A S T U D Y OF N U T A T I O N D A M P E R S W I T H A P P L I C A T I O N T O WIND I N D U C E D OSCILLATIONS by 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 de M o n t r e a l , 1979 M . A . S c , T h e U n i v e r s i t y of B r i t i s h C o l u m b i a , 1983 A THESIS IN P A R T I A L F U L F I L M E N T T H E R E Q U I R E M E N T S D O C T O R O F O F F O R T H E D E G R E E OF P H I L O S O P H Y in 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 of 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 as c o n f o r m i n g to the required standard T H E U N I V E R S I T YO F BRITISH January @ 1988 Francois Welt, 1988 C O L U M B I A In presenting degree at this the thesis in partial University of British Columbia, freely available for reference copying of department publication this or of thesis by for his this thesis fulfilment of and study. her Jfrv] may be representatives. NlECviAuicftL £kJCiMEE-RitJ Q .28 an advanced Library shall make it It is granted by the understood that head extensive of my copying or for financial gain shall not be allowed without my written The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date for I further agree that permission for permission. Department of requirements I agree that the scholarly purposes or the m% ii ABSTRACT 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 steadystate 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 threedimensional 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 TABLE OF CONTENTS Chapter Page 1 INTRODUCTION 1.1 Preliminary Considerations 1.2 Literature Survey 1.3 Scope of the Investigation 2 A N APPROXIMATE ANALYTICAL APPROACH T O PREDICT ENERGY DISSIPATION 2.1 Preliminary Remarks 2.2 Potential Flow Solution 2.2.1 Basic Equations 2.2.2 Linear Solution 2.2.3 Nonlinear, Nonresonant Solution 2.2.4 Nonlinear, Resonant Solution 2.2.4.1 No Interactions 2.2.4.2 Resonant Interactions 2.2.5 Properties of the Potential Function 2.2.5.1 Variation with Damper Geometry 2.2.5.2 Variation with the Excitation 2.3 Pressure Forces 2.4 Damping Forces 2.4.1 Effect of Viscosity 2.4.2 Energy Dissipation and Reduced Damping Ratio 2.4.3 Energy Ratio E ,i 12 12 12 12 14 15 16 17 19 21 21 24 29 31 31 33 37 3 EXPERIMENTAL DETERMINATION OF DAMPER CHARACTERISTICS 3.1 Preliminary Remarks 3.2 Test Arrangement and Models 3.3 Flow Visualization 3.4 Added Mass and Reduced Damping Ratio 3.4.1 General Procedure 3.4.2 Results and Discussion 3.4.3 Comparison with Free Oscillation Tests 3.5 Concluding Comments 41 41 41 44 50 50 53 77 81 4 WIND INDUCED INSTABILITY STUDY 4.1 General Description 4.2 Two-Dimensional Tests in Laminar Flow 4.2.1 Preliminary Remarks 4.2.2 Test Arrangement and Model Description 4.2.3 Calibration Procedure 4.2.4 Model Characteristics 4.2.5 Results and Discussion 4.2.5.1 Vortex Resonance of a Circular Cylinder 4.2.5.2 Vortex Resonance and Galloping Response of a Square Cylinder 4.2.6 Concluding Comments r 1 1 5 9 84 84 84 84 85 88 88 98 98 103 Ill iv Chapter 5 Page 4.3 Three-Dimensional Tests 4.3.1 Preliminary Remarks 4.3.2 Test Arrangement and Model Description 4.3.3 Model Characteristics 4.3.4 Results and Discussion ... 4.3.4.1 Vortex Resonance Response of a Circular Cylinder 4.3.4.2 Vortex Resonance and Galloping Response of a Square Cylinder 4.3.5 Concluding Comments 4.4 Application to Transmission Lines 4.4.1 Preliminary Remarks 4.4.2 Test Arrangement and Model Description 4.4.3 Model Characteristics 4.4.4 Results and Discussion 4.4.5 Result Summary 115 115 115 117 122 CONCLUSIONS 149 122 127 134 136 136 136 139 141 145 BIBLIOGRAPHY 153 APPENDICES 162 I NONLINEAR F R E E SURFACE CONDITION 1. Basic Equation 2. Perturbation Series Expansion 3. Free Surface Equation 162 162 162 163 NONLINEAR, NONRESONANT POTENTIAL FLOW SOLUTION 1. Second Order Terms 2. Stability and 3rd Order Equation 164 164 165 III NONLINEAR, RESONANT POTENTIAL FLOW SOLUTION 1. No Interactions 2. Resonant Interactions 2.1 Second Order Terms and Detuning Parameters 2.2 Third Order Equation 2.3 Solution Stability 167 167 168 168 171 175 IV ADDED MASS AND DAMPING RATIOS: DETAILS OF T H E ANALYSIS 1. Added Mass 1.1 General Procedure 1.2 Higher Order Terms 2. Damping Ratio 2.1 Correction Velocity U 2 2.1.1 First Order 2.1.2 Second Order 2.2 Reduced Damping Ratio II 177 177 177 178 179 179 1 7 9 181 183 V Chapter Page 2.3 V VI VII 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 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 Energy Ratio 183 190 190 U S E F U L BESSEL 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 DEFINITIONS 1. Bessel 1.1 1.2 1.3 2. Hyperbolic Functions 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 2.2 S i m p l e C r o s s - P r o d u c t s 2.3 C o m b i n a t i o n s of H y p e r b o l i c a n d Bessel Function Cross-Products 193 Functions Orthonogality Condition Cross-Product Integrals Simple Cross-Products M E C H A N I C A L A N A L O G Y 1. One-Degree-of-Freedom 2. Two-Degree-of-Freedom 193 193 193 195 196 196 197 197 O F A L I N E A R S Y S T E M 198 198 System W I N D - I N D U C E D OSCILLATION A M P L I T U D E C A L C U L A T I O N 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. V o r t e x R e s o n a n c e of a F u l l S c a l e C h i m n e y F i t t e d Nutation Dampers 199 202 202 with 202 vi L I S T O F T A B L E S Tables I II III Page Details of the d a m p e r models used i n the test p r o g r a m 44 P h y s i c a l description of the two-dimensional aerodynamic models 87 P h y s i c a l description of the three-dimensional a e r o d y n a m i c m o d e l s . . . 117 vii LIST O F FIGURES Figure Page 1 Wind-induced instabilities of bluff bodies undergoing: (a) vortex resonance; (b) galloping 2 Several typical devices providing: (a) external damping; (b) internal damping 2 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 6 by the damper geometry Nonplanar coefficients —(KK\ — 2K\) and K\jKK\ as functions of h for: (a) a = 0.308; (b) a = 0.608 25 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} i versus u> rt 36 10 Typical curves showing variation of E ,i with frequency and amplitude 11 E*i versus CJ as affected by: (a) h\ (b) a 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 r 39 40 viii Figure Page 18 Close-up view of mode (1,4) 49 19 Calibration curves to determine: (a) M o ; (b) slope for the response using both static and dynamic procedures; (c) slope for the excitation; (d) phase angle between response and excitation 52 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 \M /Mi\ 59 24 Resonant behavior of damper#5 with: (a) h/d = 0.5 and u < 1; 20 a and rj i for half-full damper#l r> (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 27 heights at e /d = 0.046 Variation of the peak response for damper#l with liquid height at e /d = 0.105 0 0 28 Variation of the peak damping ratio for damper#7 with liquid height at € /d = 0.091 0 29 64 66 67 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 32 with: (a) D/d = 1.89; (b) D/d = 4.10 Effect of internal configuration on r\ \ and \M /Mi\ (a) c o M (b) CJ r a 71 versus: 75 33 Proposed sloping cross-section 77 34 Apparatus used for free oscillation tests 78 ix Figure 35 Page D a m p i n g c h a r a c t e r i s t i c s v e r s u s e o / d as o b t a i n e d b y s t e a d y - s t a t e a n d f r e e - o s c i l l a t i o n e x p e r i m e n t s f o r : (a) d a m p e r w i t h b a f f l e s ; (b) p l a i n d a m p e r 79 A m p l i t u d e decay for the half-full d a m p e r # l oscillating at 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 o f eo/d = 1.28 ( c h a r t recorder T y p e T R 3 2 2 , G u l t o n Industries) 81 37 W i n d t u n n e l set-up for 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 u s e d d u r i n g t h e t e s t s for: (a) c h a r t r e c o r d e r ; (b) s p e c t r u m a n a l y s e r 89 36 39 E f f e c t o f e n d p l a t e d i m e n s i o n o n Cf y for: (a) m o d e l # 2 ; (b) m o d e l # l ..90 40 M a x i m u m displacements of m o d e l # 3 undergoing vortex resonance 92 41 S y s t e m d a m p i n g for different electromagnetic d a m p e r settings 92 42 Cfy 94 43 G a l l o p i n g r e s p o n s e f o r (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 Response of two-dimensional square cylinder w i t h o u t d a m p e r f o r : (a) m o d e l # l ; (b) m o d e l # 2 96 Vortex shedding excitation on two-dimensional models showing: (a) f r e q u e n c y s p e c t r u m o f t h e r e s p o n s e ; (b) S t r o u h a l n u m b e r f o r t h e 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 f o r square cross-sections 97 45 46 47 48 49 v e r s u s a f o r : (a) m o d e l # 2 ; (b) m o d e l # l V o r t e x r e s o n a n c e r e s p o n s e 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 h e i g h 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 V a r i a t i o n o f r/ / w i t h a m p l i t u d e d u r i n g free o s c i l l a t i o n s f o r d a m p e r # l w i t h h/d = 1 / 8 a n d 1 / 4 101 C o m p a r i s o n between experiments and predictions based on the energy balance m e t h o d 103 P r e d i c t i o n s o f t h e 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) d e t e 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) r e s p o n s e 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) r e s p o n s e w i t h Cjo = 0.5 104 r> X Figure 50 Page Galloping response of model#l showing: (a) effect of frequency; (b) effect of internal configuration 106 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 51 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 66 67 at various frequencies Galloping response in 3-D for model#2 with nutation dampers in: (a) laminar flow; (b) turbulent flow Galloping response in 3-D for model#l with nutation dampers in: (a) laminar flow; (b) turbulent flow 129 130 131 xi Figure 68 Page Effect of l o w liquid heights o n the 3 - D galloping response of model#2 132 69 E f f e c t o f D/d 70 S k e t c h of the horizontally m o u n t e d w i n d tunnel set-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 s e t - u p s h o w i n g : (a) F r o n t v i e w o f t h e o s c i l l a t i n g s y s t e m ; (b) c l o s e - u p v i e w o f t h e d a m p i n g device 138 72 a n d CJ o n t h e 3 - D g a l l o p i n g r e s p o n s e o f m o d e l # 2 133 E v a l u a t i o n of the secondary system d a m p i n g ratio showing: (a) c a l i b r a t i o n p r o c e d u r e ; (b) n 2 a n d r/ / v e r s u s a m p l i t u d e 140 Response of the system w i t h o u t d a m p i n g liquid showing: (a) effect o f a u x i l i a r y d e v i c e ; (b) b e a t i n g m o t i o n 142 74 Effect of liquid height o n the system response 143 75 F r e q u e n c y s p e c t r u m o f t h e r e s p o n s e f o r : (a) / „ « a 73 (b) 76 r) f t; ro / „ » / r o t 144 R e s p o n s e of t h e m o d e l as affected b y t h e parameters: (a) a>2/u>i; (b) mi/m\ 146 Sketch of the d a m p e r arrangement useful to control roll 147 VI-1 M e c h a n i c a l representation of a nutation damper 198 VJ.-2 Forces acting o n the m o v i n g base 198 VI-3 One-degree-of-freedom system characteristics at resonance 77 s h o w i n g : (a) \M /M \; (b) t | , i ; (c) E ,i 200 VI-4 M e c h a n i c a l representation of the transmission line arrangement 199 VJ.-5 Force d i a g r a m for the damping device 201 VII-1 Steel chimney w i t h 6 nutation dampers 204 VII-2 Steel chimney w i t h nutation damper ring 204 a t r r xii L I S T a O F S Y M B O L S 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 lift oscillator m o d e l oi Bessel f u n c t i o n relation defined i n A p p e n d i x III, p. 170 aj dimensionless expression for a\ defined i n A p p e n d i x III, p. 170 57 a m p l i t u d e coefficient of the correction velocity u 2 defined i n A p p e n d i x I V , p. 180 A A d a m p e r w a l l area higher order terms of t h e added mass ratio 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 e /d i n C h a p t e r 3 , a n d Cf versus a i n C h a p t e r 4 n 0 y At w i n d - t u n n e l test section area AQ aerodynamic model frontal area AA AAA higher order terms of t h e reduced d a m p i n g ratio T7 r > j, n = 1 , 2 , 3 n higher order terms of t h e energy r a t i o E j, n AK r n = 1,2,3 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 n b n u t a t i o n damper baffle w i d t h bh free parameter of the H a r t l e n - C u r r i e lift oscillator m o d e l &i Bessel f u n c t i o n relation 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 &ii2 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 nonplanar s o l u t i o n , eq. I H . 4 , p. 168 B n BB3 higher order terms of the added mass ratio M /Mi, n = a higher order t e r m of the reduced d a m p i n g r a t i o r) i Tt 1 , 4 BBB3 C Cd> Cdi 5 Cd2 higher order term of the energy ratio E i r> stability coefficient for the nonplanar solution, eq. 111.4, p. 168 absolute damping coefficient of the one-degree-of-freedom, primary and auxiliary system, respectively e equivalent absolute damping coefficient y aeorodynamic static side force coefficient, F /[p d L (V C Cf Ci r y a m cos a) /2] 2 m aerodynamic lift coefficient normalized as C\ = Ci(U/U ) , where Ci is equivalent to C/ for the moving circular cylinder 2 r r y C/o aerodynamic static lift coefficient C aerodynamic moment coefficient for 3-D square prisms m damper geometry dependent variables defined in Appendix III, pp. 172-173 CK n C 4 C (A f) n m d di dm higher order term of the added mass ratio M / M / defined in Appendix IV, p. 179 0 combined 1st and 2nd kind of Bessel function, damper cross-section width or diameter damper inner tube outer diameter structure or aerodynamic model cross-section width or diameter d 2nd order mode shape coefficients for the potential function $ defined in Appendix III, pp. 169-170 d\ 3 characteristic dimensions of a damper's sloping cross-section as defined in Fig. 33 mn D D r D\ 10 damper outer diameter aerodynamic drag force 2nd order amplitude coefficients for rj i defined in Appendix V, p. 195; also used as 3rd order coefficients for $ in Appendix II (n = 1,2), pp. 165-166 r) xiv DDi 3rd order coefficient for $ defined i n A p p e n d i x II, p. 165 DK damper geometry dependent variables defined i n A p p e n d i x III, p p . 172, 174 n A 2nd order mode shape coefficients for the potential function $ defined in A p p e n d i x III, p. 169-170 e mn en, e 2 2 Ed amplitude coefficients of the 2nd order component of u in A p p e n d i x I V , p p . 181-182 dissipated to total energy ratio, E*j energy ratio accounting for Reynolds number, Et E\,E2 / f n fnii fn2 f as defined total dissipated energy by the moving fluid per cycle Ej r 2 Ed/Et E iVRe r> total energy of the fluid m o v i n g with respect to the damper, eq. 37, p. 37 nonlinear coefficients of the 3rd order equation for fu, f i defined in A p p e n d i x III, p. 174 2 exciting frequency fundamental natural frequency of the system 1st a n d 2 n d natural frequency of system i n translation, respectively 1st or 2 n d order planar mode shape coefficients for the potential mn function $ ft ro 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 i n A p p e n d i x I V , p. 190 F sloshing force transmitted f r o m the fluid to the damper walls g F y aerodynamic static side force w i t h respect to the model, S cos a — p D sin a T XV Fijki ijp 2 n d order a m p l i t u d e coefficients for 196 n i r defined i n A p p e n d i x V , p. Fo i n e r t i a force generated b y the s y s t e m w i t h o u t d a m p i n g liquid; also used as t h e a e r o d y n a m i c force 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\ linear coefficient III.2, p . 167 g 9nm G " of the 3 r d order n o n l i n e a r e q u a t i o n f o r fu, eq. acceleration due to gravity hyperbolic function relation defined i n A p p e n d i x V , p. 196 hyperbolic function cross-product integrals defined i n A p p e n d i x V , p. 196 m Gl,..,5 d a m p e r g e o m e t r y d e p e n d e n t variables defined i n A p p e n d i x III, p p . 167, 175 G G " hyperbolic function cross-product integrals defined i n A p p e n d i x V , p . 196 m h damper liquid height h dimensionless damper liquid height, h/Ro b! n u t a t i o n d a m p e r baffle m i d - h e i g h t p o s i t i o n H full-scale structure H3,4 damper geometry 175 height dependent variables defined i n A p p e n d i x III, p . I total inertia for the rotating system 11 liquid inertia for system i n rotation ii 9 IA IDD* m Bessel 193 function cross-product integrals defined i n A p p e n d i x V , p. Bessel 195 function cross-product integrals defined i n A p p e n d i x V , p. Bessel function cross-product integrals defined i n A p p e n d i x V , p p . 193-195 12 combination of hyperbolic a n d Bessel fined i n A p p e n d i x V , p . 197 function cross-products de- XVI Bessel 195 function cross-product integrals defined i n A p p e n d i x V , p. Bessel function cross-product integrals defined i n A p p e n d i x V , p. 194 c o m b i n a t i o n s of h y p e r b o l i c a n d Bessel fined i n A p p e n d i x V , p . 