16JZ)2.2]}; ^222 = f 2 i { c 2 2 ( A 2 1 ) c f (A 2 1 ) [4G 2 2 + G 2 2 ]+aC|(A 2 ia)C2 2 (A 2 ia)[4^| 2 -+ G 2 2 l + X ^ [ 3 I D 2 2 + 4ID22}}] A 2 i / i * ' 2 cosh M u i V u = 2 / 2 1 f 2 2 1 { c 1 2 ( A 1 1 ) G 2 ( A 2 1 ) G l 2 ( A | l + 1)(A2 X + 1) + a C 2 ( A „ a ) C 2 ( A 2 1 a ) G l 2 ( *f + ^ ( A 2 , + ^ ) + ^ID21 + IDf + 4ID\2 + ID\2 j j 4cosh 2 Aiincosh 2 A 2 i « ' M„Nlt = -2/ ucr 2 2 1 { c 1 (A 1 1 )Ci ' (A 1 1 )C 2 2 (A 2 1 )G^( A i l + 1) + a C 1 ( A 1 1 a ) C { ' ( A „ a ) C 2 ( A 2 1 a ) G ^ ( A | i + 1 ) 2 [78(2,1,2,1)] 1. 4 cosh2 A11 h cosh2 A 21 M„AT 2 2 = -/ n f2 1 { c 2 ( A 1 1 ) C 2 ' ( A 2 1 ) C 2 ( A 2 1 ) G ^ 2 i A i ^ t i i + aC 2 (Aiia)C 2 (A 2 io)C 2 (A 2 ia)G2 2^ 1 1 ^—-+ A r V 2 2 2 [7 8 ( l ,2 , l ,2 )+ / 6 ( l ,2 , l ,2)]y cosh2 \\\h cosh2 X2\h > M22N22 = ^ - { [ C 1 ( A 1 1 ) C i ' ( A 1 1 ) C 2 ( A 2 1 ) C 2 ' ( A 2 1 ) + aC 1(A 1 1a)C{'(A 1 1a)C 2(A 2 1a)C 2'(A 2 1a)]Gl A 2 A 2 ^ 2 1 7^ 2}; 5 cosh An/icosh A 2 i / i NnN22 = <T241{C23(A21)C2'(A21)[16G22 + G 2 2 ( ^ - 1)] + a C f ( A 2 1 a ) C 2 ' ( A 2 1 a ) [ i ^ + G 2 2 ( ^ A 2 1 - 1)] _ A 2 [77?22/2 + 876(2,2,2,2) - 2ID\\ - 8//?22] |, cosh4 X2ih > i V 1 1 £ 7 = - < r 2 2 1 X;{ c'2 2(A2i)[d4n [ 8 G G 2 4 ( l , l , n ) - 8 G G 2 4 ( l , l , n ) n + GG2 4 (l , l ,n)(2 - A| i)]C 4(A 4 1) + C 4 n G G 2 0 ( l , l , n ) ( 2 + ^ - ) G 0 ( A 0 n ) ] +aC 2(A 2 1a) [d 4 n [8GG 2 4 (l , l ,n) - * G G ? £ > *» + G G 2 4 ( l , l , n ) ( l - Mi)]C 4(A 4 na) + e 4 n G G 2 0 ( l , l , » ) ( J ^ + ^ ^ ( A o n a ) ] | 1 , [ d 4 n . [ ^ ^ ( M . n ) 2 J cosh2 X2ih '•coshA4nn 2 + 47DP^4(1,1, n) - 47£>7?|4(1, l,n) - 167P7?24(1, l,n)] - J f r W l l , 1, n) + 4 / £ £ > 2 0 ( l , 1 , n)]j }; cosh Xonh J •* N22E7 = cr 2 xX ; {c 2 (A 2 1 )C 2 ' (A 2 1 )[-rf4nG 4 (A 4 r i )[8GG 2 4 (l ,l ,n) n , G G ^ 4 ( l , l , n ) 1 _ ,_ r 8GG 2 4 ( l , l ,n) , G G 2 4 ( l , l , n + 2 J ~ d4nG 4(A 4 no)a[ 1- -+ 189 2 + 1 2 1 A d. An + e4nGGl°[C0{Xon) + aC0{X0na)] ^ J cosh A21/1 L c o s h A 4 n / i [ L L 2 4 2 9 1 , n , 1 ) - 8 J J 2 « ( l , » , 1)] + — ^ - L W l , n , 1)]] }. * coshAonn J ' Now, recognizing gx g2 « 4t7x <72 due to the resonant interaction gives: G\2 Nh = / i a i ? a 2 i {c 2 (A a i ) C a (A n ) [ 2 G » + ^ - + (50 + 32A n)G I2} + a C 2 ( A 2 1 a ) C 2 ( A n a ) [ 2 ^ + ^ + ( j £ + 32A 4 1 )^ 2 ] + 1 ^(1,2,1,2) | /8(2,1,2,1) cosh2 Xnh cosh2 X2ih 8 2 + /8(1,1,2,2) + |/9(2,2,1,1) + 1079(1,1,2,2) - 4/ 9(l,2,l,2) + ^ID? - 8ID12 - ID21 + AID21 + + MD72}}; ( o C12 -I- C12 « C12 N?2 = fl^{cf{X2,)C2{Xu)\^^ + ^ ] + A4j^5(l , l ,2,2)+/ 2 (1,2,1,2)]|, 2 2 cosh2 X2%h cosh2 Xnh ' Ni2 = / 1 2 if 2 2 i{G 2 (A 2 1 )Cl' 2 (A 2 1 )(8Gl 2 + 2G*2) + a C a ( A a i a ) C ; ' 2 ( A n a ) ( ? | £ + 2G*2) + A ^ [^(2,2, l , l ) + 2/ 2(l,2,l,2)]| ; 2cosh2 A 2incosh2 Xnh > N 7 1 N 7 2 = / 1 2 1 f 2 2 1 { c 2 (A 2 1 )C 2 ' (A 2 1 )C 2 (A 1 1 )Gl 2 ( i + 4A21) + aC 2 (A 2 1 a)C 2 ' (A 2 1 a)C 1 2 (A 1 1 a)Gl 2 (^ 2 + 4A 2 J _,_ , 2 [/6(1,2,1,2) - 2/7(2,1,2,1) - 4/ 8(l, 2,1,2)] ^ T A 2 1 n s 5 31 I ) 2 cosh Ana cosh A 2 i / i -1 N 7 1 N 8 2 = / 1 2 1<r 2 2 1{c 2(A 2 1)Ci'(A 1 1)C 1(A 1 1)(4Gl 2 + 40G^2) + aG 2 (A 2 1 a)Ci ' (A 1 1 a)C 1 (A 1 1 a) (^_ + ^ - ) 2 [/6(2,2,1,1) + 5/ 6(l, 2,1,2) - 8/ 7(l, 2,1,2) - 4/ 4(l, 2,1,2)] ^ ' 1 1 o s o * I' 2 cosh A2i/icosh Ana ' N 7 2 N 8 2 = / 1 2 1 f 2 2 1 {[C 2 ' (A 2 1 )C; , (A 1 1 )C 2 (A 2 1 )G 1 (A 1 1 ) + aC 2 ,(A 2 1a)Ci'(A 1 1a)C 2(A 2 1a)C 1(A 1 1a)]8G} 2 + \lL\2Ll h{W,2) | ; 2 cosh2 A 2 i h cosh2 A11 h > El = EEfW^^P) + 2/on/oP/,*(0,n,p)}; 190 where F* is a function defined as: \n,p) = Ci{\in)Ci{\iP)[i20ii{n,p) + #t-(n,p)] + aCi{Xina)Ci{Xipa) [ t 2 / ? " ( 2 n , P ) + /%(n,p)] + i2„(n,p) + i2J2ii(n,p); Of El n p El = Y, Y , l d i n d i P F * C1' N> P) + ei« eip^* (3> N> P)1; n » El = Y^2[d3nd3pF*{3,n,p) + e3ne3pF*(l,n,p)}; n p E2 = ^2^2[d4nd4pF*(4,n,p) + e4ne4pF*(0,n,p)}; n p axE5 = / i i X ) d i „ J ? , * ( l , l , n ) . n /? and f3* functions are defined in Appendix V . 2.2.2 Free Surface Contibution The vector n = (———, —f, l) is normal to the free surface. Expressing or r do r)f in terms of $ and d2x/dt2 using relation (1.2), and substituting into the second term of equation (32) yields Ed = H J™* { J rdOdr}dt + higher order terms. (IV.21) Integration of the above equation gives the leading order terms presented in (35). 2.3 Energy Ratio The procedure outlined in section 4.3 of Chapter 2 leads to the following equal-ities: CJ flT ( f , d m dr dr o d m « d 2 m + / V m • VmrdOdrdzldr; {IV.22) J V 6 A A A n = 27 /o \J8 a i l A l l K - 3 r ~ ) + ( a i l A l 1 ^ dzdr - j ( ^ ( 1 ) ) 4 - 2 ^ - ^ w + « l l A l l ( _ ) W > ]r<Wdf + f {Vm)2rd6drdz\dT, {IV.2Z) d { v m ) dz For the particular cases (n = 1,2,3): (i) BBBx = 0; AAA\ = 2VS(f0n,fom) + VS(f2nJ2m) - (a i iA n ) 2 a ; 3 E E E ai . -Ai .7 i .7 iy[2 /o»^no( l , 1, n) + f2nKK112{l, 1, n)] t j n a i i A n E E E E fuMikfu {won (t, j, k, i) 64 t J K I + ID\l(i,j,kJ)}+6ID\l(i,j,kJ)+ID\l(i,^ aifcauAi,AiyAijfcAii) - 6w2(ai tAi,A2y + anAiyAiJanAu - ^{anXuyiauaijXHXij)} + 6ID\l{i,j,k,l)[3alkXlkctuXu - w 2 a 1 1 A 1 1 ( a l t A l i + axiXxj\ + 2ID10{i,j,k,l)[3ctlkXlkauXu - w 2 anAu(ai t A l t - + axyAxy)]}, (ii) BBB2 = 0; AAA2 = 2VS{fon,fom) + VS(f2n,f2m) + VS{$2n,t2m) - ( a n A n w ) 3 [Von{fu - cn)KKno(l,l,n) + f2n(f^ + f? 1)Jirjr l i a(l,l,n)] " ^^{Mfti + iii) + + ID\\) + 