1 9 7 one-degree-of-freedom function cross-products de- s y s t e m stiffness s p r i n g stiffness of the m a i n s y s t e m s p r i n g torsional stiffness of the secondary system combinations of hyperbolic a n d Bessel function cross-products fined i n A p p e n d i x V , p . 1 9 7 nonlinear coefficients A p p e n d i x III, p . 167 of the 3 r d o r d e r e q u a t i o n f o r fu Bessel cross-product integrals defined i n A p p e n d i x V , p. square root of R e y n o l d s n u m b e r , i.e., de- defined in 194 y/~R~e length of the 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 . VI-5 combinations of hyperbolic a n d Bessel function cross-products fined i n A p p e n d i x V , p . 1 9 7 de- distance of d a m p e r center of gravity to p i v o t i n g point for the ondary system, Fig. VI-5 sec- a r m length of the p i v o t i n g system, F i g . 63 aerodynamic model length Bessel 194 function cross-product 2 n d order components for rj i r> integrals defined i n A p p e n d i x V , defined i n A p p e n d i x I V , pp. m a s s per u n i t lenght of the full-scale structure d a m p i n g liquid mass of a single, full-scale d a m p e r unit mass of the d a m p e r s u p p o r t i n g p l a t f o r m p. 184-185 XVII mi mass of the primary system i n translation m<i equivalent mass of the secondary M M total mass of the oscillating system system added mass due to sloshing liquid a restoring m o m e n t of the torsional spring Mi 1st m o d a l m a s s of t h e s t r u c t u r e , e q . V I I . 6 , p . 202 total mass of the sloshing liquid M lm> Mi M Mj, , M Npqi M Ep nm q nrn nm n N 2 N N NnmEp 2nd order components for rj i v e c t o r n o r m a l t o t h e free surface 2nd order components for rj i d e f i n e d r> defined i n A p p e n d i x I V , p p . 185-188 rt i n A p p e n d i x I V , p p . 188-189 pressure exerted b y the sloshing liquid A Pnm P* • M l Qi Q 2 2 3 r d o r d e r coefficients o f t h e p o t e n t i a l f u n c t i o n $ , e q . LT.4, p . 1 6 6 coefficients of the 3 r d order e q u a t i o n for $ (» = 1 , 2 ) , p p . 1 7 1 - 1 7 2 d a m p e r inner t u b e hole size defined i n A p p e n d i x III diameter coefficients of t h e 3 r d order e q u a t i o n for $ defined i n A p p e n d i x III, p.173 damper based m o v i n g coordinate i n the radial direction dimensionless moving coordinate, A Tin 2 n d order coefficient for $ r/Ro as defined i n A p p e n d i x I V , p . 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; also u s e d as 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 Re sloshing liquid Reynolds number, Ri damper inner radius u R$/vf e 192 X V 111 Ro s Si, damper outer secondary t o primary mass ratio, so ul -&^ (l 5 fii ~ 2 s radius + s) 2 ?ii a mi/mi n d fii + ?2i> respectively area of the sloshing liquid's outer boundary 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 St Strouhal number damper geometry 167, 174 SUM1,...,5 t dependent variables defined i n A p p e n d i x III, p p . time Bessel function cross-products To u' u defined i n A p p e n d i x V , p . 196 m e a s u r e o f t h e p h a s e a n g l e tp 0 air stream turbulent as defined i n F i g . 19 intensity sloshing liquid velocity vector free 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 layer profile Uf Bessel function cross-products defined i n A p p e n d i x V , p . 196 correction velocity d u eto liquid viscosity t?2 air stream dimensionless velocity, u U r V/u d ; V Vnm, VV nm m also used as t h e average potential energy o f t h e sloshing l i q u i d in eq. 37, p . 37 vortex resonance dimensionless velocity, dimensionless galloping onset velocity, Ul n flow l/(2irSt) r) /(7rAi) rta m a g n i t u d e ofU 2 volume of the damping liquid Bessel 196 function cross-products integrals defined i n A p p e n d i x V , p . xix V air stream velocity —+ V velocity of the damper walls in translation —• V n VS w component of V in the n direction damper geometry dependent variables defined in Appendix IV, p. 192 Bessel function cross-products defined in Appendix V, p. 196 nm WS x X y y n damper geometry dependent variables defined in Appendix IV, p. 192 position of the damper in the direction of the excitation output voltage variation in x direction output voltage due to the system response 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 d Y dimensionless deflection of the full-scale structure along its height p m Yi, Y2 amplitude response of the primary and secondary system, respectively 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 (3 hyperbolic function of the damper liquid height, tanh A n m h excitation and damper geometry dependent variable defined in Appendix III, p. 171 /?i 1st order detuning parameter for /?2 2nd order detuning parameter for A Pnmt fin •7„ m hyperbolic functions defined in Appendix V , p. 197 2nd order mode shape coefficient of potential function $ defined in Appendix III, pp. 169-170 A A n 2nd order component of the potential function $ m V V f n gradient operator 2nd order component of the potential function $ m e amplitude of the excitation velocity, eou e dimensionless amplitude of the excitation velocity, e/R 0 Co amplitude of the displacement excitation Co dimensionless amplitude of the displacement, £O/RQ n m 77 1st or 2nd order mode shape coefficients of the potential function $ damping ratio of the oscillatory system, C / C , where C system critical damping coefficient e r}2 damping ratio of the secondary system rj / free surface elevation of the sloshing liquid f\f dimensionless free surface elevation, rjf/Ro rj rta aerodynamic reduced damping ratio, 4irrj c C is the M Pad^Lm reduced damping ratio for the nutation damper, 2u Mi e maximum value of rjrj over a range of frequency for a given amplitude of excitation V B, VS2 e 0 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' (X a) n nm = 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_. e ® , nq n q= 1/3,1 n potential function relative to the moving coordinates r, 0, z dimensionless potential function, 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 2nd order component of the potential flow solution $ inherent phase angle between excitation and sloshing response dimensionless exciting frequency, 0J /u)u e 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/ and 27r/ , respectively nl n2 ratio of a^/u^ fundamental angular frequency of the primary and secondary system, respectively sloshing liquid natural frequencies, \l ^ tanh\\\h power coefficient of the exponentially decaying boundary layer correction velocity u 2 A 2nd order coefficient for the potential function <& defined in Appendices II, p. 164, and III, pp. 167, 169-170 xxiii A C K O W L E D G E M E N T 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 as w e l l as f o r p r o v i d i n g m u c h of the i n s p i r a t i o n d u r i n g the course of this thesis. T h e models used d u r i n g the experiments were constructed i n the machine shop a n d special thanks are d u e t o t h e technicians for 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 of 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 was b o r r o w e d . H i s generosity i n p e r m i t t i n g its use is 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 of 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 the realization of this project t h r o u g h the discussion a n d sharing of technical i n f o r m a t i o n . T h e author wants to express his appreciation for the research assistantship awarded f r o m the N a t u r a l Sciences a n d Engineering Research C o u n c i l of 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 is also v e r y g r a t e f u l t o L o u i s e for her moral support. l I N T R O D U C T I O N 1. 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 structure, 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. M o r e recent occurrences involving tall smokestacks have been reported by Hirsch and Ruscheweyh , and C h a u l i a . Conditions creating the presence of an asymme1 2 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 Fig. 1 v i e w W i n d induced instabilities of bluff bodies undergoing: (a) vortex resonance; (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 damping. 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 counteracting 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 . More sophisticated arrangements of counteracting masses have resulted in 3 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 l a t s S h r o u d s S t r a k e s D a m p e r rri / M, / / / / 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 H y d r a u l i c D a s h p o t (b) 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 significant displacement of the sloshing fluid. This thesis proposes axisymmetric torus shaped containers, also called nutation dampers, as a means to suppress windinduced 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 encountered 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 Walshe made a significant contribution to the suppression of 4 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 Price intro5 duced the perforated shrouds. A number of other aerodynamic devices, such as the slat configuration , were subsequently proposed. A comprehensive classification of 6 the devices and comparative assessments were later undertaken by Zdravkovich , 7 as well as Every, King and Weaver who discussed the instabilities of immersed 8 marine cables. Wong and Cox drew a less extensive comparison scheme based on 9 systematic wind tunnel tests. However, only a few studies such as the exploratory 6 work by Naudascher et a l . 10 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, Reed investigated the applicability of 11 impact dampers to lightmasts and antennas. The installation of hydraulic dashpots on guyed structures was well illustrated by Den Hartog , and more recently the ad12 dition of viscoelastic material into the walls was modelled analytically by Gasparini et a l . , with Ogendo et a l . 13 14 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 Cooper as well as Hunt . Performance 15 16 of the device on steel smokestacks was evaluated and compared against that of helical strakes during wind tunnel tests in smoothflowby Ruscheweyh , and results in 17 turbulentflowwere given by Tanaka and M a h . Stockbridge dampers, bearing the 18 name of the inventor , are used extensively for controlling transmission line oscil19 lations. Their application to this class of problems is still under investigation Several analytical schemes have also been developed by Schafer and others 22-24 20-21 . 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 forces . They also have been considered 25 for earthquake applications as indicated by Chowdhury et a l . et a l . 2 8 - 3 0 26 and Yang . Hirsch 27 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. Brunner studied a full-scale tank 31 containing viscous oilflowingthrough stacks of perforated plates on a smokestack, and Berlamont considered the water tank of a tower fitted with baffles. However, 32 it is Modi et a l . 33 who first carried out wind tunnel tests to validate the idea. Very recently, Bauer 34 proposed utilizing the sloshing motion of two immiscible liquids within a rectangular container, while Kwek 35 used a tank of water to provide the auxiliary mass with energy dissipation taking place in the shock absorbers supporting the system. Liquid sloshing has had limited success with ship stability , however, it has 16 been used extensively to control nutation motion of satellites. Although many studies have dealt with its effect on satellite dynamics 36-41 , relatively little is known about the damper behavior. A few experimental investigations have been reported by several authors 42-43 . Alfriend tried to theoretically analyse theflowas a rigid 44 slug moving inside the ring and Tossman predicted damping characteristics for a 45 tube fitted with a solid rolling ball. However, one has to turn to the early research efforts at agencies such as NASA or A B M A (which were concerned with fuel-rocket 8 interactions), to find important information about liquid sloshing theory. These contributions are reviewed by Cooper neers such as Jacobsen 48 and Abramson . Interests of civil engi- 46 47 and H o u s n e r 49-50 , 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 Rodriguez , 51 and C h u 52 for rectangular and elliptical containers under harmonic excitation, re- spectively. Bauer 53 derived a theory for the straight wall torus. More complicated problems started to be examined, such as the compartimented cylindrical tank 54 or the flexible wall interaction based on a variational approach , sometimes re55 quiring numerical procedures simplify analysis Bauer 61 59-60 56-58 . Equivalent mechanical models also emerged to and reached a high degree of sophistication with models by and the pendulum analogy by Sayar . Meanwhile, nonlinear free surface 62 conditions were included with Hutton's theory 63 for circular cylinders in particular, to be followed by Woodward and Bauer's approach 64 for the torus case. These formulations were then applied by Abramson and Chen 65 66 to derive sloshing gen- erated pressure forces on container walls, and were substantiated by experiments. Recently, Miles 67 derived a fairly general theory based on the variational principle proposed by Whitham and others 68 69-70 , and verified Hutton's results for circular cylinders . He also included a solution procedure for the case of resonant interac71 tions encountered in liquid sloshing . This typically nonlinear behavior has been 72 9 found in many ocean wave problems 77-78 , as summarized by Philips , as well as 76 in other areas of research Consideration for the damping terms came from Ocean Engineering in the 1950's, with Hunt 79 and Ursell linearizing the momentum equations account80 ing for viscosity. Case and Parkinson applied the theory to cylindrical containers 81 undergoing small oscillations, while Miles modified it to include surface tension ef82 fects. The method is frequently used nowadays often given to nonlinear terms et al. 87 and Stephens et al. 88 85-86 83-84 , with additional consideration . Experimental results were obtained by Silviera for circular tanks with and without baffles, respec- tively, while Summer and Stophan found damping characteristics for a spherical 89 container based on a dimensional analysis. More recently, torus shaped nutation dampers were investigated during free vibration tests in a preliminary study at 90 the University of British Columbia. 1.3 Scope of the Investigation Optimal efficiency of nutation dampers isfirstsought through a combination of theoretical and experimental procedures aimed at providing a better understanding of the energy dissipation mechanisms during liquid sloshing. Relatively low viscosityfluidsare investigated using a nonlinear potentialflowmodel 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 for the resonance of the higher order terms f o u n d to be present for a certain of dampers. T h e m e t h o d thus predicts the pressure a n d b o u n d a r y layer d a m p i n g forces a n d provides 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 the controlling parameters, nant conditions, kinetic energy, to supply more reso- etc. T h i s is f o l l o w e d b y a n e x t e n s i v e as w e l l as class test p r o g r a m to assess v a l i d i t y of t h e accurate data theory needed for practical applications. p h a s i s is p l a c e d o n t h e c o n d i t i o n s for 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 Em- operating at the n a t u r a l frequency of the first a n t i s y m m e t r i c sloshing m o d e d u r i n g free 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 , as s u g g e s t e d by previous investigations 9 0 . Performance 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 as baffles is reassessed i n t h i s T h e m a i n objective is 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 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, 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 (u ) e energy below. through liquid sloshing. of process. parameters Vf), a n d e x t e r n a l variables leading to m a x i m u m dissipation of S o m e of the variables are indicated i n the sketch 11 Application of the concept to control vortex resonance and galloping types of wind-induced oscillations is subsequently investigated during wind tunnel tests. A l though a successful model to approximate response of circular cross-section geometries is not yet available, considerable experimental data have led to well established empirical procedures 91-92 . Furthermore, the galloping theory has shown to accurately predict oscillations of a square prism with viscous d a m p i n g 93-94 . 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 turbulent winds rising the closed circuit laminar flow and the boundary layer wind tunnels of the Department. The models underwent either two-dimensional plunging 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. 12 2. A N A P P R O X I M A T E A N A L Y T I C A L APPROACH TO PREDICT E N E R G Y DISSIPATION 2.1 P r e l i m i n a r y R e m a r k s T h e velocity field w i t h i n a simple r i g i d torus damper oscillating harmonically 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 potential flow solution w i t h the assumptions t h a t viscous effects are restricted to a small boundary layer region and 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 accounting for the velocity profile at the walls is introduced to assess energy dissipation through the action of the viscous forces. T h e procedure is s i m i l a r to the one adopted by Case and P a r k i n s o n 8 1 . Although 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 potential f u n c t i o n , a conventional E u l e r i a n approach is used here to exploit some of the results found by previous investigators. It should be noted that the study is restricted to straight w a l l dampers to facilitate understanding of the p r o b l e m , a n d the pressure forces are calculated 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 function $ ( r , 0, z,t) represents a solution of the differential equation, V 2 $ = 0, (1) 13 w i t h the b o u n d a r y conditions: dn d $ 2 dt 2 = at the V n wall; (2a) a$ a$ a $ 2 a$ a $ + ff^- + 2 — — — + —2 — - — — + ... = 0; 2 * dz dr drdt 2 r (26) dO d6dt representing c o m b i n e d kinetic a n d k i n e m a t i c conditions at the free surface. Here $ = $ + $/; $ = 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\ $/ = potential function for the d a m p e r solid 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 are i l l u s t r a t e d i n F i g . 3. 0 x 1n e r t i a I R e f e r e n c e € sincu t 0 Fig. 3 e G e o m e t r y of the square section d a m p e r selected for analytical study A t t h i s stage, it is c o n v e n i e n t t o define t h e d i m e n s i o n l e s s parameters: 14 • potential function : $ = • moving coordinates : f= Rfan' z = • excitation amplitude and frequency : Ro co = 4°-, • time : r = u>n t. Ro CJ = 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 Bauer , and is 53 presented here in a dimensionless form: 5 » V ^ r r< i \ coshAit(2 + fc) <P = e 2_j yitW (Ai,r) — — - cos 0 cos UT; . coshAi,7i M (3a) s where: e = dimensionless amplitude of the disturbance, c w; 0 fu = amplitude coefficient of mode (l,i), i.e. 1st circumferential, ith transverse mode, ^}^ \ U ^y.^ ^. *^ 1 [(wii/w) - 1 ', (36) IjAi,- with: a=g, (jju and A - = j[C (A )(A? . - 1) - C2(A «)(A? .« - 1)]; lt 1 2 li 1 a M = liquid natural angular frequency in mode (l,i), i.e., f -^-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 of: C((A ) = lt and Note J i 0, ai,- = with C (X f) 1 = li tanh(Ai,7i), h = J^Xuf)-^ n(Ai.T); ——. RQ 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 , tively, a n d p r i m e denotes 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 course, respec- the linear 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 is n o t v a l i d n e a r r e s o n a n c e as t h e e x p r e s s i o n f o r $ i n f i n i t y f o r u> = e the nonlinear 2.2.3 uu. T w o cases are therefore considered to e x t e n d the analysis 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 is a s s u m e d , $ = $(D + 6 for 96 w i i ) e p e r t u r b a t i o n m e t h o d is a p p l i e d u s i n g t h e p r o c e s s of i t e r a t i o n v a l i d f o r s m a l l parameters Here ?2$(2) g»$(S) + + is t h e l i n e a r t e r m of s e c t i o n 2.2.2, a n d i n the ( A p p e n d i x I I . 1) second order free surface i.e., _ (4) can be derived b y substituting c o n d i t i o n ( A p p e n d i x 1.2). It is f o u n d that, + / nC (A f) 2 where: to range. Nonlinear, Nonresonant Solution (w A becomes v e r y large a n d goes to 2 2 n cosh A 2 n (z + cosh A 2 n /i h) c o s 26 sin 2dr, 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 (Ao a) = 0 0 n and C' (\2nO) — °. 2 $^ ' can similarly be obtained although it was not considered here. The stability condition is presented in Appendix II.2. The solution is not always valid due to the resonance of the nonlinear higher modes at certain values of CJ, as discussed in section 2.2.5 later. The occurrence of such singularities is, however, localized to a small range of excitation frequencies and is generally not dealt with in this study. One case of interest involves the first transverse, second circumferential mode responsible for the resonant interactions around CJ = 1 , and is treated in the next section. 2.2.4 Nonlinear, Resonant Solution (w « w ) e lt A different expansion is required for the solution around the first axisymmetric circumferential mode (CJ « 1) by taking the excitation to be of the order of the nonlinear terms in equation (2b), i.e., (5) where q < 1, determined through the iterative process. A detuning equality required to eliminate secular terms is defined as (6) 17 The first order equation reduces to the free vibration linear case, i.e., d*$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, f n 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; appear in equation (8). nl « nwn), additional significant terms This is the condition for resonant interactions treated separately in the subsequent analysis. 2.2.4.1 No Interactions (w nl 96 n«„i) The analysis is similar to Hutton's theory for circular cylinders with a major 63 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 / of (6) as well 2 as the coefficients fu, f lx 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 a n d 3rd order free surface conditions while using a multiple time-scale analysis. T h e expansion T = T + Ti + T + 0 where ro = T, T\ = e ^ r , and r 2 planar solution: (9) 2 = e / r , yields two limit cycles: 2 3 f n = <Pu = tu = 0; and / u satisfying hi{K f^ + u ) + F = 0; 1 1 2 (10a) 1 stable for : - ^ - ( ^ - - 2^/^) fii hi nonplanar solution: fu[-{KK 2 a (10c) with ch = fii + function of the excitation. fn - 2KJ& x stable f o r : Here i / is (106) hi V i i = £ u = 0; and < 0, In + v \ + JZ±-F 2 Fi X 1 ^ ^ Fi, satisfying (116) ^ < 0,real. K\ a n d KK\ dependent parameters while B a n d C vary w i t h fu, K, (11c) are damper geometry KK x = 0, (11a) X ( A p p e n d i x 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 - fll)Co(Aon^) V'2n = c n 2 n (/ 1 2 1 + 1 2 1 )c (A 2 2 n f) coshA 0 n («+ h) cosh A o n ^ coshA 2 n (£ + cosh A 2 n /i h) (126) (12c) 19 2 . 2 . 4 . 2 Resonant Interactions (oJ i nwn) n It is found that, for dampers w i t h 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 = 1.99wn, u> = 31 2.96u;ii, etc., for a = 0 . 9 and h = 0.10. T h i s particular situation leads to relatively large terms i n the nonlinear solution described i n the previous section which would make the expansion invalid. The 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 ( z + h) r =Ci(Aur) / n c o s ( g + ¥?ii)cosb;r coshAn/i n v L + fiisin(0 + (fn)sinwrj _ c o s h A i ( z + h) r , , „ + CMA if) —L / cos(20 + coshA i/i «• 2 2 2 1 v?2i)cos2wr 2 + f i s i n ( 2 0 + £ i) 2 2 sin 2 w r J „ ., ^. cosh A 3 i ( z + h) r . . „ + C (A if) « ' /3icos(30 + coshA3i/i >• 3 3 u v?3i)cos3wr + c r 3 1 s i n ( 3 0 + £ 3 1 ) sin 3 w r ] + etc., (13) where the number of interactions strictly depend on the damper geometry parameters a a n d h. T h e problem, however, becomes quite complex w i t h additional terms in 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. T h i s 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. T h i s study is also restricted to the planar mode, and contributions f r o m nonplanar f u a n d / i coefficients to stability requirements are ignored. 2 $ ^ thus reduces t o jk(D $ l > = t fu ra, f\ *x cosh A n (z + h) cos(d+v?ii)Ci(Anr) + -—s—-coswr cosh A n ^ ^ . , . . . cosh A i ( z + h) . f 2 i C ( A i f ) sin(20 + 6 1 ) - sin2£>r. cosh A i / i . . (14) 2 ; 2 2 2 A second detuning equality as i n (6) is now introduced, r.2 11 = 4 - foe* - 0 e * 2 " 2 1 (15) 2 2 u> a n d a procedure similar to that of the previous case is used t o solve for u , i / , x 2 Pi, P2, / i i i f 2 i , ¥>n a n d £ n , with secular terms i n cos(0 + <p\\) coswr as well as sin(20 + £21) sin2<2>r set to zero i n b o t h 2nd a n d 3 r d order free surface conditions. T h i s approach is similar to the one employed b y Bajkowski et a l . . T h e coefficients 9 5 fiii f i> a n d phase angles <pu a n d £ n are now function of the slow time scales T\ 2 and r 2 as defined i n (9). Details of the analysis are given i n A p p e n d i x III.2. T h e exponent of the p e r t u r b a t i o n is taken t o be q = 1/3 as it is desirable t o obtain a limiting solution that tends towards the previous case when the interactions become small. T h e results are as follows: <Pu = 0, w i t h fu and f 2 i £21 = - j ; solution of the 3rd order nonlinear system of equations: / i i ( # i / i i + # i f i + V2) + F 2 2 2 x = 0; (16) f2l(if2f|i + ^ 2 / n + )92) =0. 21 Here u = 2 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 excitation £21 a n d d a m p e r geometry. T h e s t a b i l i t y c o n d i t i o n is r e p r e s e n t e d by (/»)• > Z ^ A , as s h o w n i n A p p e n d i x III.2.3. ( 1 7 ) T h e second order terms are of the f o r m ( A p p e n d i x III.2.1) $( ) =(^ 2 + l n cos0 + (03n A i n cos 30 + cos30) coswr + A3 (fan c o s 2 0 + c o s 0) c o s 3 w t + n A 2 n (V>4 c o s 4 0 + n ) sin2u>T A 4 n ) s i n ACJT, ( 1 8 a ) where: c o s h A „ ft. m A m n = COShX Y. ^ C (X r) n pn P with p = 2.2.5 4 — m , for m = [f coshA n 1,3,4; and p = p n /i + 0, for m = (18c) 2. P r o p e r t i e s of the 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 The a m p l i t u d e coefficients Geometry / m n and f m n of the various modes depend on the A excitation a n d the damper flow boundary characterized by the variables a and T h e effect o f g e o m e t r y o n t h e first m o d e is c o n t a i n e d i n t h e t e r m s as g i v e n b y r e l a t i o n s 3(b), A 10(a) o r 16. Fi is t h e coefficient F\,K\, KK\, h. etc., of the linear solution. ^ 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 1.44 at a = in Fig. range, 4. 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 1.17 A l l dampers should therefore i.e., a w a y f r o m resonance. planar mode, 0.5, similar properties as shown i n the linear 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\ o r a c o m b i n a t i o n o f K\, i n t e r a c t i o n s , w i t h KK\ exhibit at a = K, Ex, 2 and E for the 2 case of for the resonant contributing to the stability of the m o t i o n (relation 10c). 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 the resonance of the nonlinear modes i n c r e a s e s as i t a p p r o a c h e s o n a a n d h. i n t h e p r o x i m i t y o f CJ = 1. In general, the natural frequency of one of the higher order 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) 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, the latter parameters T h i s is m a i n l y d u e Kx terms. dealt w i t h i n this study, w i t h l a r g e Ex, E 2 and K. the C o n t r i b u t i o n of 2 to the overall nonlinear behavior m a y however be modest the a m p l i t u d e of the interacting m o d e to proved to be usually m u c h smaller as than fu- F, 3.0- 2.0- 1.0- 0 Fig. 0.2 0 4 0.4 0.6 0.8 1.0 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 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, negative 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" characteristics, 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 sloshing and can have interesting implications for maximizing energy dissipation, 66 as the absence of nonlinearities theoretically yields infinite response amplitudes and 24 A damping at CJ = 1.0. T h i s condition is met here for h = 0.32, 0.62 a n d 0.79, although the motion is unstable for the last two points due to small KK\. A t higher a, resonant interactions w i t h the mode (2,1) are present at low h a n d the trends are similar to those mentioned earlier, i.e., small K\ a n d growing E\, E , a n d K 2 2 i n the interacting region i n contrast to a decreasing K\ away f r o m resonance ( F i g . A 5c for a = 0.608). However, fluctuations across the range of h are smaller here, a n d 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\ a n d [KK\ — 2K\), suggests that this mode is equally sensitive to liquid height at a = 0.308 ( F i g . 6a), a n d to the resonance of the higher modes as shown i n F i g . 6(b) for a = 0.608. 2.2.5.2 V a r i a t i o n w i t h the E x c i t a t i o n 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 i n two ways. Firstly, it generates higher nonlinear terms as it grows larger, usually resulting i n lower amplitude coefficients / i i or f u as well as important shifts i n 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 2 1 / 3 + higher order, (19) non-interacting solution non-interacting solution resonance of mode (2,1) -100- \ h r l . O a r 0 . 3 0 8 resonance of mode (0,2) 1^ resonance of mode (2,2) I // -2- (b) -100- \i 0.2 (d) 0.4 0JB 0~!i iTfJ ~o^2 b74 Fig. 5 0^6 oTi A h a 1*0 Planar mode coefficients Ku KK , E E a n d Ki as affected b y the damper geometry X u 2 to 26 (KK,-2K, ) K,/KK, a=0.308 6- 2a = 0.608 0 1 0 0.2 1— 0.4 r — I— 0.6 0.8 1.0 A h Fig. 6 N o n p l a n a r coefficients —{KK\ — 2K{) and K\jKK\ as functions of h f o r : (a) a = 0 . 3 0 8 ; (b) a = 0 . 6 0 8 from the detuning equality (6). T h i s p h y s i c a l l y m a k e s sense as a l i q u i d originally sloshing i n the nonresonant region under a small excitation m a y literally resonate at higher amplitudes, w i t h t h e occurrence o f the nonplanar m o t i o n . It is interesting to note that t h esame principle holds for t h e higher nonlinear modes. the 2 n d order terms of t h enonresonant F o r instance, solution should resonate w h e n 27 | ^ - 2 | < ^ , u 4 where: / = 0,2; (20) n = 1,2,...; and UQ is of order 1 ( A p p e n d i x I I . l ) ; a condition gen- erally easier to meet w i t h larger c. Experiments, however, suggest that UQ does not need to be as high as 1, as discussed i n C h a p t e r 3. T h u s more resonant interactions are expected at higher excitations. T o illustrate how the various regions overlap, F i g . 7(a) a n d (b) show / malized b y c / 2 3 n , nor- for the resonant solution to account for different expansions, versus CJ. T w o amplitudes e a n d damper geometries are considered. T h e coefficient / o i (now normalized b y e at resonance) for the first configuration is presented i n F i g . 0 4//3 7(c). Noteworthy are the hardening characteristics w i t h increasing e displayed i n Fig. 7(a), a n d the corresponding reduction i n fu, 0 as opposed to more linear be- havior of the damper with resonant interactions ( F i g . 7b). equilibrium positions is usually present w i t h fu A region with two exhibiting the j u m p phenomenon as one branch ceases to exist or becomes unstable. In the case of a = 0.608, a n d h = 0.196, b o t h branches collapse near resonance (CJ = 0.98) while another loop develops at a higher frequency. T h e interacting coefficient fu is shown as a fraction of i n F i g . 7(d) a n d stays small due t o the stability requirements ( | f 2 i / / n | < 0-32 here, f r o m 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 = Jj 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 $ ) + * * + ^ = 0. 