2[3(/141 + 2 / i 2 i f i 2 i ] / ^ + (aiiA 1 1) 4/P 1^[3(/ 1 4 1 + ^ ( 3 - ^ - 12a;4) + 2 f t f? " i i Q/\2 (3 - - 5 - - 12a;4)] + ( a u A n ) 2 / ! ? } * ^ + f l\)(3 - 2a;2) + 4/iVi 2i(3 - 4a;2)] + (a 1 1 A 1 1 ) 2 /D 1 ^[2(/ 4 1 + C4a)(3 - 2a;2) + 8/2 1Cu62(l + 2a;2)], (iii) BBB3 = 2[fnWS(dln,dlm) + f 2 i( / 2 n , / 2 m)](l + a n A „ w 2 ) - 6(a11A11u;)3/i2i<r2iii'ii'i2i(l, 1,1); AAA3 = AAA\ + VS{dln, dlm) + VS{eln, elm) + VS(e3n,e3m) + VS(d3n, d3m) + 2VS{e4n, e4 m) + VS{d4n, d4m) - (anA11w)3^{32c:|1[2e4nii'ii'22o(l,l,n) + d 4 n i t t r 2 2 4 ( l , 1, n)]+ n \hi$2i[{dln-Ze3n)KKl2l{\,l,n) + (e l n - 3d3n) KK123(1, l,n)]} - £ ^ { c ^ [ 9 ( J £ > 2 2 + 16/Z?22) + 2 4 J £ > 2 2 + ( a 2 1 A 2 1 ) 4 / £ > 2 2 ( 1 8 - 12a;2) + ( a 2 1 A 2 1 ) 2 / £ > 2 2 ( 2 4 - 16a;2)] + fxx&W&xl + 4ID\1) + SID\27 + 2ID\\ + K>(anXn)4ID\l(2 - 34a;2 - ^a; 4) + 4 ( a 1 1 A 1 1 ) 2 / £ > 1 1 2 ( 8 - 2a;2) + 4 ( a 1 1 A 1 1 ) 2 J ^ ( i 4 I - 32a;2) + 40(a 1 1 A 1 1 ) 2 /D 1 1 2 ( l - tf) + lS{ct^ii?ID\l\, 192 where: AAA\ = AAA\ for the case where i = j = k — I = 1; VS(rin,rim) = (auAu^ 2 ) £ [ ' " 2 r ? n ^ f ] + WS{rinirim); WS(rin,rim) = ^^rinrirnpij{n,m)[IAjj{n,m) + j2J A}j{n,m) + a 2 n Ay n ] ; n m with j = i when rin = fon, f2n, dln, d3n, d4n, and j = 0,1,3 for r,„ = e 4 n, e3rv, and ei n , respectively. A P P E N D I X V : U S E F U L B E S S E L A N D H Y P E R B O L I C F U N C T I O N R E L A T I O N S A N D D E F I N I T I O N S 1. Bessel Functions 1.1 Orthonogality Condition C „ ( A n p f ) C m ( A m g f ) f c i f = 0, if A n p ^  A m , ; A ^2 ' ^ A np A m g , where A n p = i { ( A 2 p - n 2 ) C 2 ( A n p ) - ( A 2 p a 2 - n 2 ) C 2 ( A 1.2 Cross-Product Integrals ,j,k,l) = / ' c n A a r j C ^ A y i f j C U A f c x f j C ^ A n f J f c i f ; J a np ,J,k,l) h(i ,j,k,l) h(i ,j,k,l) h(i j,k,l) j,k,l) Hi j,k,l) h(i j,k,l) h(i j,k,l) In the following, (i,j,k,l) is omitted when i = j = k = I = 1. IDr(i,3\k,l) = / 1 C ; ' ( A m f ) C ; ( A n y f ) C ^ ( A m f c f ) C ^ ( A m / f ) f c i f ; J a ID\ ID? I D n m , ID?™' IDnm' ID%m> ID%m' TTjnmt ID™> TDnmt ID™> TTjnmi jr\nmi J A /16 ID i™' i , j , k , l ) i,j,k,l) i,j,k,l) i,j,k,l) i j , k , l ) i,3,k,l) i,j,k,l) i,j,k,l) i,j,k,l) i,j,k,l) i,j,k,l) i,j,k,l) i,j,k,l) i j , k , l ) i j , k , l ) C"(\nir)Cn(\n]r)C'm(\mkr)Cr ^ n m p ( ' i J> kt •I Jnmp{})3, kt KKnmp{i,j\ kt -f J a J a -i: -C • i = / C'n{Xnif)Cn{Xnjf)C'm{Xmkf)C„, J a = / 1 c ; ( A « < f ) c : ( A , , y f ) C ^ ( A m * f ) c ; J a -i: = / C^Xnif)Cn{Xnjf)Cm{Xmkf)Cr J a -i: •i. -i: C'n (\nif) Cn ( A n y f ) C'm (Xmk f ) C„ Cn(Xnif)Cn(Xnjf)C'm(Xmkf)C„ C'n ( A n i f ) Cn ( A n j r r ) Cm (Xmk f ) C„ Cn{Xnir)Cn(Xnjf)Cm(Xmkf)Cn C n ( A n ^ ) C ; ( A n y f ) C r n ( A m f c f ) C r C'n{*ni?)Cn{Xnir)C'm{Xmkr)Cn Cn{Xnir)Cn{Xnjf)Cm{Xmkf)Cr C'n(Xnif)Cn{Xnjf)Cm(Xmkf)Cr Cn{Xnif)Cn{Xnjf)Cm{Xmkr)Cn Cn(Xnif)Cn{Xnjf)Cm(Xmkf)Cr Cn{Xnif)Cn{Xnjf)Cm{Xmkf)Cr \ ^ d r Xmir)—; r , ~dr Am/rj—; r A ,f)--A m / r ) ^ ; A m f j — ; Xmif)fdf; Xmif)fdr] Xmlf)^-; r A m / r ) ^ ; X m i r ) — ; r Xmlr)rdf; Xmlr)rdr; Xmlr)rdr; \ *\df Amir)—', r C'n{Xnif)C'm{Xmjf)Cp{Xpkf)fdf; 1 jj. Cn[Xnif)Cm{Xmjf)Cp(Xpkf)—; Cn(Xnif)Cm(Xmjf)Cp(Xpkr)fdf; 195 IDD\p{i,j,n) IDDlP(i,j,n) IDDq/{iJ,n) IDDq/{i,j,n) IDDqp{i,j,n) IDDq6p(i,j,n) IAnm{i,j) = JAnm(i,j) = 1.3 Simple Cross-Products D i { i J , k , l f C;'(A g tf)c;(A wf)C p(A p nf)fdf; J a / ' ^ ( A ^ C ^ A ^ f j C ^ A ^ f ) ^ ; J a ' / ' c j C A ^ C ^ A ^ f j C . l A ^ f ) ^ ; / ' c j A ^ C ^ A ^ f j C p C A ^ f ) ^ ; / ' ^ ( A^C^A^fjC^Apnf)^; J a -I ' ^ ( A ^ C ^ A ^ f j C p C A ^ f ) ^ ; f Cn{Xnif)Cm(Xmjf)fdf; J a I C n (A m f )C m (A m yr )—. J a r D 2 { i , j , k , l D 3 { i , j , k , l D 4 ( i , j , k , l D 5 ( i , j , k , l D 6 { i , j , k , l D p ( i , j , n Dl(i,j,n D p 0 { i , j , n = E E E E /i./iy/iJk/uC^AxOC^A.yJdCA.OCiCAw); i j k I = E E E E /i.7ij/i*/i^ i(Ai»a)Ci(A»ya)C'i(Atfca)C1(A,,a)a; » y * j = E E E E /w/iy/iJk/ii^AiOCiCAyjC^A.-OCtCA,-,); = E E E E hihjhkhiC'l{\ua)Cx (Xi^C'^Xi^C^Xua)^ i j k l » y fc t = E E E E /i.7iy/i*/iiC'i(Alia)C1(A1ya)C;/(Alfca)C1(A1,a)a; i y fc J = E E E /i»7iy/pi^ 'i(Ai»)gi(Aiy)^ 'p(Api)! » y i = E E E /i«7iy/pnc'i(Aita)Ci(Aiya)Cp(Apria)a; * y n = E E E /i.7iy/pnCi(Alt)Ci'(A1y)Cp(Apn); i y n = E E E /i.7iy/pnCi(A1,a)Ci'(A1ya)Cp(Apna)a; * y n tnm{i,p) — Cn(Xni)Cm(Xmp) - aCn(Xnia)Cm(Xmpa); i; „\ — n (\ \r> (\ \ C w ( A n t a ) C m ( A m p a ) CL ttnm[i,p) = C ' „ ( A r i t ) C m ( A m p ) + aCn(Xnia)Cm(Xmpa); c \ n t\ \r> (\ \ r C n ( A n t a ) C m ( A m p a ) wnrn[i,p) = Cn[Xni)Cm(Xmp) + Unm{i>J,P) = C , n ( A m ) C n ( A „ y ) C m ( A m p ) -CL Wnm{hj,p) — Cn(Xni)Cn(Xnj)Cm(Xmp) - aCn(Xnia)Cn(Xnja)Cm(Xmpa). a C n ( A m a ) C n ( A n y a ) C m ( A m p a ) 2. Hyperbolic Functions 2.1 Definitions and Cross-Product Integrals 9nm Fijki F G l m ( i , J , k , l G ^ m ( i , j , k , l G l m { i , j , k , t GGl*(i,j,n GGlP(i,j\n GGlP(i,j,n GGlP(i,j,n _ cosh A n m ( z + h] cosh Xnmh = E E E E fufijfikfii i j k i c o s n A\%h cosh Xijh cosh Xi^h cosh Xnh = E E E *' j P flifljflp cosh Xnh cosh Aiyft. cosh Xiph f ° J ^ 9ni9nj9mk9mldz', k9midz; / 9ni9nj& J-k f ° / §ni9nj9mk9mldz\ J-h f ° / 9qi9qj9pndz\ J — h /9qi§q]9pndz; -h f° / 9qi9qj9pndz; J — h / 9qi9qj9pndz. J-K 197 2.2 Simple Cross-Products Xni&ni XmpCtmp Pnm{hP) = ^2 _ ^2 ' A m ^ A m P ' + "t—]j for A m - — A m p 5 2 