2 (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 = ^ ) { ? / , — €QCJ{AI ' ^ — I C l ( A " ) Bi)\ ainuT > - — o C , ( A " < ) 1 -x c w(Ai + B\) sinwr; (24a) h{\ — a ) 0 2 — > sin UIT — — sin3wr; h{l - a ) Cf 2 , (24o) K 1 30 , 1 (eo^) / 1 3 sin 3£>r >Ul J . . fc(l-o ) W C 4 1 2 1 (cow) / fr 1 1 3 . . \ sin wr >. (24c) Here cases (i), (ii) and (iii) represent solutions for the nonresonant, resonant without 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 theflowis dominated by the mode shape of M the first order term for low amplitudes, | ——| essentially follows the variations set Mi a 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 M until the motion no longer exists. The nonlinear component of \-rj-\ and the 3rd Mi a 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 is greatly affected 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 here. a=0.40 hr0.30 o= 0.036 M /M, a € Planar Nonplanar -4 Fig. 8 T y p i c a l added mass characteristics at low amplitude 2.4 D a m p i n g F o r c e s 2.4.1 Effect of Viscosity F o r incompressible flow, t h eNavier-Stokes equation reduces t o 8 1 —* (u.V)u + | ? = -V(gz + I) + ^ + v V*u, (25) f 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 —• velocity due to viscosity, and recognizing that V — V $ / , [ ( V $ + u2) • V ] ( V $ + u 2 ) + | - ( V $ + u2) at = -V(gz + - + ^f) V + t / , V 2 ( V $ + u2), i.e., ( V $ • V ) V $ + (u2 • V ) V $ + ( V $ • V ) u 2 + (u2 • V ) u 2 = -V{gz + + —) + dU2 + i/,V(V2$) + ^/V2u2. (26) U s i n g B e r n o u i l l i ' s equation, i . e . , - ( V $ ) 2 + gz H 1- —— = 0, as well as V 2 $ = 0, 2 p at reduce the above equation to (u2 • V ) V * + ( V $ • V ) u 2 + (u2 • V ) u 2 + ^ = i/jV2^. (27) T h e corresponding continuity equation is V • ( V $ + u 2 ) = 0, i.e., As u 2 = V • u 2 = 0. (28) — V $ at the w a l l , the correction is of the order of the potential flow s o l u t i o n , and a n expansion of the following f o r m is assumed tT = *u 2 e l) 2 + c '4 2 2) + e 3 ? «2 3 ) + • • • > (29) where g — 1/3 a n d 1, at and away f r o m resonance, respectively. S u b s t i t u t i n g into (27) yields, up to the 2nd order: ^ - v^u™ = 0; (30a) 33 a-(2) - v V u[ f 2 + 2) • V J t Z ^ + ( V ^ ) • V)4 1 + (4 X) • V)V$< > = 0. X) x (306) The linear differential equation (30a) leads to a relatively simple solution for under simplifying assumptions , i then derived by substituting for 80 s in (30b). Details of the analysis and expressions for the correction velocities are presented 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 , 96 dEd dt /x / |V x u\ dv + f (n- V)\u\ dS-2 2 2 f n • u x (V x u)dS. (31) Setting u = V $ + u , considering V x V $ = 0 (irrotational flow), and u — 0 at the 2 boundaries gives dEd dt d f(V x u ) dv -j 2 v 2 + j z f = (n-V)|V$| dsl. 2 (32) 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 where : E C = equivalent absolute damping coefficient, d e ~; 7TWeC5 Ed = total energy dissipated per cycle. rir i is therefore the damping ratio 77 of a single degree of freedom system of rigid t mass M, damping coefficient C , and natural frequency w , divided by the mass ratio e e Mi ——. Recognizing that the total energy of a solid mass M oscillating harmonically M with a displacement x = tosmu t is -Me^oj , it is also the ratio of dissipated to 2 2 e 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: + I2 (m,n) + J 2 ( m , n ) ] + e £ A A i } ; n n 0 -i&hjimMw^ (34a) 2 [fh+A)I ** ' u(1 1} + Z / n ( l , 1) + 7 2 ( l , 1) + J2 (1,1)] + n ( m ) ( i 3 v a | 4 * " ( i + J 2 ( l , 1) + J 2 u ( l , 1)] + n - " 21 + " » ( v/S&Ifcfc^l, 1) + J 2 ( l , 1) + 4J2 (1,1)] + 21 }; U + m (346) ) + 4 / / ( l , 1) 22 (34c) kk's ,//'s, J2's and J2's are combinations of Bessel and hyperbolic function crossproducts (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 / W l n A i n O = l n [ / A i i v m / n + JAii(m,n) + Am]}; (35a) + J 4 ( l , l ) + An]; (356) i h ( i U ) ( ^ l h (m, n) ( i 11 ^ + A ] + $ i A a [ M ( l , 1) + 4 J A ( l , 1) + A ]}. n 2 2 21 21 22 22 22 (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). rj i clearly reaches a maximum at resonance, with magnitude rt 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 rj i versus CJ r> 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 rj i with CJ beyond resonance for the nonplanar motion. r> 37 2.4.3 Energy Ratio E ,i r 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, (37) E = T + U, t where: T = average kinetic energy, — / 2n J U = average potential energy, - p / (V$) dv dt; 2 L2 J 0 J v — / \p I gzdv dt. 2TT LJ J 0 V 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 E = 4h(l - a )^-, (38) 2 r>l where E is as follows for the various cases of nonresonant, resonant without and t with interactions corresponding to (i), (ii) and (iii): (i) E = ^ { a „ A 1 1 w V i 2 i ^ + /ii^/iy/?ii(»,i)[/i4 (t,j) 11 t + JAuihJ) + altlLu]}+(eoCj) AAAi; (39a) 2 (") E = t 7 r ^ ( / ? 1 + f f ){|ll«& A 1 a + J A i i ( l , l ) + a^An]} + 11 + /? [7A (l,l) 11 11 * AAA ; /a 2 (396) 38 + JA {1, 1) + a ^ A u ] + ? XX + aliAai]} + JXTBBBS 2 2 + 1 ^ (l, 1)[/A (1,1) + 4 J A ( l , 1) 22 M * 22 , 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 E j r is inde- pendent of the amplitude of excitation for a given ij i. However, it is a function of r> frequency as the potential energy contains acceleration terms. For a given damper liquid and geometry, the Reynolds number increases with CJ, which further contributes 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 variation of the Reynolds number is defined as E; A = E VRe~, rA (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 a=0.40 hr0.30 Re/d>= 2.497x10 0.04 4 planar nonplanar e r 0.36, planar € r0.036, € r0.036, o 0.03 o o 0.02 0.01- 0.7 Fig. 10 0.8 0.9 T 1.0 1.1 1.2 -lh 1.4 A 1.6 Typical curves showing variation of E i with frequency and amplitude r> 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* for h < 0.26 in Fig. 11(a) may therefore not occur in practice. t It may also be noted (Fig. l i b ) 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 compared to the planar mode, as illustrated in Fig. 10. Ej r 40 8 « =0.032 a = 0.308 o 6 4 ^—---ir---- A hr h = 0.692 0.260 (a) i i • i • 4 . •-^4^0^40 € -0.036 c h-0.200 (b) 0i8 0.9 i!o i'.i A ^'.2 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 D E T E R M I N A T I O N OF DAMPER 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 further evaluate performance. The controlling dimensionless variables discussed in the previous chapter were varied and the effect of internal devices such as baffles investigated. 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. Spectrum Analyser Oscilloscope 1 L~ Filter •Damper BAM BAM J L Strain Gauge Strain Gauge {I Moving Frame Scotch-Yoke Mechanism —VWH L . ~ ...J 77TTT7777Tr Bearings 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 sensitivity 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 platform 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 natural 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 excitation needed to calculate pressure and damping forces. 44 Small scale transparent plexiglas torus shaped dampers with square or rectangular 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 excitation 90 (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 # (cm) 1 D Capacity Internal Cross- (cm) (ml) Configuration Section 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 b a f f l e s : z 8 hVd-0.50 b / d = 0.50 (b) Fig. 13 No of holes = 34 d | / d = 0.39 q; / d = 0.11 (c) 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 elevation 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 C i ( A i i r ) dependence of the 1st order potential flow solution (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 = 9 0 ° , 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 f ) dependence of relation (3a), and are shown in Fig. 16, n 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) Theoretical Modes Ci(A„f) Free surface discontinuity cos 6 Fig. 14 1st planar mode exhibiting: (a) antisymmetric motion about circumference; (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) perpendicular 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 theoretical 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 F a 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 Co cos wt, 2 (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 bending 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 w , for the case of dynamic testing, while the oscilloscope e provided the results for the static procedure. The other channel recorded the displacement 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 frequency 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 F , ama plitude eo and frequency w of the excitation, and the corresponding phase angle tb, e supplied the necessary information for calculation of the added mass and damping ratios 1 " <S - * c 1o 1s and : where Mi =5 & ^ (42) and \F \ refer to the magnitude of the component oscillating at the B exciting frequency. The higher harmonics of the added mass can similarly be found as ,Af . a Mi \F.\ M/cow?' n n (43) 2^,, cm Fig. 19 Calibration curves to determine: (a) M ; (b) slope for the response using both static and dynamic procedures; (c) slope for the excitation; (d) phase angle between response and excitation 0 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 ~j a C o = D i — ; ~~T 1— a a 1+ a = i — ; 1—a h a n d h 7 =;—; a ( ) 44 1— a were adopted, and tests conducted through their systematic variation. Additional information concerning the effect of internal configuration and corresponding parameters was also included in the study. Results mainly describe the damper behavior under excitations near the first sloshing frequency uu, as it is the condition of maximum damping, with the accuracy of the phase angle measurements diminishing 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 significantfindingsof 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., 3u t, 5u t, e e 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 frequency 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 characterizing resonance was generally observed. The maximum r) j usually coincided with r 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} i (Fig. 22). The stability boundaries for the r< 55 (b) F . mV D/d 4 0- =1.84 h / d ; 0.500 <iu : 0.924 « /d=0.0 7 0 3 0- o 20resonance 1 o- 3rd beam harmonic I 5th of d a m p e r support harmonic \ 1o 15 f. Hz (c) 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 halffull damper#7 Fig. 22 Variation of damping and added mass ratios with frequency for halffull damper#l 58 planar mode are realistic, but the nonplanar region extends beyond what is experimentally observed with underestimated n i. Predictions for the added mass are, in r> 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 rj i suddenly drops at CJ = 1.14. The accurate Ti prediction of the resonant point quite close to CJ = 1.0, indicative of weak nonlinearities, 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, erj/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 n / and |—— | as the nonplanar mode takes over ' Mi r (Fig. 23). A gradual reduction in damping then accompanies a further increase in 59 60 eo/d whereas very large rj i can be obtained with decreasing amplitudes. Ti 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 amplitude) 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 indicates 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 € / d = 0.046 4.0 0 3.0* /d o = 0.091 \. D/dr4.10 h/d r 0.500 R e / c u r 3.35 2.0 x10 4 € /d =0.475 —' ' 0 1.0 0.0 0.5 0.7 0.9 1.1 i />. • i 1.4 i 1.6 A CU Fig. 25 Peak damping ratios as affected by amplitude (iv) Effect of Liquid Height This geometric parameter significantly affects the position of the resonant region, 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=1.89 h/d = 0.500 * / d = 0.046 0 P l a n a r h/d U n s t e a d y or 0.7 N o n p l a n a r 0.9 1.1 * 0 . 2 5 0 * 0.3 75 * 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 / d = 0.046 0 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 rj j is around 3.0 and is r essentially somewhat insensitive to liquid height in the range h/d < 0.625. In general, 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. A t 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 h/dr0.500 D/d = 1.89 € /d = 0.105 o 2A- 1.6 0.8 P l a n a r U n s t e a d y or 4— IM /M 0 0.9 h/d ( resonant Theory 1.12 1.50 0.37 5 1.03 1.30 0 . 5 0 0 0.93 • 0 . 6 2 5 - • 0.7 5 0 — 0.2 5 0 resonant 0.0 freq Expts • 1.0 TT4 ill ).T 4.0- N o n p l a n a r 1.6 A — frequency 1.4 1.6 A CU -1.0 -4.0- Fig. 27 Variation of the peak response with liquid height for damper#l at e /d - 0.105 0 67 D/d = 4 . 1 0 € /d = 0 . 0 9 1 h/d 0 0.375 0.500 • 0.625 • 0.750 3.2 1.6 —r-z^- 0.9 o.r Fig. 28 1.1 1.4 i;e A Variation of the peak damping ratio with liquid height for damper#7 at e /d = 0.091 0 (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 50- h/d = 0 y. h/dz0.500 mV D/d = 1 . 8 4 40- CJ r 0 . 9 2 4 € /dr0.158 0 30H Re=3.22x10 Ay /Ay, - 0.57 4 3 20H 1<H (a) Cb) 030H h/d = 0 y, mV h/dz0.500 D/d = 2 . 4 0 20H Uj = 0 . 9 2 4 Ay /Ay, - 0.37 * /d=0.198 3 o Re= 2 . 9 4 x 1 0 ' 1(H CO (d) 5 10 15 f, H z Fig. 29 0 1 5 10 15 f, H z 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 , promising a very efficient design. 90 Theoretical predictions are not always straightforward as the transition point from the no interaction to interacting solution has to be arbitrarily chosen. Both formulations 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 / . The present tests however indicate this may not be the case near - 1 2 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 D/d 1.00 1.89 A 2.40 • 4.10 3.0 • • 2.0 1.0 0.2 Fig. 30 0.4 0.8 1.2 € /d D Maximum damping ratios versus amplitude for various D/d M Reynolds number. Results show that r) j and r a Ij^rl remain generally unaffected by a change in Re in the nonplanar mode (for Re as low as 1.73 x 10 4 in Fig. 31a), with a slight downward trend in the planar mode (Fig. 31b). A t very low Reynolds number (580 or 700 in Fig. 31), the curves drop significantly with the planar motion 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 nonlinearities 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 D/d =1.89 h/d = 0.500 cu r1.15 Re(x1 0 ) 4 3.0A A \ 1.6- •—. • Non-planar 2.66 • Planar o X 0.8- 4.01 2.82 A» D 1.73 o 0.070 A & A a 4 » ..•_'.J° 0.0 0.0 4 oT -*H— 0.4 0.8 1.2 € /d 0 lM /M,l a 4.0; 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/dz0.500 to Re(x1 0 ) Unsteady A Planar 2.8- A A A» A 2.0- o \ . =1.02 o 3.42 2.41 1.47 0.058 A 1.2A 0.8- 0.4- i 0.2 Ll i 0.8 0.4 1.2 € /d Q IM /M| a 4.0- 1.0- 9v2? • -1.0^ 1 /4 i — —i—r 0.0 A 0.4. A __ • A ^ » j 2 € /d D A »-••- A A • -4.0Fig 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 considered 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 investigations . 90 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 rj ,h A similar r picture is obtained by varying the frequency (Fig. 32b), with lower maximums for the added mass and damping ratios. The peak response for ij i appears to be wider r> 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 maximize 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 analyse 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 ri i = 2.2 at resonance, for an amplitude eo/d = 0.06. The same trends were rt 75 D / d =1.89 h / d = 0.500 U) r1.15 3.0 Non-planar 1.6 Planar • a * Plain Baffles Tube o.e-\ Re=2.66x10 EI---J 0.0 0.0 0.2 ~t^~i— 0.4 0.8 1.2 € /d 1.2 € /d 0 0 M Fig. 32 Effect of internal configuration on nr>i and |-r-^-| versus: (a) erj/d 76 D/d = 1.89 h/d=0.500 € / d = 0.049 o 3.0-^ Non-planar Planar • ° A 1.6H / P fi : Plain Baffles Tube Re/oj=:2.31x10 4 0.8 —r-z^0.7 i 0.9 i 1.1 -r—tA r 1.4 1.6 A IM/M,I Q 4.0 A •a 1.0H » 0.0 lit- 0.7 A —i— —I 1.1 0.9, I—— 1.4 1 .6 A CO -1.0H ,-' • \ r -* -4.0 Fig. 32 Effect of internal configuration on rj i and r Mi versus: (b) £> 77 observed during free oscillation experiments, although here resonance was more difficult 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 decrement 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*— Fig. 33 -I d i—H Proposed sloping cross-section 3.4.3 Comparison with Free Oscillation Tests The distinct character of the free oscillation tests and associated apparatus 90 (Fig. 34) provided a means to verify the steady-state excitation results. The additional 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 RECORDER OR OSCILLOSCOPE n BAM SPRING MOVEABLE STRAIN GAUGE COLLAR ADJUSTABLE SUPPORT AND HEIGHT TO CONTROL NATURAL FREQUENCY BEARING 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, n i r> 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 rj i Ti 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). damping ratio was taken as The reduced 79 Baffles D/d Non-planar • Planar • Steady-state =1.89 h/d = 0.50 cu •—•* Free oscillations =1.15 Re=: 2.66x10' 0.8- 0.0 0.0 Plain 3.0- D/d 3 1.89 h/d r 0.50 cur 1.39 R e r 3 . 2 1 x10 4 1.6- 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: x m = 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 \ *' or I ( X ) lnxi/x Jj v m ^ o - 2 ^ 7 ' = ( 4 6 ) * M — - - L — j , (47) where x is the amplitude function, x{m) = x . (48) m A polynomial fit for the envelope of amplitude decay was then applied to facilitate the analysis, x(m) = A + Aim + A m 0 2 2 + (49) as shown in Fig. 36, which yields 1 A + 2mA2 + ... + x ^ 1 ( X ) = ~2^ [ A 0 + pm^-^A^Ii Aim + ... + A mP p ] T ( 5 0 ) 81 x =2.50 - 0.075m + 0.0019m 0 40 80 m, Fig. 36 2 c y c l e s 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 rj i. Any configuration restricting the r> 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/ j and |-r-p|) while promoting r> 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 field 67 is another possible avenue of research. • Finally, turbulence was never found to be beneficial as rj i did not change its r> trends at higher amplitudes or frequencies of excitation (with the transition from laminar to turbulent flow). 84 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 4.1 General Description Effectiveness of the dampers in controlling vortex resonance and galloping instabilities 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 produce desired boundary layer thickness and turbulence intensity, were used to simulate 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 setup 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 results velopment of a successful galloping theory 93 91 combined with the de- permit an approximation of the cross- flow oscillations, provided the aerodynamic reduced damping 77,. , also called mass>a damping or stability parameter , is known. These tests are therefore well suited for 6 8 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 rj Tta 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 rj r>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 Transducer Damper \ i Vv End Plate Air Bearing Shaft •V\A- Function Generator Conditioner Air Bearing Block I i—r t i i—i Dead Weight 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 1 MODEL# CROSSSECTION * 102 m m • 1 IX 51mm 673 LENGTH(mm) 3 2 C"~^| 0 1 m 2 673 673 BALSA & BALSA & ALUMINUM ALUMINUM MATERIAL MASS (g) 786 END PLATES 204 mm PCV 349 ! 745 102 m m "1 m t 1 305 1 mm 251 m m i • 204 mm t 305 | mm Static force measurements on the square cross-section models were carried out with the six-component pyramidal strain gauge balance (Aerolab). Drag D r forces perpendicular to the flow S p and were recorded over a range of angle of attack a giving the side force, F y = S cos a — D sin a, p useful in prediction of the galloping response. r (51) 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 configuration , 97 are desirable to reproduce conditions of two-dimensionality , the drag and weight 98 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 / versus a, for both the no end plate and the large end plate y 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 R = 1.23xio Model#2 4 • • 0.4 A- dm bm/dm 0 2 3.5 • 0 • 5 -A 5 (no plates) (med.) (large) 0.2 -0.2- 0.6 Model# 1 R r 1.17x10" am/d . • 0.4 £r > o -A o 3 5 m bm/d m 0 2 3.5 0.2 -0.2 a, Fig. 39 o Effect of end plate dimension on C / for: (a) model#2; (b) model#l y 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 resonance response for a circular cylinder , as shown in Fig. 40. 91 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 n (i.e., no damper, 0 mA) was also found to be constant e at low Y but quickly increased beyond a certain threshold. This is in agreement with earlier studies using such test facilities , and is beleived to be caused by the 99 higher bending stresses at larger amplitudes. The deformation during the initial displacement may however be different from that applied by the distributed loading of the air flow pressure field and such a rise in r} may not occur during the 8 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 Fig. 41 0.5 0.6 y 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 B r o o k s 100 . The higher blockage ratio in Fig. 42(b) causes the Cf values to be larger at small y angles of attack. The dC/ /da slopes at a = 0° are approximately 2.6 and 4.6 for y the two models, respectively. The galloping response prediction, obtained using a 13th order polynomial approximation for C / , agreed with the experimental results y for the electromagnetic damper set at higher n (i.e., galloping onset velocity L7 ria 0 far from vortex resonance U ), as illustrated in Fig. 43(a) and (b). For a lower r value of the aerodynamic damping, both vortex resonance and galloping regions overlap, and the response is rather insensitive to rj r>a until the latter is large enough for a distinct peak near U (Fig. 43c). The data were taken for a model natural r 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 2.60a - 0.0075a - 5.93 x 1 0 Q 3 3 + 2.34 x 1 0 a s 5 7 - 2.25 x 1 0 a + 1.23 x 1 0 a 6 0.6- 9 9 n A 0.4- s ' A A Model#2 9 0.2- A /A =5.5% 0 -0.2- t R (x10 ) 1.23 1.74 2.74 3.88 4 (a) • 0.6- 4.60a - 0.018a - 1.96 x 1 0 a + 8.56 x 1 0 a - 7.99 x 10 a - 9.63 x 1 0 V + 1.23 x 1 0 a 3 q. 0.4- 5 3 7 6 6 1 9 9 13 • A i f 0.2- A Model#1 A • A /A =11.1% e t • -0.2 (b) i— 8 Fig. 42 A 16 T 24 Cfy versus a for: (a) model#2; (b) model#l 32 a, o 95 Y/IL u /u 4.63 10.7 15.7 0.8- c • 0.44 1.04 0.6- * 1.50 a ? (%) 1 > r • 1.20 • 1.50 * 1.72 47.4 112. 162. U /U 1.98 2.47 2.83 0 r ^ , 0 32.7 40.9 46.9 from ref. 93 from polyn. fit 0.4 in F i g . 41 0.2 (b) (a) 0.5 1.0 1.5 2.0 0.5 U/U 0 Y Model 2.86 • 5.73 o 1 1.7 12.8 A • 21.5 • 0.8 0.6-I 1 1 2 1 1 u /u 0 r 0.17 0.35 1.14 0.77 1.30 0.4 0.2 (c) 0.6 Fig. 43 1.4 2.0 2.8 U 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 circular 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) ,i (liquid reduced damping r ratio) through ,1 V where L m and d m 47T 1 , 1 M/p L d a m Mi/M m are the model length and diameter, respectively; p is the air a density; M is the total system mass; and rj is the inherent damping determined to 8 be 0.105% for small oscillations during the experiments. Using the model characteristic 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 rj T>a « 12.5, 20.0, and > 40.0, re- spectively. This in turn requires rj i to be of the order of 0.36, 0.31 and > 0.35, rj respectively. This is indeed the case with h/d = 1/2 as observed in Chapter 3 (p. 59) where r} i was found to be greater than 2.0 in the nonplanar mode at low excir> 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 e /d = 0.25 and 0.45, for h/d = 1/4 and 1/8, respectively, due 0 to the hardening characteristics at low liquid height. Larger values of r) j = 0.50 r 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) j « 0.31 r and 0.36 estimated from the response during the wind tunnel tests. Possible innaccuracies 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 thefluctuationsin damping with eo/d precisely (section 3.4.3), therefore rj j is likely to be lower prior r 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] j. One way would r 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 — no damper • plain (#1) • baffles (#2) A tube (#3) D/d - 1.89 CJ =1.15 Re=2.66x10 0.8 Fig. 46 1.0 4 U Vortex resonance response on model#3 showing: (a) effect of liquid height and w; (b) effect of internal configuration 101 ^ , 0 Y 0.4 = 2.92 Damper#1 Damper Parameters h / d = 0.500 & 13 M i / M =0.038 limit D/d • A — - A 1.89 15.2 \ u > R e (x10 ) 4 6.70 2.66 \56.8 0.2 0.0 (c) 0.6 Fig. 46 1.0 0.8 Vortex resonance response on model#3 showing: (c) effect of diameter ratio and CJ h / d = 0.125 0.6 0.4 0.2 0.4 Fig. 47 0.8 € /d 0 Variation of ri j with amplitude during free oscillations for damper#l with h/d= 1/8 and 1/4 r 102 hensive study by Diana and F a l c o 101 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. Noticeable discrepancies are apparent although the general trends are well reproduced. A n alternate approach is the partly successful Hartlen-Currie oscillator m o d e l requiring the empirical determination of the variables 102 and 6^, using the lightly damped response here (Figs. 49a, b), and applying it to the model fitted with nutation dampers (Figs. 49c, d). The static lift coefficient CJO when taken to be the same as that determined by F e n g 103 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 rj i = 0.36 and 0.31 for h/d = 1/8 and r> 1/4, respectively, as found earlier from Fig. 40. Furthermore, the half-full damper corresponding to rj i > 0.35 was found to suppress the oscillations (not shown). A ri higher C j of 0.5 to account for the larger blockage ratio was then considered and 0 peak amplitudes closer to the experimental results obtained (Fig. 49d). A minimum t] i of 0.62 was then needed for h/d = 1/2 to bring Y down to near 0. Overall, rt 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 werefirstmounted 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 parameter 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. b =1.0 0.03 0.05 / 0.10 h C| 0. 8 i o = 0.3 ' a =0.05 C| o 0.3 = 0.20 ' 0.6 0.5 limit 0. 0.7 h limit 0.2 (a) 0.6 (b) 0.8 1.0 1.2 0.6 0.8 U * • h/d 0 0.125 0.250 Expts Hartlen-Currie a = 0.05 b 0.6 C| =0.3 h 0.2 H h/d h = a -0.15 h o b =:0.5 h C| =0.5 0-125/ o 0.250X, A (d) 0.0 1.0 0.8 U 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 / = 0.3; (d) response with C/o — 0.5 0 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 rj j exhibits a jump at r 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. O n the other hand, a gradual increase in response began around Uo = 2.0 for CJ = 0.92, with rj i reaching a maximum at higher amplitude (Chapter Tt 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 governing 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 characteristics, 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 Damper = 2.92 Par. D / d =1.89 h / d =0.500 M,/M = 0.038 Damper #1 R e (x10 ) 2.13 2.66 3.22 6.41 4 — Damper No U R e = 2.13x10 0.92 — A—-A 4 No D a m p e r Plain (#1) — -• B a f f l e s Q Tube (#2) (#3) 0.0 Fig. 50 Galloping response of model#l showing: (a) effect of frequency; (b) effect of internal configuration 107 Damper Par. limit U limit 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 rf r>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 galloping 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 U is much lower than U " (galloping onset velocity). With the knowledge r 0 of r) j versus Y and the necessary modification to the stability analysis (Appendix r 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] j is high at low amplitudes. Moreover, there r 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/ / further dropping at higher amplitude, thus r> 109 leading to an even larger response. Results for the 5.1 cm diameter model show better agreement with the theory (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 rj i varies Ti from a very low value (< 0.1 at Y = 0.1) to about 0.21 at e / d = 0.26 (Y = 0 0.15) before the system starts to gallop. This is compatible with the upward trend for r/ / r> until e / d = 0.5 (Fig. 47) combined with its value of 0.36 for eo/d = 0.35 estimated 0 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 electromagnetic dampers (Fig. 43) however shows the half-full configuration oscillating 110 -Theory Expts Y Ir.l model 1 0.6- h/d=0.500 0.4- OJ-1.39 \ \ : 0.2- Y w=1.15 T) ig.23 r | : F Fig-35 model 2 1.2 h/d = 0 . 1 2 5 ^ Tl , :Fig.47 r ^ h/d 0.8-I = 0.250 TV-Fig.47 • ^ 0.4 =0.500 h/d ^r,! : F i g . 2 3 (b) 20 Fig. 52 30 U Predicted galloping galloping response response ol of square square prisms fitted with nutation rreaictea 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 damping. 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) j is much larger. This again r Ill 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., rj i = 0.46), and r< 6.94% (r) i = 1.8) for CJ = 0.92, which are close to the values of the steady-state exrt 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 C, =1-4 — a = 0 . 1 3 , b =2.50 ah=0.30 , bh=11.0 0 0.3 h 0.2 • HartlenCurrie h Expts 0.1 77 = • 0 i Y 0.3 1.70% 1.85 — 6.94 C = 1.4 a -0.30, I )o h b =1 1.0 h A GO 0.2- 0.1- D (b) 0 0 9 Fig. 53 0.92 • 1.39 A 1.3 1.7 U Hartlen-Currie model predictions for model#l with: (a) netic damping; (b) nutation damping electromag- 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, liquid sloshing resonance with CJ close to 1 resulted in maximum efficiency, while baffle and inner tube arrangements were less effective at reducing vortex resonance and galloping instabilities. Low liquid heights with CJ > 1 are better suited for restricting the response at high amplitudes, due to their hardening 113 Damper Parameters D/d = 1.89 h/d = 0.50 M|/M =0.038 UJ-0. 92 Re=2.13x10 Y Damper# 1 limit 0.4 %,a = 4 2.92 •—• A—-A End Plates p| tes N o E n d a 0.2 0.0 1 .5 0.5 3.5 2.5 U Y 0.4 limit •—• A A 2.92 •( ! . 7 0.2 00 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 rj i versus amplitude curves rt 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 mass ratio less than 1% (damper#l, h/d — 1/8). < 0.1 with a liquid to total 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 , was 94 modified to position the damper outside the wind tunnel thus avoiding interference 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 Damper Recorder or Oscilloscope 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 significant 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 MODEL* 1 CROSSSECTION I • 102 m m 1 * LENGTH(mm) MATERIAL 2 508 | | J_5I 3 mm ^ 508 BALSA & BALSA & ALUMINUM ALUMINUM MASS (g) 644 244 ^ 102 m m 4 | 11 5I m m 508 483 PCV BALSA 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 Cf . The higher blockage continues to show a larger excitation with y 118 a slope of 1.4 at a = 0 ° and an improved curve for Cf until a = 10° (model#l, Fig. y 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 stiffness were used to arrive at a system inertia of 0.2021 K g - m 2 for a total mass of 1.044 K g . 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] generally increased with amplitude in free vibration. B Earlier work by Sullivan 94 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 loading). 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 investigations . The vortex shedding excitation away from 105 119 0.6• • 0.4- 0.2^ Model#2 dCf /da v A /A =5.5% A« • = 0.60 D •a t Ln/dmZlO.I \., L»-BA -0.2R (x10 ) 4 (a) • o * • 1.23 1.74 2.74 3.88 0.6- 0.4" dC /da fv = 1.40 • • ° °• Model* 1 U A • 6B # A • OA" A / A = 1 1.1% 0 0.2 U/d • 4»i t m = 5.03 A OA" -0.2 (b) —i— 8 Fig. 56 16 —i— 24 32 a, Static side force for three-dimensional square prisms as: (a) measured on model#2; (b) on model#l; 120 • • D 0.12 • • 2 • 4 R r 1.23x10 0.08 • • 0.04 • 8* • • • • model 1 • model 2 • • i 1 i 1 i i 1 -0.04 (c) 8 Fig. 56 16 24 32 a, o 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 procedure 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 structure. 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 before 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 Fig. 57 0.8 0.4 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 / . Fig. 59 illustrates the effect in laminar y 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 (b) 0.0 0.6 Fig. 60 U 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 . 106 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 = 1.13 D/d =1.89 h/d = 0.500 CUr1.15 Re=2.66*10 M./M z 0.067 1,0 4 Y — 0.4- No D a m p e r 0.2- 0.0 0.7 Fig. 61 0.9 1. 1 1.3 1 .5 Effect of internal configuration on the 3-D model response U 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 (laminar) 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 decreases with the amount of rotation as defined in Fig. 63(d). rj j is fairly constant r € /d D Fig. 63 Damping characteristics as affected by liquid height for nutation dampers 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 laminar 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 ^ Damper Par. = 1• 1 3 h/d: Damper#1 & 13 0.500 M,/M r0.067 D/d * 1.89 — - - 15.2 6J Re (*10 ) 4 0.46 3.10 1.06 22.7 NoDamper D/d » — * 1.89 — - • 15.2 — Fig. 64 to Re (x10 ) 1.15 6.70 4 2.66 56.8 NoDamper 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 galloping oscillations with the boundary layer pofile (Fig.67b). This agrees with other studies 107 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 laminar 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 P a r . D/dr1.89 Re=2.66x10 M|/M=:0.134xh/d 0.5 Fig. 67 1.5 2.5 3.5 4 U Galloping response in 3-D for model#l with nutation dampers in: (a) laminar flow; (b) turbulent flow 132 V r Fig. 68 n - 3.1 7 Damper Parameters 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 = and 1/8 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 rj i with displacement. r< 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 Fig. 69 1.5 2.5 3.5 Effect of D/d and CJ on the 3-D galloping response of model#2 134 of Fig. 5 6 , the first term A \ of the polynomial fit, found to be 0.316 and 0.251 for 94 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 e /d 0 > 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) j « 0.031 in r 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} i improves with amplitude (Fig. 63c). r> 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. rj i was, however, estimated to be rj 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 instability 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 approximation 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 Stockbridge 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 beamlike 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 cylinder, 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 displacements. E n d 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 D e v i c e 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 ) . With the installation 2 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 k — 0.285 N-m. The inherent damping ratio 2 of the system (i.e., no damper) was estimated to be 0.04% corresponding to a very low rj r>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} 2 (rj 2 = 8 , where C<j is the damping coeffi2 B 2ra2U>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 ij j. r 140 x/d Fig. 72 Evaluation of the secondary system damping ratio showing: (a) calibration procedure; (b) r/ and r) j versus amplitude a2 r 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 / / m /m —0.12 O J / C J l = 0.96 2 1 2 freq.(Hz) • • 2.08 • • 1.68 • • 2.40 meas. calc. "2.5 0.5 1.68 1.71 no damper j j empty damper 1st Peak 2nd Peak 0.64 0.37 2.40 2.43 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 h/d " n l CUn2 0.4- data points only \ Trans. • * rolling o 0 0.25 0.50 Damper #1 D / d = 1.89 m /m =:0.12 2 1 c u / ^ i = 0.96 2 0.2- 0.5 Fig. 74 Effect of liquid height on the system response vortex resonance peaks, with the half-full damper performing better at u 2, where n 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 mechanism 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). 