cosh2 A m / i A a* t: ~\ A n i A m p ( A m a m p — A m p a m ) f ^ , ^  Pnm\l>P) - 72 \2 ' I O r A™ ^ A m p , Ant A m p - 4- ^ 211 for A - A ^ cosh A„i/l Ant 2.3 Combinations of Hyperbolic and Bessel Function Cross-Products knm(i,p) = Pnm(i,P)tnm(i,P)', lnm{i,P) = Pnm[i,P)vnm{i,p); kknm(i,p) = Pnm{i,p)ttnm(i,p); UnmihP) = 0nm(i,p)wnm(i,p); 12 y) = ^nm(')j) cosh Xn{h cosh A m j/i 1 «72 n m (t ,j) JAnrn(i,j) cosh Ant'/i cosh Amy/i 198 APPENDIX VI: M E C H A N I C A L A N A L O G Y OF A LINEAR S Y S T E M 1. One-D egree-of-Freedom It is assumed that the action of the sloshing liquid within the nutation damper is modelled by a mass-spring-dashpot system as shown in Fig. VI-1. d a m p e r x - € 0sincj Pt d a m p e r I k H A A H i m Fig. VI-1 Mechanical representation of a nutation damper This common approach, incomplete as nonlinear effects are not included, can be a useful tool for understanding the more complex fluid mechanics problem. The motion of the liquid, represented by y here, imposes inertia and damping forces on the moving base (Fig. VI-2). They can be nondimensionalized in terms of an added mass and reduced damping ratio \Ma/Mi\ and r/rjj, respectively. k(x-y) •C d(x-y) 1 Fig. VI-2 Forces acting on the moving base Standard vibration theory gives (x — y), and in turn yields: . Ma CJ2 — 1 T]CJ ^,1 = 773 -,\2 , fo,.M2' ( V L 1 ) Mi ( w 2 - l ) 2 + (2r 7 u; 2 ) 2 ' / r > ' (o>2 - l ) 2 + (2r?u;)2' where: — ; wn = y/k/m, and rj = ~ — (VI-1) is plotted below against CJ for various n. The added mass is always zero at resonance (CJ = 1.0), with diminishing maxima at larger damping (Fig. VT-3a), 199 while rjr,i shows higher and narrower peaks for decreasing n (Fig. VI-3b). The energy ratio is also derived as 8TT Er>l = r'KTTl (1 + l/w 2)w' showing an increase with CJ for a given 77 (Fig. VI-3c). (VI.2) 2. Two-Degree-of-Freedom System The aerodynamic model of section 4.4 fitted with the rotating damping device can be represented by the vibration absorber configuration shown below: F 0 e i w e ' ]1 m. aero, model w_t , - Y , e ' w e / T V 2 =Y 2 e i V Fig. VI-4 damping device Mechanical representation of the transmission line test arrangement {VI.3) The standard formulation for such problems leads to the following eigenvalue equa-tion 1 1 0 , - U?{UJI + 1 + s) + w* = 0, , m2 <*>1 2 *1 2 k2 , W N where s = ; Uo = — ; with UJX = , u 2 = ; and u>n = —. mi u)2 mi m.2 u2 With the experimental determination of uj\, u>2, and wn, the inertia ratio 5 can subsequently be determined from (VI.3) as, , = - A - l ) ( £ f - 1 ) . Assuming C\ to be small, the response of the model is, _Fg , r ( i-P a) + 3 a y / a (VIA) (VI.5) Ijj, where u> = — ; A is the left hand side of relation (VI.3) when substituting ojn by U2 Cd2 U; r\2 = and s 0 = wo ~~ **%(! + s). At resonance, (VI.5) reduces to m2W2 Y _ ^ 0 [ ^ + ( l - ^ ) 2 ] 1 / 2 L J 2 *i — 1 T=—i w o> ki \ns0\ (VI.6) Fig. VI-3 One-degree-of-freedom system characteristics at resonance showing: (a) | M a / M , | ; (b) i , P | , ; (c) Er>l to o o 201 as A = 0 and w = Un. With the parameters of (VI.3), Y\ is found to be a function of 57 2 5 w o a n d s, for a model with given m i , k\, and exciting force Fo. Thus the damper design can be optimized. Note: For the experiment, m i = l.GQQKg. m 2 = Md + m P ^ ' where: Md = damper mass; m p , lp = damper plate mass and length, respectively; L = distance from damper center of gravity to system center of rotation. According to the inertia forces in the vertical direction (Fig. VI-6), F = m 2 y 2 = MdLO + m p ^ 0 . (VI. 7) F m p lp0/2 Fig. VI-5 Force diagram for the damping device 202 APPENDIX VII: WIND-INDUCED OSCILLATION A M P L I T U D E C A L C U L A T I O N 1. Galloping Theory with Equivalent Damping According to Parkinson's theory111, the amplitude Y is given by where: ^ = n ^ 1 ( 1 _ £ ) y 2 + |y< + ... + ^ , ( , m ) N = 1,3,5,..; n = ^ ^ - ; U0 ^ 2M ' nAi A i = 1st coefficient of the polynomial fit for Cfy; I?3, . . . , B N = integration constants times higher coefficients of C / y . A limit cycle is reached for dY2 t l e d[dY2ldT) —r- = 0, stable for 1 '—s- < 0, i.e., dr dY* (1 - |-) + 2BZ(^)2 + ... + ^ f ^ ) " " 1 < 0. (F//.2) Based on the dissipated energy per cycle, the equivalent damping ratio is a function of Y (average amplitude for the cycle). A polynomial fit to follow its variation is used , rj = D0 + DXY + D2Y2 + ... + DMYM. {VII.Z) Relation (VII.l) is still valid as it also represents an average quantity for the cycle. UQ is however no longer independent of Y and the stability equation becomes, [Left hand side of (VII.2)] - —^[DiY1 + 2D2Y2 + ... + MDMYM\ < 0. (VIIA) 2. Vortex Resonance of a Full Scale Chimney Fitted with Nutation Dampers Recently, mathematical models for circular cross-section structures have been developed to predict full- scale response 1 1 2 - 1 1 3 . The case of a uniform 5m diameter steel chimney with a height of 80 m, mass density of 1500 Kg/m, and structural damping of 0.3%, was considered by Vickery et a l . 