1' r o t i i I i rot y, mV In, Fig. 75 ' I I I 1 2 3 A 4 Frequency spectrum of the response for: (a) f v I ' 5 f, Hz w f t', (b) / „ >• f ro rot 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 / w i = 0.96 shown earlier (Fig. 73a). 2 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 2, Fig. 76b). This overall n behavior seems to agree qualitatively with the vibration absorber relation (VI.6) that predicts the resonant amplitude under a constant excitation F: the larger windinduced 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 m /m zz 2 ] A (Jj h/d 0.12 • a^/cu,^ 1.32 0 0.25 0.50 o A 1.33 0.94 Damper# 1 D/d =1.89 <Z\ | - Trans rolling w i ^n2 2.04 0.877 1.98 n 0.5 1.5 2.5 m /m - 0.078 2 1 OJ2/CU1 = 2nd Peak 1.76 2.98 3.5 7.0 U h/d 1.00 0 0.72 A 4H ri(xJW*b) 1st Peak Damper#8 D/d = 2.77 meas. calc. <*>i w2 1.72 2.28 1.74 2.30 n n 1st Peak 2nd Peak 0.73 0.32 (b) 0.5 Fig. 76 1.5 —1 2.5 1 *—1—XX 3T? 1 1 — 7.0 U Response of the model as affected by: (a) w /u>i; (b) m / m i 2 2 147 Fig. 77 Sketch of the two damper arrangement useful to control roll interaction can disrupt the first vortex shedding lock-in region. O n 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 frequency 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 u 2n • 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. 149 5. CONCLUSIONS This investigation has provided information useful in the design of nutation dampers for controlling wind-induced instabilities. With the objective of optimizing the energy dissipation parameters, it has also contributed to the understanding of nonlinear liquid sloshing problems using both theoretical and experimental procedures. Extensive tests with two and three dimensional models in laminar and turbulent flow wind tunnels suggest that the concept of nutation damping can effectively 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 approach in conjunction with the boundary layer correction, although predicting the correct trends, does not account for several mechanisms for energy dissipation. 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 sloshing 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 oscillation 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 dependent parameters related to the acceleration of the structure proved to be significant for the case where the aerodynamic model is unstable and the damping ratio is strongly dependent on amplitude. (vi) Relatively small nutation dampers were usually adequate to suppress the vibrations. 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 T7,o) were necessary to r significantly delay galloping of the square cross-section. (vii) The weaker aerodynamic excitation associated with the three-dimensional models 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 laminar 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 frequencies. 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 according 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 characteristic 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 investigating. 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[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 CONDITION 1. Basic Equation In polar coordinates, the kinematic boundary condition: drjf_ 1 d^drjj_ dt r dr dr dr\j_ _ d§_ dO dQ 2 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 dt 2 + + ( — ) dz y ) 9 d z dr drdt + 1 2 dz* ( — ) r* d9' r + dz dldi + [ — ) 2 dQ y dQ dOdt 2 r dr dO 2 z K + ( 1fr> "cV " 2 J 2 r } 2 dr dQ dQdr a * a$ t9 $ 2 t9$ d$ a $ + 2dr— — — — + r 50 dz dfldz dzdr 2 2 2 dx _ 3 r c o s d x.\d§ . dh _ _ _ _ , _ _ 2 , n + [ s n m c o s , ] . t r . .3) (J 2. Perturbation Series Expansion Introducing: „, = + £ *r,( 2 2 ) + e ^} 3 3) + (7.5) and substituting into (1.2) gives d in terms of By using a Taylor series expansion around z = rjf as a n * * t9$. and $ ( \ 3 1 ,d $. , . 2 and replacing for rjf (section 1.3) gives an expression for r $ =rif in terms z of $*=o, needed to get the full nonlinear free surface conditions. Relation (1.3) then reduces t 0 e«AW + e *AW + e *A< ) = 0 +... 2 3 3 163 at z = 0, or expressed in dimensionless form: * a $< ) 2 A m = w dr 2 12 i a$< ) <*iiAn dz 2 1 df 3 Q!iiAn f ^~dl r d t """onAn 1- a ^ ) 1 dz a*( )a *w, 2 2 a 1 + dfdr df dfdr d$Wd $W 2 d& >d $W 2 2 + * w 3 ( 1 as / 2J + 2 2 + a i l af ' l A„ 2 dz af 2+ 2 2 J f dr + ^ a* 4 3 dzdr [ ae J 2 I 2 + 1 3 2 + H r 2 H ( + ^2 ( - 3 5 - ) 2 + H 1 r H 2 1 )a $( ) afas A„ a $(*) 2 * asar i 2 1 3 ttll 2 az ar r dQ dz r a$( ) a $ ( ) a $ ( > a & w a *^) 1 a *( )a $( ) 1 dr dfdz dfdr df dzdfdr f dzdd dBdr i # ) a $ w a $( )a $( > a&^a ^ ) ^ f de dzdedr*~dz~ alaT as as arU r 1 x2/ ^ 1 a $( ) a$( >a $( ) .a ^ ) \ aI55 5" o„A a s ar a s a r al a7 " 2 2 x 1 1 2 1 2 1 3 2 1 1 2 2 1 2 1 3 2 r ( a i l A l l J r ttuAn 1 2 J 2 3 H 1 ( 1 + — 1 — ^ 2 - + 2 1 aiiAn 2 1 2 2 + + 2 2 a ^ ) aa ^^) ia i w a$( )a$( as ' a s f a f * de ' af as 2 a$( )a$( )a $( ) 2a s ^ a s ^ a ^ ) _ f df de dfde f a s a * asa# 1 a ** ) .add) i,a* \ / ^ \ 1 2 + 2 a + 2 ~dl dddr * ^ dz dzdr ~dl dldT a $( ) i a $( > a$< ) a $(*) + dr dzdr 1 1 d6dr 2 onAn 2 2 df 2 d$dr 2 2 d& 'd $W a a i l A l l x dz 2.d& >d $W 1 dzdr 2 50 2 i | r ; dr f a$( ) a$( )a $( ) 1 2 1 1 1 dr i 3 a^c ) a^ ) A 2a$( '3 $( ' 1 dfdr 1 55C9T a $< ) ' = 2 2 f- 2 2 a&^a ^ ) 2 5$(i)a $(i) az M (,.7) l 2 ( 1 ) TTO— A 2 l 1 2 1 4 1 1 : dz + 1 + — 2 J 1 1 \ + ru* cos 0 cos + l 2 2 ur. (1.9) dr i ' 3 Free Surface Equation Using expressions (1.2), (1.4), (1.5) and (1.6), and using nondimensional parameter r)f = results in a$( ) —] — e a n A n [ — - — 2 fit =&a.\\\\i\siCjfcos0cosu/r ? ar ar + anAnK-Wcosflcoswr - _ ^ _ ) - ^ - + ^ V * * * ) ] + 1 where s- = 1 for q = 1 , and zero otherwise, and 2 = 0. t 2 (1.10) 164 APPENDIX II: N O N L I N E A R , POTENTIAL FLOW NONRESONANT SOLUTION 1. Second Order Terms Substituting for - E /i^i(AHr) ° " t • coshAi,7i C h S h A l t ( ) cosflcosfrr into the second order free surface boundary condition (relation 1.8) and rearranging yields, 1 a$( ) a $( ) 2 2 dr 2 + o:iiAn dz 2 =EE^r {^(Ai*f)ci(A i ^ ^ * 2 l i f) i + [Ci(A f)Ci(A f) - ^ C ( A f ) C ( A f ) lt 1 iy l t 1 i y + AX,yCi(A f)C (A f)]cos2f;}sin2u;r. lt Assuming $( ) = E E rn grating (77.1) as fnmC {X f) 2 n 1 + ^) coshA /i ^^ (* cos nm nm n (II.l) iy # inpc2;7-, c o s n and inte- s nm {II.l)C {X f)rdf / n nm Ja to use Bessel function orthogonality condition (Appendix V . l ) , it is found that ^ ( 2 ) = E { / 0 ^ 0 ( A 0 n f ) ° S h A ° f t K ) coshAon/i » ~ / , « c o s h A ( z + ^) + /2nC (A f) T cos 29 \ sin 2wr, coshA /i • > K n C 2n 2 2 n K 2n ; , (77.2) 2n where: /2n = ^ ^ / l i / i y ^ O n i f [LLuo(t,j,n) + J J i i o ( » , j ' n ) + n t ° and / 2 n AKjjKK\ 10 (i, j , n) ] 2u;[(a;on/^ -4]Ao„ n 2 5 = ^2 E / i * / i j ' 2 n , n * = n 2 n i L L n 2 ( ^ ^ n ) ~ AKjjKKii {i,j,n)] ^ii2(»'»i» ) + w 2 2Cj[(Cj /Cj) -4}A /X 2n 2 2n 2 n 165 Here LVs, J J ' s and KK's are Bessel function triple product integrals, A's are coefficients related to J C (X f)fdf as defined in Appendix V.1.2, and 2 nrn + AKij = auaijXuXij ^ - -Afy. (JI.3) It can be shown quite readily that the solution is singular for (&/„/£;) — 4 = 0 (/ = 0,2), as fon and f become unbounded. Furthermore, it is of order e for (uin/Cj) — 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 - 1 2n 2 (^) or, - 4 > u v, 2 (IIA) 0 (JJ.5) \- --2\>—. F 2. Stability and 3rd Order Equation A standard stability analysis assumes a general 1st order solution of the form = E[( i * e i c o s 108 , ^ + 3t sin0) coswr + (e2t cos 0 c cosh A u(z + h) + en sin0) sinwr]Ci(Ai f)cosh \\ih t where en, e {, e ,- and e - are function of the slow time scale T = er, thus following a procedure similar to Hutton's theory of resonant oscillations in circular cylinders . Subsequent substitutions into (1.8) and (1.9) lead to the 3rd order system of equations: 2 3 4t 2 63 {-^•jr + P2n + Di^2 1 e 2 « E ^2(«ij ik + e e + DD 2j 3i t i 2 ^2(e e y k e 3k + e je )] 4 4k cosflsinwr = 0; - t t ]} xi 3k Ak - eiye )]} cosflcoswr = 0; e 4fc y k e ,(EE( i j k + DDi 2j e4i(T] yZ( 2j sk t 2 3j j k + DDi ^ {-^- + Pin ~ 2k j k e [^2 X +e e e i 2 4 e i J e i f c + e (y] ^T{e je u 2 3k 2j'2Jfc + e ye jfc + e ye fc] e c 3 3 4 4 - eiye *)]} sintfcoswr = 0; 4 166 de^x + P3n + Vl Z ^ 3 » l e 72 i + n i ' j 3 k* D i ^ e 2 , [ ^ ^ ( e 2 j eiyc fc)]} sin 0 sin CJT = 0. e 3 t - 4 k 3 Here D\ and DD\ are complicated, frequency dependent expressions otherwise similar to K\ and KK\ of Appendix III. 1, while p i , . . . , P4 are the third order terms: n pin = Di E E E » j * e n i» iy i*; e fi (77.4) P2n = PZn = P4n = 0. The stability of the solution previously derived is studied by considering a disturbance in the ith mode such as: eu = hi + Aie 2; Xr e 2i = A e 2 A r 2; e 3i = A e 3 A r 2; and e -+ A i e 4t A r 2. Substituting into (II.4) leads to the following conditions: A = -3(Di/ ) ; (77.5) A = -(/i ) [^i(i?i-i?ii)]. (77.6) 2 2 2 2 t 2 t 2 The solution is stable for nonpositive real part of A and requires 7?i(£>i — £ > u ) > 0 in (II.6). 167 A P P E N D I X HI: N O N L I N E A R , POTENTIAL FLOW RESONANT SOLUTION 1. No Interactions Using Hutton's theory for a circular cylinder and substituting for the Bessel function solution of the torus problem, it is found that: 63 • Detuning parameter: "1 = 0; CJ - 1 = " = a5P7r UJJ.1) 2 • Coefficients of the 3rd order equation for fu'. „ Ci(An) - q C i ( A „ a ) *i = 7 ; An K = -1L[SUM1 An + GI); x KKy = -±[SUM2 An {111.2) + G2], where: SUM1 = ^ { - 1 8 / i ( l , 1,1,1) + A !(3 - 7 a ) I ( l , 1,1,1) + S A ^ a ^ 2 2 1 2 2 - a ) J ( l , 1,1,1) + 6J (1,1,1,1) + 3A (3 - 7 a ) I ( l , 1,1,1) 2 t 3 2 4 2 X 1 5 - 1 2 / ( 1 , 1 , 1 , 1 ) + 6 / ( l , 1,1,1)}; 6 SUM2 7 = - L { - 6 J i ( l , 1,1,1) + A ^ l + 19A ) J ( l , 1,1,1) + A ^ a 2 X 2 2 ^ - a ) J ( l , 1,1,1) + 2J (1,1,1,1) + A !(3 - 70^)75(1,1,1,1) 2 x 3 2 4 - 4 / ( l , l , l , l ) + 2 / ( l , 1,1,1)}; 6 7 GI =Y,{[CK KK i{l,n,l) 1 -//i i(l,n,l)]fl n + l0 0 0 \-^KK {l,n,l) l2l n - i/Ji2i(l,n,l) - JJi2i(l,n,l)]n G2 = Y^{[CKiKK i{l,n, 2 n }; 1) - 2 7 / i o i ( l , n , l ) ] 2 n „ 10 0 n + [-CK KK 2 Here I\, I ,...,Ij V.1.2 with: 2 121 (1, n, 1) + 7/1,2,1 (1, n, 1) + 2 J J121 (1, n , l ) ] n 2 n }; are Bessel function multiple product integrals defined in Appendix o - f° . **0n — f2 ' n /ll o - * 2n **2n — » j Hi 2 168 and CKi = a n a i i A A i i - - A ^ + A ( l - a ); CK a ). 0 • 0n n = a „aiiA Aii - ^ A 2 2 2n 2 n n u + A (l n (7/7.3) n Coefficients for stability relations: 5 = AKKlMflx + ^ F l ] ] r 2 (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 iC (A if) cos20sin2wr 2 +AK C (X f)} 0 +/ 1 n f 2 1 , ^ , A 2 + [C[ (X f) 2 11 { ^KAii^CUAaif) gi(Anf)C (A 2 2 1 Ci(Auf)C (A f) V 2 +AKi — 21 ——^ - n f) C l 2 2 + AK C (X f)) ( A l l f ) 0 2 l cos 20} lx sin2£r C (A f)C (A f) + 1 11 21 ^ K A n f j C ^ A ^ f ) _ C (A f)C (A f) ] c Q s 6 + 1 l 1 2 11 , r 3Cj(Aiif)C (A f) A -J cos 39j cos WT + f [—— ' 2 21 2 21 - , 3C (A f)C (A f) , C (A f)C (A f) , 3C;(A„f)C^(A f) + ^ + AK J cos0+ [ 1 11 2 21 A t r 1 11 2 21 1 a f ai 2 _3 C l (Anr)C,(A r* +fe i{[C (A f) a a 2 a i 2 1 f) + A ^ K + 4 ^ ^ 1 M X 2 l 2 r ) + AK C {X f)] 3 2 2l , , J + [C^ (A f) 2 21 +A7f C|(A f)] cos40} sin4u>r = 0. 3 Here: " AK"i = ( a 2" n 4 C *( » ) A f (777.5) 21 0 J ' - 1)A*^ - a n Q : 2 i A A i + ^ A ^ ; n 2 169 AK - (al - 1)A^ + 5 a i i a i A i i A i - AK = 2 Z Assuming x (~ 2 q : 2 + 1) 2 21 A = ( ) C » \emnj and integrating (III.5) as i 2a a A A . + ( A n r 2 2 1 e m 1 1 2 1 1 1 f ) ° t COShA /l C s h k ) cosnO nm l (III.5)C (X f)fdf n nm to use the Bessel function orthonogality condition as before, it is found that, $( ) = $( ) + $( ), 2 2 (7/7.6) 2 where: $i = no interaction solution, i.e. relation 12, with *(2) $ V ^ / r j n (\ ln n cosfl coshA /i coshA (z -I- h) v ln n in fan', ^coshA (z + A) = 2_.\ [di Ci(\ r) 2 n> 2 1 ln , ^ /\ M 3n . —s—-cos30j coswr cosh A / i r . ^, /» «\ coshA (z + /i) + [<*3nC (A f) * ' C O S 30 cosh A / i „ . cosh Ai„(5 + ^) „, + e w(Ai rj s cos0j cos3u>r cosh\\ h , rj f\ ^coshA (z + &) . + [d4nC (A r) i — x — - cos 40 + ein^3(A r) 1 3n 3n 3n 3 n l 3n 3n 3 n n n , 4n 4 4n cosh A / i 4n , with: din = /<w\ ^coshA (2 + ^ ) , . cosh Aon^ 0N ftinfutii', [7/ i(l,l,n) + J J ( l , l , n ) + 12 1 H Cln = m ~ ~ 1)A../Af „ 5 n [|77i (l,l,n) - J J ( l , l , n ) + ^LRK {1, l )] (*!» " l)A3n/AL /nf i; ftJJWl.l.n) - 3 J j W l . l . n ) + ^ K i i W l . l . n l l 23 3n 12l 7ln/llf2i; 7 l R d ^KK (l,l,n)} —0 3 r e 2 123 123 >n ^ 2 ; 170 C3n = 7 3 n / l l f t i ; [|//i i(l,l,n) + 3 J J 2 1 S T l ^4n = n (l,l,n) + ^KK {l,l,n)} 121 ("in ~ 9)Aln/A 2 4 n m 2 5 n f i; 2 _ [ / / 2 4 ( 1 , 1, n ) - 4JJ 2 2 2 4 ( l , 1 , n ) + AK3KK224{1,1, (*4 n " 1 6 ) A » / A j 2 [// 2 2 0 (l,l,n) + 4JJ 4 2 2 0 (l,l,n) + 0 n AX ^K 24(l,l,n)] 3 ~ 16)A „/A Hn l 4 n n)] 2 2 n Here d\\ = d i = 0 for the resonant interaction with mode ... = 0 with the higher modes, to eliminate secular terms. 2 (2,1), and ^ 3 1 = <f4i = Regrouping cos 0 cos d>r terms in (111.5), it follows that Integrating as f (JJ/.8)Ci(Aiif)fdf, J a leads to ui — a i f , 1 TT 2 i -Afl _ [ | J J (fl i , 11, 1l )\ +1 JTJ Ti i (ft i , 11, 1l )\ +1 *qiKK {i, W H E R E ' 0 m 2 131 1,1)] A777*T7 1 Using (6), this results in = or & - 1 aifti 2 i / = v — ajfcij with 2 oj = gl/S* Similarly, /?iftiC (A if) - -f\C'*{\ur) - £i!^llQ. + A X C i ( A i i f ) ] = 0, 2 2 f 0 which, after integrating as f\lII.9)C {X f)fdf 2 21 J a f 2 gives: /?i = &i— ?21 2 (7/7.9) 171 where, b - 1 1 JJ Furthermore, from (15): 02 1, 1) + 2 A i / A 21 V "^ ' *' *) ~ ^(h 2 1 x = P-bl&, 112 0 2 fi=%*L with, ' fai' AK KK {1,1,1)] & e / 2 2 ' 3 ; and 61= W 1 6 1 gi/3' 2.2 Third Order Equation It should be recognized that equality (6) is equivalent to » . X - g ^ - ( * + •*/«>,./._... . (//7.10) 2 2 Applying (6) to relation (1.5), (111.10) to (1.8), and substituting for $ W and $( > into (1.9) yields u) y 2 {-,2/uC l ( A f) u ^ / + l l f t l [ C +(Ag +2aj A; -4«»« 1 1 a i i " A » f ^ f f ) + A^ cos wr + {-/?2iT2iC (A f) + ^ - A M c i ^ A n f ) 2 2 - . ~ -r-a^Af ) C ( A f ) ] cos 20 sin2wr + 1 1 2 + (AlTo 21 3 $( ) 2 3 n 1 + a m dr aiiAn d£ —Pn cos 0 cos wr — Q 2 2 cos 20 sin2<2>r — P 3 1 cos 30 cos CJT —... — P cos n0 cos mur — Q m cos n0 sin mwr = 0. nm 2 (777.11) n Here P ' s and Q ' s are complicated expressions representative of the various mode shapes of $ ^ and $ ^ \ and n m R m 2 Pn = -Pii + T H - Note: Pi\ = term of the no interaction case, = -E{/»^n[C7f C (A f)Co(A 0 1 + ^[C7iT C (A 1 1 n 1 1 f)C (A 2 2 n 0 r e f ) - CKAnfJCofAon*)] f) - 2 C l ( A l i r ]f 2 ( A 2 - ) f - C{(A f)C (A f)]} + ^{-WlC^CAnfjC^Anf)] + A !(3 - 7 a + 3Af^(3 n 2 2 + 6 _ C ? ( A l l f ) 1 2 W 2n 2 + SAJ^S ^ ^ - afxJCfCAnf) 7a )C (A f)Ci (A f) 2 2 1 1 1 1 1 1 ( / / j i 2 ) CKi a n d CK a r e d e f i n e d b y (III.3); a n d 2 P n = expression d u e t o the interaction w i t h second mode, n - |c (A2if)C (A3„f)] + f2ie3n[C (A - 5 2 3 C2(A2lf 2 lf l(Alwf) - ^C{(Anf)Cl(A - 3 _,_ , + ^f C 2 ( A 2 1 > / 2 3 ( A 3 - f 2 (A 2 1 f) 4 ic (A 2 2 1 C (A 2 3 = 1A 4 = ?-A CK 5 = 1A f)C (A 2 2 1 A 3n u 2 3 ^ l ( A - l f ) 21 3 n 3 3n 6 f)]} 2 1 2 2 1 f) 2 C((A f)C 2 n f)CUA 1 1 2 (A 2 1 f) 3 f iA -«n«2iAnA 2 C 2 2 ( A 2 i f ) C i ( A n f ) M C (A 3 f) 2 2 f)C (A 1 2 2 1 f)C (A 1 1 1 f) 1 1 i + - a n a 2 A 3 2 1A l n - 2 n -- 1 A 2 l 2 2 4 *2 - 2 CK 1 ( 2 + 2 where: CA- 2 f)C/f Cj(Anf)C (A f)C '(A ~ + ^ ^ ( A n ^ C K A a i f ) + o l f)C (A 2 1 f) + M i d C A x x f j C K A ^ f ) + ^ " C 21 - f ) l n c d [C {\ f)C {\ f)CK 2 2 - 5 f)] + Ci(Aiif)C (A 1 Ci(Auf)C + l r i f)Ci(A ^(A^fjCxCA^f)] + ^ [ 0 ^ ) 0 ^ ) 0 K - 2 1 a 3 a - ana iAnA i - 3ana iAnA i 2 1 A n A 2 n A n A 3 3 — -a ia n 2 9 i + -aiiai AiiAi - n 3 n n - 3 n A iA 2 3 5 -a iai A iA 2 n ; n 2 l n ; n 2 2 + ^aiiai AiiAi 2 n r e - ^a ia 2 l n A iAx ; 2 n A 2 CK 6 = £\2 _ 1A 3n 2 2 9 + -aiia nAiiA 2 3 3 n - 1 -a io;3 A iA 2 n A 2 DKi 1 2 A -- T a n ^ i A u A f ! + (1 + a i i ) o ; 2 i Q : i i A f i A = "g«21 A 4 21 11 2 A + DK = DK = ^ 4 2 3 2 A «ii(2-^«L)A iAi 2 - a\ ) x ( 2ana iAnA i; 2 2 1 - 2a ,!) + 2a\ \\ + 2 i ; x x i a I u a 2 1 A n A 2 1 . 2 1 2 3 n ; 173 Similarly, Q contains interacting terms only as, 2 2 Q22 = -^{hie^diX^CsiXs^CK, n - CKAufjC^Aanf)]+ + Cl(All ^f l(Alnf) - 3 C l ( A l i r ]f l n 2 21 n 4 4n 3 21 4 2 4n S ^ 0 2 ^ 1 C 2 ( A 2 l f ) f 2 2 1 2 ( A l l f ) 2 1 4-,2/ 1 1 f)Ci(A 1 1 DK + C ^ u ^ ^ Q _ 6 n 2 A c' * x ,) {X t) + +DKB f)C (A 2 1 f) - 2 1 V ' U «r'*fi + £>X C (A 4 , _ „ Ci (A f)C2(A if) ^ K 1 ? C 3 { 11 Ca ai |gi(Ai^)gi(Aii^2(^)} (A ir) 3C (A f)C (A f) 2 2 f ) + DK ~ ;* '' + M C ^ ( A r 3C (A if)C (A f) 7 2 3 2 f 7 x 4 ^ \2 3 _2 \ 4 2 2 1 f) -5 + -aiiai A Ai ; 4 l n 1 = - A ( l - a i ) + -X\ n x n n u 4 ln 3 n = 16anaonAnAon — 2A Aona2iaon CK 2 — 8 a n a 21 4 n AnA 4 n n 2 4n ln AiiA — 8 o : - a iQ:4nA iA - 4 a 2 n -ct ct X X ; 1 5 ) - -A| + -aiia 4 n 5 n 2 3 n 4 1\2 2 4 X f)C (A 2 1 L2/! ~2\,1\2 ^ = - A j ^ l - aj ) + - A f ^ - - a i i a : n A i i A ; 3 CK10 = - - A i ^ l - a CKu 2 1 2 /' 2 1 9 9 2 1 2 1 Ci^s = - T A I J I - a n ) - - A 4 4 CK 1 % 2 2 CK" 2V 2 1 2 ^ 4 f + ^ C (A where: n d^AnQCaCAaif) l x 4 D 21 2 ^ lC^CAn^C^Aaif) + 1 f)] 2 {- 2 1 3ri 3n 2C (A f)C (Ao r)] - 12 3 C;(A f)C['(A f)C (A f) n + - 11 J - C (A f)C (A f)]} + / ^ 2 11 - Ci(A f)C (A f ) + f rf4n[C (A f)C (A f)C rT 2 ?2 1 21 2 1 f) CxCAnfJdCAmf) 9 + f2ie n[C (A f)Co(Aonf)Cii' 4 " 3(A3 f)] + /nd3n[C (A f)C (A f)CK o - 3 ( A 3 ^f - CKAnfjCftAmf)] 1 l f t Cl(All fuesniCiiXiiVCiiXm^CKe + /iidin[Ci(Aiif)C (A f)Ctf + - Ci(A„f)Ci(A 3 - 3 n ; 2ia:iiA iAu — A 2 1 2 a A iA n 2 n - -A 2 0 n n + 2A^; +A 2 X ; 174 DK = z-aiiAnAju + Tttiia2iAf A i - - a i i a a i A i i A f j + - a f i ^ i A f x A i ; o 4 o 4 DK 5 = -A -! - a n a 6 = i A ( l + a ); 4 M DK 1 2 2 2 1 2 1 2 2 A A i; n 2 1 DK = ^ i A ^ A ^ l + 4a ) + ^ a i i o ^ A n A ^ - 3a !) 3 5 = gA^Cl - 4a|J + - a i a A i A ; DK 9 15 = r -A (l - 4a ) + — a u a 2 i A i i A i . 7 8 9 2 3 x 1 ;2 21 2 1 1 ^cc X ; 2 2 4 21 21 2 1 21 2 Setting the terms with cosflcoswr and cos20sin2u>r in (111.11) to zero, integrating as ' / (/J/.llJdCAiifJfdf and Ja I {III.ll)C {\ f)rdf; 2 21 Ja and replacing v by a i f i leads to: x 2 / i i ( t f i / i i + ^ i f l i + v ) + 7i = 0; 2 (777.13) 2 M#2<r i + £ 2 / u + /? ) =o, 2 2 2 where JFfi and F i are defined by (III.2), and A £?i = -^-[SUMZ An A E = -™-[SUMA A i A K = -^-[SUM5 A i 2 2 2 -G3- 773]; -GA- HA}; {111.14) 2 2 2 - GS]. 2 Note: SUM3 = \h (1,2,2,1) - i / i (2,1,2,1) + M 7 2 2 (2,2,1,1) + DKJs(2,2,1,1,) 7? 7^ + 7 (2,2,1,1) + - ^ / ( 2 , 2 , 1 , 1 ) + £>7r /5(2,1,2,1) - 27 (2,1,2,1) 4 5 6 3 + 7 (1,2,2,1); ST/M4 = ~\h{2,1,1,2) - ^(1,1,2,2) + 7?7T 7 (1,1,2,2) + Z?7C,7 (1,1,2,2) 7 4 5 2 2 3 + 7 (1,1,2,2)+7J>7C 7 (1,1,2,2)+ £ > i f 7 ( l , 2 , 1 , 2 ) - ^76(1,2,1,2) 4 5 5 6 5 + I/ (2,1,1,2); 7 4 5(7 Af5 = - ^ - 7 ! (2,2,2,2) + DK*I 16 2 (2,2,2,2) + DK I {2,2,2,2) 7 3 + 37 (2,2,2,2) + M 7 ( 2 , 2 , 2 , 2 ) - ^7 (2,2,2,2) 4 9 5 6 175 + §J (2,2,2,2); 4 G3 = 7 J2{[CK KK {l,n,l) n 3 -5JJ + [CK KK {l,n,l) - JJ + [CK KK {l,n,l) - 3JJ 211 5 211 6 231 J2{[ 7KK (l,n,l) CK - 231 + [CK KK {l,n,l) 4 GA = - 9JJ {l,n,l) 231 2 1 1 2 3 lln ^II {l,n,l)] 211 l3n ^/J 2 1 1 (l,n,l)]u i(l,n,l) - i// 2 3 1 (l,n,l)]n - 3JJ 132 23 (l,n,l) - (l,n,l) - 2 1 1 ^II i{l,n,l)] 1 3 2 (l,n,l) - J/ 1 3 2 l n 3 r i (l,n,l)]*y }; l B n + [CK XK 8 1 1 2 (l,n,l) +JJu (l,n,l)-/J 2 + [CK KK {1, 9 n, 1) + JJ 112 + [CK KK {l,n,l) 10 G 5 - IJ 112 - 3JJ l32 1 3 2 n 2 1 1 2 ( l , n, (l,n,l)]7 n 3 l)]n ln l)]n }; ( l , n , l ) - II {l,n, 132 = ^ { [ ^ 1 ^ 2 0 2 ( 1 ^ , 1 ) - 277 2 0 2 3n (l,n,l)] 4n 7 n with, + [ C X 7 r X 4 ( l , n , 1) - 8 J J 1 2 2 2 lin = 7 — — , /11?21 £4n_. ' l4n 2 f l and " i n = 7—-—J f) l — ^ '4n — 3 <*1 A n r , ( 1 , n , 1) - J J 242 H4 = ±[b 4ra . 5 2 2 A ^ - + x L A 4n n = 1,2,3; = TrkiTr" + ( i i u - 2aiiaaiA A )iir/r a l ) ] n }; (1,n, /llftl o ?21 2 # 242 n 2 1 2 1 2 (l,l,l)]; « \\ KK {1,1,1)]. 2 n X 112 21 2.3 Solution Stability A general solution with nonzero time derivatives and phase angles, combined with a slow time scale TJ = c / , 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: 1 fix = fu cosv?u; 3 eJi = - / i i s i n y ? u ; f 21 = & i cos £ 2 1 ; e 21 = f i sin £ 2 2 1 , to simplify cross product expressions of the form, cos(&r + £>u) sin(wr + ipu) = ^sin2(<2>r + <pu); it cos(wr + ¥ u ) c o s ( 2 u ; r - ( - c ; i ) = \ cos(wr + £ i - <pu) + ^ cos(3wr + £ i + <£>n); ? 2 2 2 176 etc., gives: -"1/1*1 + o i ( / u f 2 i - i i 2 i ) = - 2 € -"1«11 -al(/lle21 e + ll£*l) = 2 e biiffi-ell) = 4 -Pitii + (II1.15) <*ei. 2 Ml -/3 e* -6 (2/r e ) = - 4 - ^ - . 1 1 1 i n Stability is studied in the neighbourhood of the steady-state solution derived earlier by introducing a disturbance such as: /ii = / n + A e i; 1 A r e =A e n 2 A r i; & = C i + A e i; 2 and recognizing that V\ — a i f , /?i = A r 3 and = A e i; 4 Ar Now (111.15) reduces to 21 ftl / V o 2A aif: i i —2A —2aif2i o 26i/n 0 0 -2&X/U A M iA 0 A -«l/ll 2 = 0. -4A ?21 L 4A / l l / S21 (7/7.16) VA ; 4 This then yields (determinant=0) f 4 16A + b\ -4±- + Soifc!/ ! = 0, J 2 or 2 (4A) = - ( 6 j ^ + 8o 6 /f ). a 1 1 i21 (7/7.17) The solution is stable for negative (4A) , i.e., * i $ - + 8 a i * i / i i >0, 2 (777.18) m or (^) >-8^. 2 f21 1 Ol 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 O F 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 / rrij r2w r2ir / pcos9rd9dz + / / pcosOrdQdz. {IV.l) 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 d p(r,9,z). , ~\ p(r,9,z)=p(r,9,0) + z ^ \ =o+-z U=o + • {TV.2) -h Jo 2 V K l 2 rTT 2 z 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) 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: cosh A ( z + h) coshX (z + h) Xa Xa _ dz = - r - r tor M f m ; J. -fi coshA /i coshA /i n m = I( ^ + ^ i ) 2 cosh X h A for n = m; (IVA) s i n h A ( z + h) s i n h A ( z + h) Xa Xa dz= r-5 r-= for n^m; -h coshA^/i coshA h A —A , -h Jo 0 n m n n A n m — A m 2 0 m n m L n m n m 2 m n n 2 U* cosh — +X h? n) n A for Well known trigonometric relations such as: •2»r cos n9 cos mflcW = 0 = TT for n ^ m ; for n = m , 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 EEEWi*[ ( i^ii) '"ii(»,i ':)+Uii(i,i,':)]i « J k 3 a 2 ) Bi = a n A u ^ ^ ^ / x . f / o p i i o ^ p ) + »' j (») A ^t {i,p)}; 12 > = / nE { ( / n" A ) n o p [ ^ ^ - Anan^iolLp)] + ^ [ ( / n + £ ) p 3 ^ y ^ 1 + ( " / + 4 f » ) ' » ( l , P ) " (A i + 2 ?I I)"HAII* *I (1,P)]+ 2 2 2 ^ 3 Y ^ [ ( / i i + llfx i)«n(l,l,l) + ( a n A n ^ S / ? ! + 2 B 2 = / u X:{(/i i - f ? i ) n o p [ ^ f i l ^ + a n A n ^ M l , * ) ] + ^ [ ( / I ' I p 2 2 " f n ) ^ ^ £ i i A 3 (Hi) " f » ) M l . P ) + ( / « + fi i)«nAiifi» * (l,p)]+ + 2 a a ia J ^ [ ( / i i " 7fi i)«n(l,l,l) + (aiiAu) (3/ 2 2 2 u A = / i i f t i f E n i p X ^ l C i ^ i p ) -aCx(A a)] p 3 a f l > 1 1 (l,l,l)]}; + / ( i , 1) l p l +5 2 1 p - «nAnfi»*2i(i|i)]}; B = /iifci{£3 p z 7 l p ^[C.(A 1 A 8 p ) -aC.(A ,a)] 8 | 3 p + 2 J 2 1 (l,l) 2 + «uAnwt (l,l)]}; 2 1 = P + 3/ (l,p)] + « . . A i 6 ' [ ( " " ' ' ; 23 B 2 1 + / „ &£ { P ^ [ ^ + M 3 7 ")«».(l,rt + ( ' " " " / " ' " I M l . r i ] M ! + 179 + 3/ (l,p)] + ^ £ f c i o ( l , p ) + a n A n w [ ^ 2 23 + 2 r 7 4 p 1 0 ( l , p)) + £ i l l ^ [ t t 2 1 ( i , l , l) + (l 2 1 ) P ) + ^t (l,p) 23 ( d l 3 l 4 4 + a2iA2iaiiAii)iy i(l,l,l)]}; 2 P + ^*io(l,p) + « n A n ^ [ ^ 2 i ( l , p ) + ^ _ ^ [ u 2 i ( 1 ) M ) 2 3 (1,P) +274p*io(l,p)] _ ^A^anAu)^!,!,!)]}, + 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 A and B in the expressions for A4 and B. 2 2 4 2. Damping Ratio 2.1 Correction Velocity ui 2.1.1 First Order Taking u to be harmonic, i.e., u = U e 2 2 2 , equation (30a) tnWet becomes 1 d dU^ d U^ Neglecting curvature effects, i.e., ——(r—-2-—) « ——|—, and assuming the gradir dr dr dr ents perpendicular to the boundaries to be large compared to the change in other directions, i.e., 2 2 near the vertical and bottom walls, reduces (IV.5) to: ijW = s *u v d tnu) rjW^JOJ^L tnw Setting = Ae* r e 2 e } dz near r = Ro or r = R .. frV(5 ) £ dz n l ear z =-h. (77.7) near r = RQ and r = i?,-, and substituting into (IV.6) gives 180 or 17 (1) 2 = o T e i + a^e * , A r x where r lmuj Ai = e and A = —\\. 2 Introducing the boundary conditions: u\ = — at r = Ro, u = 0 far away from the solid walls, taking the real part and defining the dimensionless velocity, 2 4 *=sib <™> yields a nonzero solution near the rigid wall for the various cases considered here, (i) Nonresonant case (TV. 10) = —c ai cos(u>r + fl), n where / ai 0 Ci(X R) R u = V Y f u C l , gu sin 9 near f = R, R = 1 or a, ( u R ) g u cose J x as expressed in the (r,9,z) cylindrical coordinates, and fl = ^ / - ^ » f ° -R = 1, and r 2 V (f — a), „ „ , /— coshAi (S + h) , \/2 coshAi,7i I, for R = a, with / = v i t e . Also $71, = —J— ? —- (Appendix 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 A 1 C{(A f) lt cosh A i,- COS0 Ci(A -f) lf cosh Ai - sin0 t k \ , a n d n = -ii±il/. \/2 t 0 (ii)Resonant, No Interaction Case (Planar and Nonplanar) u 2 x) = -e [oTcos(a>T + fl) + M i n ( w r + n (7V.11) 0)], where fl and a\ are the same as for case (i), with t = 1 only, and 61 accounts for the nonplanar terms as \ 0 h = fi Ci(A J?) n R gn cos 9 criiCifAuJE^usinfl J / c . fn n C ' l i X n f l cosh A11 h Ci(Anf) sing r n ^ . cos 0 f coshAn/i 0 181 near f = R, and z = —h, respectively, (iii) Resonant, Interacting Case fiW = -[e aTcos(wr + fl) + e ^ f i ^ s i n ^ w r + fl\/2)], (TV. 12) n where a~[ and fl are identical to those of case (ii), and 6 is given by, 2 / bo 0 \ fei 21 . s 2 -021 cos 20 n 2 ^ g 21 20 2 V f2iC (A ii2)02isin20 J 2 8 cosh A21/1 C (A f) ~cos 2<r: f coshA2i/i C2{\nR) = g (A f) 0 V 2 J near f = R, and z = —h, respectively. 2.1.2 Second Order u£} Equation (30b) becomes ^2JT(2) + (u 2 r7(2) near r = R , Ri, or near z = — h by replacingd u. 2 0 dr simpler nonresonant case where $ ^ -e 2 with 2 1) -V)V*] d 2r7(2) u. 2 dz 2 (7^.13) Considering the = ^^/iiCi(A f)c7i,cos0cosit;r and lt d 1 d d 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*] = 7E u; {e T^ -[l+ 0 2 1 1 2 cos2(u;r + fl)] + ej—[cos fl + COS(2U>T + fl)] f, (7F.14) where: / ell = - ^ E hihiCx{\xiR)C {\ R) x xi 0 sin 20,01,-ffiy - g " ^ \—J2 - sin 9 2 , \ -jjj-gugij \ , , . SU9W 2», , + cos Ogxigxj J 182 62 = €21 + C22; \ sm2d gugij , ~2~R~^ ~f2 susui fn 2 eir= - ^^/i.7iyC i(Ai,-f)Ci(Aiyi2) , sin fl , 2 , R„ h + y)\ 1 f -75—I0it<7i/ + JJitfiy^] » 3 +2cos 0 7 (j y V 2 t lt J 1 sin 0 -^ltAly + cos Ogugij 2 sin 20 2iE * 3 V - 0 i t 0 i y1 *"" / 2/1 cos / J v^ ^ J Ogugij-^ Upon substitution of (IV.14) into (IV.13), an analytical solution for ?7 ^ can be found by introducing the simplifications of the Taylor series expansion near the boundary, i.e., 2 Ci(Ai.-f) = CxiXuR) + (f - R)C[(X R) U + Uf - i?) C('(A iE) + 2 lt 1 C i ( A f ) = Ci(A iE) + (f - J2)c;'(A -J2) + -(f - i2) C{"(Ai<i2) + ... lt lt (^-15) 2 lt 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 -I from previous development. Recognizing that C'^XuR) = 0 and / y/2 is a large number leads to: «Ci(Ai,-J2); Ci(Ai,r)«(f-iE)Cl'(A i2). CI(AI.T) {iv.ie) lt Taking f « J? and substituting (IV.16) into expressions for en and e yields a 2 closed form solution for u 2 . Neglecting terms of order / , applying the boundary condition u ^ = — V$^ ^ at f = R, and nondimensionalizing acccording to (IV.9) gives 2 2 fi( ) = 2 2 I{ flL (1 - e ) - S i in fl + '-f [l + [(1 - 0) sin fl - cos fl]] 2n S - sin2(u;r + f l ) ^ 2 " + sin(2u;r + fi)[% + % f l + 2_ 2_ -cos(2wr + n ) - ^ e + sin(2a;7- + n v 2 ) [ - | 2 n - l a l e " ^ + cos(2u;r + / fiv^^-e ^ }, 2 ^ 0 2 21 e 22 (iv.n) 183 where e , e n 21 and e 22 are equivalent to e n , t \ and e 2 when Ci(Xuf) 2 2 is replaced by Ci(AiiJ*), C{(A f)//v^ by C['{\ R), and f by R with U lt t 0 C (A .R) - 2 / : 2p" 2 2p sin 26g 2p \fopCo{Xo R)9op + h C {X2pR)92p cos 20 J P P 2 u 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 l a y e r " . Solving for the corresponding velocity profile however adds another degree of complexity and is not considered in this analysis. 2 109 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; ' , ( ) s i n h A ( z + h) « (z + h); / K 1 8 lt near z = —h. This leads to an expression for identical to (IV.17), with now different vectors e , e , e , and E . 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. n 2l 22 3 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: ^^I {h ^ ^ } *»J ^Uj ^ ^ } > Ed Ed u 0 0 { { )2+{ )2+{ )2]dv 2]dv dt dti near r=iE near °' * - ; (/y i9) (/y 2o) where u ^ = ( « 2 r » « 2 c 7 > « 2 « ) in the (r,0, z) coordinates. B y nondimensionalizing the above relations, taking the derivatives with respect to f and z, squaring and integrating over v, the reduced damping ratio n j is obtained as per relation (34), where the nonlinear terms are listed below for the various cases of resonant and nonresonant conditions: 2 2 r 184 , (382y/2-567) H — + 9 1 ( U ) M l 1 + / ~T~ 17^2 1 2 12 + (18^-40) + G 1 M l l M 2 2 L M ^ i 2 E ^ 42 3 + (76- *+ ( 2 2 4 5 < + 103 , 125^ + S . N u 2782>/2. „ , -—)M 1 4 M i 3 ^ M 5 2 9 w - —M N 22 g 1071 Ho"""" " ~ l25 N l l N 8 2 5 2 8 « + * )}; + 4 2 JV 2 22 w AT -NUN22 + 1231 M (199- 1 3 5 v ^ ) „ + * - — ^ M u ^ loUU « Izo + r 9 ^ 2 g »r n 3 ^ (1647 + 66y/2) n E 2 1 1 6 , (400-253v/2) , (5608 + 425y/2) + — '-M22N22 + z^rz 54 S ^ b w 3 6 2 - f 2 2 (178>/2- 105) + ^ '-MnNn M 6 32V2) JV + g V ^ (175-112^) 18 ,272 + {— f(872-225y/2) 1 + — + 1 2 4 2 S 3 30 (51U/2-657) 3 2 A A , - AA* + AA -AA ^ w 3 ~ 25 2 2 18 (382\/2 - 450) , , , , + '-M M frnl (m) Lll&Z (155 - 43V2) 8 r Li\\L>\2 H H \ ^ , f9\^w2 47!i"lO~ = , (18y/2-40) T + o 597 + To" 7 2 " 8 2 + 2 \ / 2 £ ? + 4v 2^f + 4 V 6 " £ | } ; / BB3 = 2oi£ 5, l where A A J = A A i when considering the first mode only, and L , , L , L11L22,—, aiEs are complicated expressions involving Bessel and hyperbolic function crossproducts as defined in Appendix V , such as, 2 2 2 I ! = D ! ( t , j,Jk,0i[G (.-,i fc 0 + G ( , , i , f c , 0 ( l + 3 A A , + 2A ,) 1 1 2 1 > 11 f 2 + G»(i,j,k,l)} + D2(iJ,kJ)j[ *% ' '' a Clll{l J k y 2 2 + G?(i,j k,l)(± CL l) 6 t + SA^-A , + ^-) + " ( ' ' ' + ^ y ^ / ^ " ( t , j,fc,0 a a 4 + 2J£> (t,y,A:,/) + / ^ ( t . i , * , / ) +2JD (»-,y,fc,/) 2 C H J n - 2W\ l £l 2 11 (t, y, * , /) + 2iD\ (.-, y, * , /) + I D } (t, j, K 01; l = Bs(»\ J, *> 0 j l G } (•*. i . . 0 + s ^ 4 1 f c 1 1 (», J. M R ^4(», 3, k, I) 1 G J ^ M sG^f.-.j,*,/)] + a' 4 F + ^ [ 3 I D ? { i J , k , t ) 4 m + /i?JS(«,i, *,/)]; £n£ 2 2 = D {iJ,k,t)±[G\%j,k,t) (3X - s 2 xj + D (t,y,fc,/)i[^MM _ 6 ( 3 A ? y + 2)G\ (i,j,k,l)) 1 ^ G5 (.-,i,*,0] + 1 ) iD\\{i,j,k,i) + J i i w ^ l s / u J ^ , i , * , 0 - «>ii(«\i,M 1 + L X X E 2IDH(i,j,k,l)}; = £> (t,i,2)[GGl (t,i,n) - G G 5 ( t , j , n ) 2 3 + I>8(t,J,2)[ 1 a2 A G G i 2 ( t < J > ) ] 8 E 3 y 7 + A ,)] - + IDDl°(i,j,n)} - ^[WD\ (i,j,n) - 2IDD\ {i,j,n) - 2IDD\ {i,j,n) + 2 IDD\\i,j,n) + 2 0 i3 - 2 = D {iJ,2)[GG\ {i,j,n) 9 F o[IDD\ (i,j,n) 2 t 2 2 U (,-,y,0)[GGl°(i, j,n)(l + Aj )] _ - D (i,j,0)[GG\°(iJ n){\ 2 a ^ - ( / + 0) L L £-L^GGj (.\j,n)] 2 7 2IDD {i,j\n)]; 12 (», j, n)] + £ > ( t , j, \GG\ 2 1 0 2) [ ° " C J ' n ) + i G G i ( t , j , n ) ] + [D (.-,j,0) + Z? (t,y,0)]GG (t,y,n) 2 10 9 - Fi LLioi{i,n,j) - F \^LL {i,n,j) jo M 2 X = 10 ij2 + X2X {G\U\ + G M(2, - * ) ( 1 + 2X 2 2 + aCl4( 4 Alia) { g a 2 2 XX JJi i{i,n,j)}; 2 ) + Xt (2s + s )} + G | * s ^ + G i M M - * )(^r + ^ i 2 x 2 + n( 2 A 2 5 l + * )1 2 + "4} + , ^ t o ( 2 ^ 1 + •?) + 2IDl\2s - s ) <*• 4 cosh X h G + 2s J P * 2 I £ } 1 2 2 K xx + s {IDl - 2ID\ + 2ID\ + ID] )). 2 l X l 1 Here: «i = fii ~ & M _=C ( A 2 2 2 1 1 a n M 22 *2 = fh + fiiJ )G ) , +, G i i o( 2, s + , s» - j ( A n [G s )\|+agi^AuttjCy'CAna) 2 / 2 [G»4 + g . a Mxx d [ r l l x 1 2 r 2 ^ ! + 4)) + f 2 2 j 8 x \\ Ai^l + A)™? + s IDll}; 4cosh X h A X 2 xx = i g i ' ( A ) g i ( A i i ) { G J ^ - G\ \2s {\ + X u 3 2 l 2 ) + s X }} 2 XX 2 2 xx 186 4 cosh An/i = 4f &{c«(A„)(G» + 2G» + G* ) + + 1 n ^ + Gi )}; 1 M| = 2/? ? {Gi'[C?(A 2 lf 1 At: f cosh Ax + 2 13 c f (A,,) + C } ( * . . « ) C f ( * » « ) ) a J i y 4 M Il) 1 0 f) = / l A { c f ( A n ) [ G » + G^CA , - 2A 4 + G^Af, - - 1) + % o a a + - 1) + G ] + aCf ( A „ « ) [ ^ 3 h X \ AID? cosh An/i - 2ID? + J D l - 2ID\ + 2ID\ + ID} ]}; 1 l l 1 Mh = fhtii{lCl{Xii)CZ (\n) + Cf (AnajC^CAnoJajGj a cosh A n / i M 1 3 M 5 2 } = / ? f ? i { c J ( A „ ) C ( A ) G | ( l - A ) + aCf (AnajCflAna) 1 l w 1 11 2 ^ T Au/i T T ^ a - Ah) + -cosh M , M% and M12M42 are identical to M permuting 52 and si; 2 2 2 M E 12 3 1 1 2 X X " / , M £ 2 2 " " > ^>} 7D ; and M11M22, respectively, after = X;{/2nS2[C2(A )C (A )[GGl (l,l,n) - GG* (l,l,n) 2 2n 2 n 2 n + g g P(M.") a ( 1 - A?,)] + a t 7,(A .«)C?(A a V 2 2 )[°S^lil=) l l 0 GGj°(l,l,n)(l + A J + aCo(A a)C (A a)GGi (l,l,n)(4 2 0n 2 0 11 2 cosh An/i 2coshA n'i - 2IDD\ (\, l,n) - 2mE>4- (l, l,n) + 2JZ>£>| (1, l,n)] L 2 2 + [ °" [IDD\ (l,l,n) coshAon'i 2f Sl t 2 2 0 + IDDl°(l,l,n)}} }; J 187 + 6 ? G i ( l , l , n ) ( l - X )\ + a C ( A a ) C ( A a ) [ a + 2 G G i ' ( l , l , . ) 2 + n G 2 , G 2 ( l l n 2n ) ( 1 l _ A 2 11 h ) "( » . 2 C g 1 1 w ) 1 T [ J ^ 1 ( 1 , 1,n) - m D $ ( l , 1 , n ) 2 i 9 2 cosh An/icosh A / i 2ri + 2IDD {l, 1,n) - 2 7 Z ? £ > ] ( 1 , 1 , n) - 2 / £ > £ > i ( l , 1,n)}}; = ^{^[^'(AujC^AxOdCA^tGGl ^,!^) + 4 (M,")j l2 M 4 2 ^ 3 2 2 2 G G 2 n + aCriAxx^C^AnajdCA^a)!^ ^ '^ 2 + 1 + [Ci'CAujC^AujCoCAon) + aCKAuaJdCAnajCoCAona)] GG 4 (l,l,n)/ 0 0 r t 3x- * " J »* [|LL cosh A n / i cosh A i / i * / a 2 L + J J m (1, n, 1)] + M 5 2 ^4 1 2 1 (l,n,l) 2 ° L L i (1, n, 1)1; COSliAon" /ll?llX)f2l{[ l( ll) 'i ( ll)C2(A n) n = C / A w S l C 1 0 , A 2 + aC (A a)Ci'(A o)C (A a)]GGl (l,l,n) 1 11 n 2 2 2n ^2, t" J £ W l , n , l ) + 2JJ cosh A n n cosh A a A 121 2 n ^11 = ?2 i{c (A )[76G 4 2 4 21 /~<22 [76% + G a ^2 2 2 = f i{c 2 + G (h 22 2 2 2 (A on ^ , + 2 4 a 2 1 X 3 2 1 2 2 1 22 2 22 2 + 2 1 1 2 2 + ^ID 2 22 21 22 ] A 2 + IDf + 4ID\ + ID\ j j 2 2 4cosh A i i n c o s h A i « 2 2 2 2 1 2 2 2 2 2 1 22 + 16JZ) . ]}; 2 2 X + 1) + a C ( A „ a ) C ( A a ) G l ( *f + ^ ( A , + ^ ) 2 o G ]+aC|(A ia)C2 (A ia)[4^| 4ID }} ' + 2 1 21 l—^\llD cosh A /i4 4 2 { c ( A ) G ( A ) G l ( | l + 1)(A 1 + 2 4 i /~<22 4 2 MuiVu = 2/ f - 32) + 4 G ] + a C ( A a ) X - 21D\ + 8IDl 22 21 2 2 2 ™) + 4%) a' a 2 ) c f (A )[4G ^ [ I D + G l + 2 cosh A i / i * 2 21 4 + IDf + SID + 2A 4 \2 q 2 6 + 2/P| 22 (l,n,l)]}; ' ' 2 2 2 M„N = -2/ cr lt 2 u 2 { c ( A ) C i ' ( A ) C ( A ) G ^ ( i l + 1) 1 1 11 11 2 2 A 21 + aC (A a)C{'(A„a)C (A a)G^( |i + 1 ) 1 2 11 1. [78(2,1,2,1)] 2 A 21 4 cosh A11 h cosh A 1 2 M„AT 22 2 2 = -/ 2 {c (A )C '(A )C (A )G^ i i^tii 2 1 n f 1 1 2 2 1 2 2 2 1 + aC (Aiia)C (A io)C (A ia)G2 ^ 2 +A MN 22 2 2 2 2 2 A ^—- 1 1 [7 (l,2,l,2)+/ (l,2,l,2)]y 2 8 6 cosh \\\h cosh X \h 2 2 > 2 = ^ -V{ [ C ( A ) C i ' ( A ) C ( A ) C ' ( A ) 2 + aC (A a)C{'(A a)C (A a)C '(A a)]Gl A A ^ 7^ }; cosh An/icosh A i/i 5 = <T {C (A )C '(A )[16G + G ( ^ - 1)] r 22 2 1 11 1 11 11 2 11 2 21 2 2 21 21 2 21 2 2 1 2 2 NN n 4 2 1 22 2 3 21 2 22 21 2 2 + aCf(A a)C '(A a)[i^ + G (^A 2 1 2 2 2 2 1 2 1 - 1)] _ 2 [77? /2 + 87 (2,2,2,2) - 2ID\\ - 8//? ] |, 22 22 6 A cosh X ih 4 iV £ = 1 1 7 -<r 2 2 X;{ '2 (A2i)[d4n[8GG (l,l,n)-8GG (l,l,n) c 1 > 2 2 24 24 n + GG2 (l,l,n)(2 4 A | )]C (A ) + i 4 41 C4n GG (l,l,n)(2 20 + ^ - ) G ( A ) ] +aC (A a) [d [8GG (l, l , n ) - * 0 + GG 2 4 2 0n 21 24 4n G G ? £ > *» ( l , l , n ) ( l - Mi)]C (A a) + e G G ( l , l , » ) ( J 4 + ^^(Aona)] 2 | J 4n ,[ 1 2 0 4 n d 4 n .