1 1 4 They determined that a damp-ing ratio n8 = 2.2% is required to keep Y < 0.1, with the response approximately proportional to nj1/2 in the range of interest here. With a natural frequency of 0.3 Hz, such structure can be fitted with a nutation damper at the tip so that, Mi ^lWe = where the modal mass M e = i m ^ ^ r - ^ d z (F/J-6) (VI1.5) 203 is found to be 30 000 Kg. Here: m(z) = chimney mass per unit length; z = vertical axis with origin at the ground level; H = height of the chimney; Yp(z) = horizontal deflection at height z; Y = tip deflection. Considering a nutation damper similar to model#7 used in this study with h/d = 1/2 and D/d = 4.10, the conditions of sloshing resonance requires: ue « w n . (VII.7) Now, we = [ ^ t a n h A u f t ] 1 7 2 , (V//.8) thus, R0= A l i y t a n h^f. (VII.9) (2TT/II) 2 5.1 JRo = 0.836 m as A n = 1.255 for this damper, which yields d = 0.327 m with a container liquid mass mj given by, m / = p n r ( j ) ( ^ ) d 8 . (7/7.10) for oil, mi = 180.2Kg. From Chapter 3, rjrii > 1.0 for to/d < 1.0 and CJ « 1.0. At Y = 0.1 corresponding to rjg = 0.022, eo/d = 1.53 and therefore it is assumed that rjr>i < 1.0 (the variation with amplitude is not too pronounced here and the steady-state results should apply reasonably well). Taking nrti « 0.7 leads to Mi = 943 Kg, thus requiring the use of 5 to 6 damper units (Fig. VII-1). If a lower response is needed, e.g. peak Y = 0.06, a damping of na = 0.04 is expected (from Y propor-tional to relationship), eo/d is then 0.92 and nrti is of order 1, which yields Mi = 1200 Kg, or about 7 units. This compares advantageously with the 1500 Kg pendulum tuned mass damper proposed in reference 114. For higher frequencies such as /=0.8 Hz considered in the same article, more damper units are needed as a lower D/d ratio is required to meet the condition CJ = 1.0. This further reduces efficiency and it is found that for D/d = 1.89 and h/d = 0.5, R0 = 0.302 m, d = 0.209 m, mi = 21.7 Kg (oil), and Mi = 2200 Kg. Thus installation of 102 damper units is required. A ring could easily be designed to fit all the containers, as illustrated in Fig. VII-2. damper arrangement Fig. VII-1 Steel chimney with 6 nutation dampers damper ring •frnrrtT Fig. VII-2 Steel chimney with nutation damper ring L I S T O F P U B L I S H E D A R T I C L E S 1 • Modi, V . J . , and Welt, F., "Nutation Dampers and Suppression of Wind Induced Instabilities", Proceedings of the ASME Joint Multidivisional Symposium on Flow-Induced Vibration, 1984 ASME Winter Annual Meeting, New Orleans, Louisiana, Dec. 1984, editors: M.P. Paidoussis, O . M . Griffin, and M . Sevik, Vol. 1, pp. 173-187. • Modi, V . J., and Welt, F., "On the Control of Instabilities in Fluid-Structure In-teraction Problems", Proceedings of the 2nd International Symposium on Struc-tural Control, Waterloo, Canada, July 1985, Editor: H . H . E . Leipholz, pp. 473-495. • Modi, V . J . , Welt, F., and Irani, M.B . , "On the Nutation Damping of Fluid-Structure Interaction Instabilities and its Application to Marine Risers", Fifth International Symposium and Exhibition on Offshore Mechanics and Artie En-gineering, Tokyo, Japan, April 13-18, 1986, Paper No. OMAE-1197; also Pro-ceedings of the Conference, Editors: J.S. Chung, et al., Vol. 3, A S M E Publisher, New York, pp. 408-416. • Modi, V . J . , and Welt, F., "On the Control of Instabilities in Fluid-Structure Interaction Problems", Proceedings of the 4th IFAC Symposium on Control of Distributed Parameter Systems, Pasadena, California, U.S.A. , June 30-July 3, 1986, editor: G . Rodriguez, Pergamon Press, London, in press. • Modi, V . J . , and Welt, F. , "Visualization of Sloshing Motion in Nutation Dam-pers", Proceedings of the 4th International Symposium on Flow Visualization, Paris, France, Aug. 1986, Editor: C. Veret, Hemisphere Publishing Corporation, pp. 353-358. • Welt, F. , and Modi, V . J . , "On the Control of Instabilities in Fluid-Structure Interaction Problems Using Nutation Dampers", Proceedings of the 3rd Asian Congress of Fluid Mechanics, Tokyo, Japan, Sept. 1-5,1986, Editor in Chief: T . Matsui, pp. 616-619. • Irani, M . B . , Modi, V . J . , and Welt, F. , "Riser Dynamics with Internal Flow and Nutation Damping", Proceedings of the 6th International Symposium on Off-shore Mechanics and Artie Engineering, Houston, Texas, U.S.A. , March 1987, Editors: J.S. Chung, et al., Vol. 1, pp. 119-125. • Modi, V . J . , and Welt, F. , " O n the Vibration Control Using Nutation Dampers", Proceedings of the International Conference on Flow Induced Vibrations, Bow-2 ness on Windermere, England, May 1987, the British Hydromechanics Research Association, pp. 369-376; also entitled " A n Investigation of Nutation damping and its Application to Wind Engineering", Proceedings of the 11th Congress of Apllied Mechanics, June 1987, Vol. 2, pp. E-81-E-82. • Modi, V . J . , and Welt, F., "Damping of Wind Induced Oscillations Through Liquid Sloshing", Proceedings of the 7th International Conference on Wind En-gineering, Aachen, Federal Republic of Germany, July 1987, Editors: C. Kramer and H . Gerhardt, pp. 143-152; also Journal of Wind Engineering and Industrial Aerodynamics, in press. • Irani, M . B . , Modi, V . J . , and Welt, F. , "Dynamics of Offshore Risers with In-ternal Flow in the Presence of Ocean Waves and Currents", Proceedings of the IMACS/IFAC International Symposium on Modelling and Simulation of Dis-tributed Parameter Systems, Hiroshima, Japan, Oct. 1987, pp. 329-336. 

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