[ ^ ^ ( M . n ) cosh X ih '•coshA n 2 2 ^ 2 4n + 47DP^ (1,1, n) - 47£>7?| (1, l,n) - 167P7? (1, l,n)] l l , 1, n) + 4 / £ £ > ( l , 1 , n)]j }; •* 4 4 24 J f r W + -cosh Xo h 20 J n N E 22 7 = cr X;{c (A )C '(A )[-rf4nG (A )[8GG (l,l,n) 2 2 x 21 2 21 4 24 4ri n , GG^ (l,l,n) _,_ 8GG (l,l,n) , G G ( l , l , n + 2 J ~ 4nG (A o)a[ 14 1 d r 4 4n 24 2 4 189 + e GGl°[C {Xon) 4n + aC {X a)] 0 0 J [ L L 2 4 2 d.An ^ cosh A21/1 c o s h A 4 / i 121 L n - 8 J J « ( l , » , 1)] + — ^ - L W l , n , 1)]] }. coshAo n ' 1 , n , 1 ) 9 * A2 + 0n 2 J n Now, recognizing g g « 4t7 <7 due to the resonant interaction gives: x 2 x 2 G\ = / i i ? i { c ( A ) C ( A ) [ 2 G » + ^ - + (50 + 32A )GI } 2 Nh a 2 2 a a ai n n + aC (A a)C (A a)[2^ + ^ 2 2 2 1 + ( j £ + 32A )^ ] 4 n 1 + ^(1,2,1,2) cosh Xnh cosh X ih 2 2 2 1 / (2,1,2,1) | 8 8 2 2 2 + / (1,1,2,2) + |/ (2,2,1,1) + 107 (1,1,2,2) - 4/ (l,2,l,2) 8 9 + ^ID? - 8ID N? = o + 21 + 2 cosh X %h cosh Xnh 2 12 ' 2 2 = / i f i { G ( A ) C l ' ( A ) ( 8 G l + 2G* ) + a C ( A a ) C ; ' ( A a ) ( ? | £ 1 2 2 2 2 2 21 A 2 21 2 a ^ [^(2,2, l , l ) + 2/ (l,2,l,2)]| 2 2 7 2 C 2 2cosh A incosh Xnh N 2 2 A 2 7 1 7 ] + 4j^5(l,l,2,2)+/ (1,2,1,2)]|, + 2G* ) + N MD }}; « 12 fl^{cf{X ,)C {Xu)\^^ 2 2 9 -I- C 12 2 + ^ + AID 21 C 2 Ni - ID 12 ( 9 = / 1 2 f 1 2 2 2 2 2 a i n ; > { c ( A ) C ' ( A ) C ( A ) G l ( i + 4A ) 1 2 21 2 2 21 2 11 2 1 + aC (A a)C '(A a)C (A a)Gl (^ + 4A J 2 _,_ , A T 2 21 1 2 2 11 2 2 [/ (1,2,1,2) - 2/ (2,1,2,1) - 4/ (l, 2,1,2)] ^ 2 2 21 6 7 1 8 n 5 s I ) 31 2 cosh A n a cosh A i / i - 1 2 N 7 1 N 8 2 = / <r {c (A )Ci'(A )C (A )(4Gl + 40G^ ) + aG (A a)Ci'(A a)C (A a)(^_ + ^ - ) 2 1 1 2 2 1 2 2 21 11 21 1 11 2 11 1 2 11 [/ (2,2,1,1) + 5/ (l, 2,1,2) - 8/ (l, 2,1,2) - 4/ (l, 2,1,2)] ^ o o * I' 2 cosh A i/icosh A n a ' f {[C '(A )C; (A )C (A )G (A ) 2 6 6 7 '11 4 s 2 N 7 2 N 8 2 = / + 1 2 1 2 2 1 2 , 21 11 2 21 1 11 aC (A a)Ci'(A a)C (A a)C (A a)]8G} 2 + \l \ 2 L Ll , 21 11 2 21 h{W,2) 1 | 11 2 ; 2 cosh A i h cosh A11 h > 2 l E 2 2 = EEfW^^P) + 2/on/o / *(0,n,p)}; P , 190 where F* is a function defined as: \n,p) = Ci{\ )Ci{\i )[i 0ii{n,p) in [ t2/? " ( 2 + # -(n,p)] + 2 P + /%(n,p)] + i2„(n,p) + n,P) aCi{X a)Ci{X a) t in ip i J2 (n,p); 2 ii Of El n p El = Y, Y , l n » P * C ' > P) + i« ip^* ( > > P)1; 1 F e N e 3 N El = Y^2[d3nd3pF*{3,n,p) n p + e e pF*(l,n,p)}; E + e e F*(0,n,p)}; = ^2^2[d4nd F*(4,n,p) n p 2 aE x d i n d i 4p 3n 3 4n 4p = /iiX)di„J *(l,l,n). ? , 5 n /? and f3* functions are defined in Appendix V . 2.2.2 Free Surface Contibution The vector n = (———, or r)f in terms of $ and d x/dt term of equation (32) yields 2 —f, l) is normal to the free surface. Expressing r do using relation (1.2), and substituting into the second 2 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 equalities: CJ flT ( f , d o d + 6 A A A n = 2 7 /o a i l A l l 8 - j(^ dr m « d K-3r~) ( 1 ) d { v m ) dz dr 2 m m • VmrdOdrdzldr; /V JV \J m ) 4 - 2 + ^ ( - {IV.22) a i l A l 1 ^ ^ w dzdr + « ]r<Wdf + f {Vm) rd6drdz\dT, 2 l l A l l ( _ ) W > {IV.2Z) For the particular cases (n = 1,2,3): (i) BBBx = 0; AAA\ = 2VS(f ,fom) + VS(f J ) 0n E E t 2n - (aiiA ) a; 2m 2 n 3 E a i . - A i . 7 i . 7 i y [ 2 / o » ^ n o ( l , 1, n) + f KK {l, 2n j n aiiAn E 64 E E t J + E fuMikfu K {won 1, n)] 112 (t, j, k, i) I ID\l(i,j,kJ)}+6ID\l(i,j,kJ)+ID\l(i,^ aifcauAi,AiyAijfcAii) - 6w (ai Ai,A y + a n A i y A i J a n A u - ^{anXuyiauaijXHXij)} + 6ID\l{i,j,k,l)[3a X ct X - w a A (a A + a X \ + 2ID {i,j,k,l)[3ct X auX - w anAu(ai A - + axyAxy)]}, 2 2 t lk 2 11 11 lt xj 10 = 2VS{fon,fom) + VS(f ,f ) + VS{$ ,t ) 2 (ii) BBB 2 AAA 2 li t xi lk lk lk u u u lt = 0; 2n 2m 2n [Von{fu - c )KK (l,l,n) n + f (f^ no " ^^{Mfti f + iii) + 2 1 lia 4 1 1 1 1 4 1 3 n + ID\\) + 2[3(/ 4 11 n + ? )Jirjr (l,l,n)] 2n 2 / i i f i i ] / ^ + (aiiA ) /P ^[3(/ 2 - (a A w) 2m + + ^(3 - ^ - 12a; ) + 2 f t f? "ii 4 Q/\2 (3 - - 5 - - 12a; )] + ( a u A n ) / ! ? } * ^ + \)(3 - 2a; ) 4 2 + 4/iVi i(3 - 4a; )] + ( a A ) / D ^ [ 2 ( / 2 BBB 3 3 2 11 1 4 + C )(3 - 2a; ) 4 1 2 a 2 1 ln lm 2 2n 2 2 - 6(a A u;) /i i<r2iii'ii'i2i(l, 1,1); = AAA\ + VS{d , d ) + VS{e , e ) + + VS(d , d ) + 2VS{e , e ) + VS{d , 11 AAA 11 + 8/ Cu62(l + 2a; )], = 2[fnWS(d ,d ) + f i(/ ,/ m)](l + a n A „ w ) 2 (iii) 2 2 fl 3 11 2 ln 3n lm ln 3m 4n VS(e ,e ) d ) lm 4m 3n 4n 3m 4m - (anA w) ^{32c:| [2e4nii'ii'22o(l,l,n) + d 4 n i t t r 3 11 1 ( l , 1, n)]+ 2 2 4 n \hi$2i[{d -Ze )KK {\,l,n) ln KK (1, 3n + (e l2l l,n)]} - £ ^ { c ^ [ 9 ( J £ > 123 2 2 ln + - 3d ) 3n 16/Z? ) + 2 4 J £ > + 22 22 ( a A ) / £ > ( 1 8 - 12a; ) + ( a A ) / £ > ( 2 4 - 16a; )] 21 4 21 22 2 + fxx&W&xl 21 2 22 2 + 4ID\1) + SID\ + 2ID\\ + 2 - 34a; - ^a; ) + 4 ( a A ) / £ > 2 21 4 11 11 2 1 12 7 K>(a X ) ID\l(2 ( 8 - 2a; ) + 4 ( a 2 1 1 A n 1 1 ) J^(i 4 2 I - 32a; ) + 4 0 ( a A ) / D 2 4 n 11 11 2 1 12 ( l - tf) + lS{ct^ii?ID\l\, 192 where: AAA\ = AAA\ for the case where i = j = k — I = 1; VS(r ,r ) = (auAu^ ) £ [ ' " r ? ^ f ] + WS(r ,r ) = ^^r r pij{n,m)[IAjj{n,m) in in im im 2 in n 2 n WS{r r ); ini + j J A j{n,m) 2 irn m with j = i when r = f , f , d , d , d , and j = 0,1,3 for r,„ = e , e , and e i , respectively. in on 2n 4n ln 3rv 3n 4n n im } + a Ay ]; 2 n n APPENDIX V: USEFUL BESSEL A N D HYPERBOLIC FUNCTION RELATIONS A N D DEFINITIONS 1. Bessel Functions 1.1 Orthonogality Condition C„(A f)C (A n p m m g f ) f c i f = 0, A ^2 where A n p if A ' = i{(A ^ 2 p n p ^ A ,; m A p A g, n m - n )C (A 2 2 n p ) - (A a 2 p 2 - n )C (A 2 2 1.2 Cross-Product Integrals ,j,k,l) = /'cnAarjC^AyifjCUAfcxfjC^AnfJfcif; Ja ,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) IDr(i,3\k,l) is omitted when i = j = k = I = 1. = / C;'(A f)C;(A Ja 1 m n y f)C^(A m f c f)C^(A m / f)fcif; np ID\ i,j,k,l) m m n n n n m n n n n m mk m (X ID?™' i,j,k,l) C' ( A f ) C ( A r r ) C ID ' ij,k,l) Cn{Xnir)C (Xnjf)C (X f)C i,j,k,l) TTjnmt i,j,k,l) m ID™> TD nmt n j n n m -C mk n r n (A C' {*ni?)C {X r)C' {X r)C n n ni m mk C {Xnir)C {Xnjf)C {X f)C n n m Am/r)^; mk f ) C„ A m f j — ; m f c f)C mk X if)fdf; m r X if)fdr] m n Xmlf)^-; r r • i i,j,k,l) C' (X f)C {X f)C (X f)C i,j,k,l) C {X if)C {X f)C {X r)C n n = i,j,k,l) ni n n nj n m nj mk m mk A /r)^; m r n C' {X if)C {X f)C' {X f)C„, X i r ) — ; r = / i,j,k,l) ~dr n Cn(An^)C;(A yf)C n Ja ID™> n i n m r mk C (X if)C (X jf)C' (X f)C„ ID% > i,3,k,l) d C' (\ if) C ( A y f) C' (X f ) C„ A ,f)-- -i: n m , i,j,k,l) ID% ' ^ Am/rj—; r Ja nm \ Xmir)—; r r Ja ID? D n , i,j,k,l) I C"(\nir)C (\n]r)C' (\ kr)C -f n n nj m m mk / c ; ( A « f ) c : ( A , , y f ) C ^ ( A * f ) c ; X r)rdf; 1 < ml m Ja TTjnmi jr\nmi -i: C (X f)C {X f)C (X f)C i,j,k,l) ni n nj m mk X r)rdr; ml r C {X f)C {X f)C {X f)C 16 ij,k,l) ID i™' ij,k,l) = / ^ n m p ( ' i J> k Ja J A / n n t ni n nj m mk X r)rdr; ml r C^X f)C {X f)C {X f)C ni n nj KKnmp{i,j\ k t k t mk r C' {X f)C' {X f)C {X f)fdf; -i: •i. -i: n ni m mj p pk 1 •I Jnmp{})3, m jj. Cn[Xnif)Cm{X jf)Cp(X f)—; m pk C (X if)C (X f)C (X r)fdf; n n m mj p pk \ *\ Amir)—', r df 195 f IDD\ {i,j,n) p C;'(A f)c;(A f)C (A f)fdf; gt w p pn J a /'^(A^C^A^fjC^A^f)^; IDDl (i,j,n) P Ja IDD /{iJ,n) q ' /'cjCA^C^A^fjC.lA^f)^; IDD /{i,j,n) q /'cjA^C^A^fjCpCA^f)^; IDD {i,j,n) qp /'^(A^C^A^fjC^Apnf)^; J a IDD (i,j,n) q p 6 -I '^(A^C^A^fjCpCA^f)^; = f C {X f)C (X f)fdf; IA {i,j) nm n ni m mj J a JA (i,j) C (A f)C (A yr)—. = I nm n m m m Ja r 1.3 Simple Cross-Products Di{iJ,k,l = E E E E /i./iy/iJk/uC^AxOC^A.yJdCA.OCiCAw); i j k I D {i,j,k,l = E E E E /i.7ij/i*/i^i(i»)i(»y)C'i(A a)C(A,,a)a; » y * j D {i,j,k,l = E E E E /w/iy/iJk/ii^AiOCiCAyjC^A.-OCtCA,-,); D (i,j,k,l = E E E E hihjhkhiC'l{\ua)C i j k l 2 3 4 A a C A a tfc 1 (Xi^C'^Xi^C^Xua)^ x D (i,j,k,l 5 » y fc t D {i,j,k,l 6 = E E E E /i.7iy/i*/iiC'i(A a)C (A ya)C; (A a)C (A ,a)a; i y fc J / li 1 1 lfc = E E E /i»7iy/pi^'i(i»)i(iy)^'p(pi)! » y i A D (i,j,n p g A A = E E E /i«7iy/pn'i(Aia)Ci(Aiya)C(A a)a; * y = E E E /i.7iy/pnCi(A )Ci'(A y)C (A ); i y n c t p pri n Dl(i,j,n D p 0 {i,j,n lt 1 p pn = E E E /i.7iy/pnCi(A ,a)Ci'(A ya)C (A a)a; * y 1 n 1 p pn 1 1 tnm{i,p) — C (X i)C (X ) n n m i; „\ — n (\ - mp \r> (\ aC (X ia)C (X a); n \ n m C (A a)C (A w n t m mp m p a) CL ttnm[i,p) c w [i,p) = \ C'„(A rit )C (A m mp ) + aC (X ia)C (X a); n m mp Unm{i>J,P) \ r C (A a)C (A a) + a C n(A a)C (A ya) C ( A = C (A )C (A„y)C (A ) - Wnm{hj,p) — nrn n t\ \r> (\ = C [X i)C (X ) n n , n n m m n n t m m p mp n m m m p n n m m p a) CL C (X i)C (X j)C (X ) n n n n m - mp aC (X a)C (X ja)C (X a). n ni n n m mp 2. Hyperbolic Functions 2.1 Definitions and Cross-Product Integrals _ cosh A 9nm Fijki n m ( z + h] cosh X h nm = iE jE E iE c o s n k F G^ (i,J,k,l m m (i,j,k,l A\%h cosh Xijh cosh Xi^h cosh Xnh flifljflp = E E E cosh Xnh cosh Aiyft. cosh Xi h *' j Gl fufijfikfii p P f° J ^9ni9nj9mk9mldz', k9midz; 9ni9nj& / J-k Gl {i,j,k,t m f° / §ni9nj9mk9mldz\ J-h GGl*(i,j,n f° / J—h GGl (i,j\n P GGl (i,j,n P GGl (i,j,n P / -h 9qi§q]9pndz; f° / J —h / 9qi9qj9pndz\ J-K 9qi9qj9pndz; 9qi9qj9pndz. 197 2.2 Simple Cross-Products Pnm{hP) = Xni&ni ^2 2 cosh A 2 a* t: ~\ Pnm\ >P) l A m m ' /i m A A m m 72 A ^ m p a m nt A m P ' A - — A p m ) m ^ f ' \2 A ^ m for + "t—]j niA p(A amp —A - X pCt p _ ^2 I O r 5 ,^ ™ ^ mp, A A mp cosh A„i/l 4- ^211 A nt for A -A 2.3 Combinations of Hyperbolic and Bessel Function Cross-Products k (i,p) nm lnm{i,P) kk (i,p) nm UnmihP) y) 12 = P (i,P)tnm(i,P)', = Pnm[i,P)v m{i,p); nm n = P {i,p)tt (i,p); = 0nm(i,p)w (i,p); = nm nm nm ^nm(')j) cosh X {h cosh A m j /i n 1 (i,j) JA nrn «72 nm (t,j) cosh Ant'/i cosh A y/i m 198 A P P E N D I X 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. damper damper k H A A H I i m x - € sincj t 0 Fig. VI-1 P 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 \M /Mi\ and r/ j, respectively. a rj k(x-y) •C (x-y) d 1 Fig. VI-2 Forces acting on the moving base Standard vibration theory gives (x — y), and in turn yields: .M CJ — 1 Mi T]CJ 2 a (w -l) 2 2 + (2r u; ) ' 7 2 2 fo,.M2' (o> - l-,\2 ) +, (2r?u;) ' ^,1 = 773 / r > 2 ' 2 2 ( V L 1 ) where: —; w = y/k/m, n 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 rj ,i shows higher and narrower peaks for decreasing n (Fig. VI-3b). The energy ratio is also derived as r 8TT Er > l = r (VI.2) 'KTT (1 + l / w ) w ' 2 l showing an increase with CJ for a given 77 (Fig. VI-3c). 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: 1 ] F e e' m. i w 0 aero, model ,- , ' Y w_t we T / V =Y eiV 2 damping device Fig. VI-4 e 2 Mechanical representation of the transmission line test arrangement The standard formulation for such problems leads to the following eigenvalue equation , 110 - U?{UJI + 1 + s) + w* = 0, {VI.3) , 2 <*>1 2 *1 2 2 , W where s = ; Uo = — ; with UJ = ,u = ; and u> = —. mi u)2 mi m.2 u With the experimental determination of uj\, u>, and w , the inertia ratio 5 can subsequently be determined from (VI.3) as, , = - A - l ) ( £ f - 1 ) . (VIA) m k N X 2 n 2 2 n Assuming C\ to be small, the response of the model is, _Fg ,r(i-P ) + a 3 y a /a (VI.5) Ijj, where u> = — ; A is the left hand side of relation (VI.3) when substituting oj by n U2 Cd2 U; r\2 = m2W2 and s = o ~~ **%(! + s). At resonance, (VI.5) reduces to w 0 _ ^ 0 [ ^ + (l-^) ] *i — 1 T=—i ki \ns \ 2 Y 0 1 / 2 2 o> L J w (VI.6) Fig. VI-3 One-degree-of-freedom system characteristics at resonance showing: (a) | M / M , | ; (b) i , , ; (c) E a P| r>l to o o 201 as A = 0 and w = U . With the parameters of (VI.3), Y\ is found to be a function of 57 5 o d s, for a model with given m i , k\, and exciting force Fo. Thus the damper design can be optimized. n w a n 2 Note: For the experiment, m i = l.GQQKg. m 2 = M + d m P ^ ' where: Md = damper mass; m , l = damper plate mass and length, respectively; L = distance from damper center of gravity to system center of rotation. p p According to the inertia forces in the vertical direction (Fig. VI-6), F = m y = M LO 2 F 2 d p m l 0/2 p Fig. VI-5 + m ^0. p Force diagram for the damping device (VI. 7) 202 A P P E N D I X VII: WIND-INDUCED OSCILLATION AMPLITUDE CALCULATION 1. Galloping Theory with Equivalent Damping According to Parkinson's theory , the amplitude Y is given by 111 ^ ^ = n 1 ( 1 _£ | < ... ^ ) y 2 + y + , + ( , m ) where: N = 1,3,5,..; n = ^ ^ - ; U ^ ' nAi A i = 1st coefficient of the polynomial fit for Cf ; . . . , B N = integration constants times higher coefficients of C / . 0 2M y I?3, y A limit cycle is reached for dY 2 —r- = 0, t l dr (1 - |-) d[dY ldT)s 1 2 e stable for + 2B (^) i.e., + ... + ^ f ^ ) " " 2 Z '— - < 0, dY* 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 = D + D Y + D Y 0 X 2 2 + ... + D Y . M {VII.Z) M 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)] - —^[DiY + 2D Y 1 2 2 + ... + MD Y \ M M < 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 . 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 . They determined that a damping ratio n = 2.2% is required to keep Y < 0.1, with the response approximately proportional to nj / 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, 112-113 114 8 1 2 Mi ^W (VI1.5) = l e where the modal mass M e = i m ^ ^ r - ^ d z ( -) F/J 6 203 is found to be 30 000 K g . 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: u « w . e (VII.7) n Now, w = [^tanhAuft] thus, R= e A l i y 0 (2TT/II) JRo = 0.836 d = 0.327 2 1 7 2 , (V//.8) tanh^f. m m (VII.9) 5.1 as A n = 1.255 for this damper, which yields with a container liquid mass mj given by, m/ = pnr(j)(^)d . (7/7.10) 8 for oil, mi = 180.2Kg. From Chapter 3, rj i > 1.0 for to/d < 1.0 and CJ « 1.0. A t Y = 0.1 corresponding to rjg = 0.022, eo/d = 1.53 and therefore it is assumed that rj i < 1.0 (the variation with amplitude is not too pronounced here and the steady-state results should apply reasonably well). Taking n i « 0.7 leads to ri r> rt Mi = 943 K g , 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 n = 0.04 is expected (from Y proportional to relationship), eo/d is then 0.92 and n i is of order 1, which yields Mi = 1200 K g , or about 7 units. This compares advantageously with the 1500 K g pendulum tuned mass damper proposed in reference 114. a rt 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, R = 0.302 m, d = 0.209 m, mi = 21.7 K g (oil), and Mi = 2200 K g . 0 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 Fig. VII-1 arrangement Steel chimney with 6 nutation dampers damper ring •frnrrtT Fig. VII-2 Steel chimney with nutation damper ring 1 LIST O F P U B L I S H E D ARTICLES • 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., " O n the Control of Instabilities in Fluid-Structure Interaction Problems", Proceedings of the 2nd International Symposium on Structural Control, Waterloo, Canada, July 1985, Editor: H . H . E . Leipholz, pp. 473495. • Modi, V . J . , Welt, F., and Irani, M . B . , " O n the Nutation Damping of FluidStructure Interaction Instabilities and its Application to Marine Risers", Fifth International Symposium and Exhibition on Offshore Mechanics and Artie Engineering, Tokyo, Japan, April 13-18, 1986, Paper No. OMAE-1197; also Proceedings 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., " O n 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 Dampers", 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 . , " O n 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 Offshore 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 Engineering, 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 Internal Flow in the Presence of Ocean Waves and Currents", Proceedings of the IMACS/IFAC International Symposium on Modelling and Simulation of Distributed Parameter Systems, Hiroshima, Japan, Oct. 1987, pp. 329-336.
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A study of nutation dampers with application to wind induced oscillations Welt, François 1988
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Title | A study of nutation dampers with application to wind induced oscillations |
Creator |
Welt, François |
Publisher | University of British Columbia |
Date Issued | 1988 |
Description | 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. |
Subject |
Oscillations Winds |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-10-21 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0098275 |
URI | http://hdl.handle.net/2429/29451 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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