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A study of nutation dampers with application to wind induced oscillations Welt, François 1983

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A PARAMETRIC STUDY OF NUTATION DAMPERS by FRANCOIS WELT B.Sc.,Ecole Polytechnique De Montreal,1979 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department Of Mechanical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July 1983 © Francois Welt, 1983 In presenting this thesis in p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t freely available for reference and study. I further agree that permission for extensive copying of th i s thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It is understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Mechanical Engineering The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: July 26, 1983 i i ABSTRACT Performance of a set of torus-shaped nutation dampers, suitable for arresting r e l a t i v e l y low frequency o s c i l l a t i o n s , i s studied experimentally using a simple test f a c i l i t y . More important parameters a f f e c t i n g the damper performance are established and their influence assessed for a variety of configurations with the aim to arrive at more promising geometries. Results suggest damping c h a r a c t e r i s t i c s to be p a r t i c u l a r l y sensitive to physical properties of the l i q u i d used, i t s height in the torus, damper geometry, and dynamical parameters representing amplitude and frequency. Among the configurations studied, dampers with perforated inside tubes, b a f f l e s , horizontal layers and f l o a t i n g rectangular pieces of wood in flow showed most favorable performance in terms of energy dissipated per unit volume. Such nutation dampers are l i k e l y to be suitable in tackling a variety of v i b r a t i o n a l problems of i n d u s t r i a l aerodynamics, earthquake engineering and off-shore structures. i i i TABLE OF CONTENTS Chapter Page 1 INTRODUCTION 1 1.1 Wind-Induced O s c i l l a t i o n s 1 1.2 Suppression of Wind-Induced O s c i l l a t i o n s 4 1.3 Scope of the Investigation 11 2 NUTATION DAMPERS 13 2.1 Damper Description 13 2.2 Parameters 14 2.3 Dimensionless Parameters 16 2.4 C r i t e r i a 19 3 TEST PROCEDURE 2 3 3.1 Test F a c i l i t y 23 3.2 Models 26 3.3 Instrumentation and Calibration . . . . 29 3.4 Experimental Procedure . . . . . . . 30 4 RESULTS AND DISCUSSION 36 4.1 Common Parameters 36 4.2 Specific Parameters 53 5 CLOSING COMMENTS 72 5.1 Conclusions 72 5.2 Recommendations for Future Work . . . . 74 BIBLIOGRAPHY . 7 6 APPENDICES 79 I SYSTEM CHARACTERISTICS 79 i v Chapter Page APPENDICES 7 9 II ENERGY DISSIPATION VERSUS DAMPING 86 III DETERMINATION OF LIQUID NATURAL FREQUENCY . . 88 V LIST OF TABLES Tables Page I Damper models used in the test program . . . 27 II Spring and system s t i f f n e s s e s 81 III Variation of l i q u i d natural frequency with l i q u i d height for dampers 2 and 8 91 v i LIST OF FIGURES Figure Page 1 Wind induced i n s t a b i l i t i e s of bluff bodies: (a) vortex resonance 2 (b) galloping 3 2 Several t y p i c a l devices aimed at modifying character of the flow to minimize i n s t a b i l i t y 5 3 Maximum amplitude of o s c i l l a t i o n (peak to peak) against the damping c o e f f i c i e n t for c i r c u l a r cylinder f i t t e d with several damping devices 8 4 Several devices used in practice to promote energy d i s s i p a t i o n 9 5 A schematic diagram showing the nutation damper and i t s application 13 6 A schematic diagram of the s t a t i c stand test f a c i l i t y 24 7 Static stand test f a c i l i t y used during the wind tunnel tests 25 8 Typical free o s c i l l a t i o n signal traced by a chart recorder at a frequency of 3Hz . . . . 29 9 Calib r a t i o n plots showing displacement against e f f e c t i v e spring s t i f f n e s s . . . . 31 10 Variation of the damping r a t i o with amplitude of o s c i l l a t i o n s as affected by the frequency and l i q u i d height. Note the effect of turbulence at higher Reynolds number 37 11 Energy d i s s i p a t i o n as affected by the damper amplitude and Reynolds number . . . . 38 12 Ef f e c t of i r , = a/d as recorded during an a i r bearing s t a t i c stand test 39 13 Typical amplitude decay plots for the system f i t t e d with a nutation damper . . . . 41 V I 1 Figure Page 14 Variation of the damping r a t i o with the l i q u i d height (h/d) at several d i f f e r e n t frequencies for c i r c u l a r (l) and square cross-section (2) dampers. Note the square section damper with lower frequency attains peak value at lower h/d 42 15 Effect of frequency on c r i t i c a l h/d corresponding to peak energy dissipation . . 44 16 Effect of surface roughness on the damping r a t i o 45 17 Variation of the damping r a t i o and energy di s s i p a t i o n as functions of the Reynolds and Weber numbers 47 18 Effect of ring diameter on dissipation c h a r a c t e r i s t i c s in high and low frequency regimes 49 19 Plots showing the influence of the Weber and the Froude numbers on the dis s i p a t i o n parameters 51 20 Effect of frequency and damper diameter r a t i o D/d on the d i s s i p a t i o n functions rj and E af 52 21 Representative results showing the influence of spheres in the f l u i d f i e l d on the damping r a t i o 54 22 Typical results showing the effect of screens on the damping r a t i o 57 23 Plots showing the influence of inside perforated tube on the damping r a t i o : (a) high frequency case with inside tube without holes; (b) low frequency case with inside tube without holes 58 (c) number of holes; (d) tube location, high frequency; (e) tube location, low frequency; (f) hole size at high frequency . . 60 24 Eff e c t of r i g i d and f l e x i b l e baffles on the energy d i s s i p a t i o n process. Note the staggered arrangement of baffles on inside and outside walls at high frequency represents a favourable arrangement. Baffles are not p a r t i c u l a r l y e f f e c t i v e at low frequency 61 v i i i Figure Page 25 Pa r t i t i o n i n g of the damper to provide two layers of l i q u i d s i g n i f i c a n t l y improves the damping at low frequency 63 26 Effect of f l o a t i n g wooden pieces on damping ch a r a c t e r i s t i c s 65 27 Comparative performance of several more promising damper configurations 66 28 Relative energy dis s i p a t i o n c h a r a c t e r i s t i c s of four more favourable damper geometries . . 67 29 Plots showing comparative performance of the more successful configurations when tested using the a i r bearing f a c i l i t y . . . 69 30 Plots showing comparative performance of combined devices in high and low frequency regimes: 2 (plain damper); 21 (layers with tubes); 22 (layers with b a f f l e s ) ; 23 (layers with pieces of wood); 24 (tube and baffles) 71 1-1 System representation 80 1-2 Variation of system damping ra t i o (without nutation damper) with excitation frequency over a range of values of system mass. Note the effect of mass i s r e l a t i v e l y small . 83 1-3 Typical plots showing e s s e n t i a l l y exponential decay of amplitude with time over a range of values of system parameters . 84 1-4 Force diagram 82 111- 1 Test arrangement used to determine natural frequency of damping l i q u i d s 89 III-2 Close-up view of a damper under test . . . . 90 III-3 Motion of water in damper 2 during f i r s t mode; h/d =3/4, f = 0.87Hz 92 III-4 Water in damper 2 executing 2nd mode; h/d = 3/4, f = 2.15Hz 92 ix ACKNOWLEDGEMENT The author wishes to thank Dr.V.J.Modi for his generous assistance during the course of thi s research work. His help and the sharing of his experience were very much appreciated during the conduct of the experiments and the preparation of this thesis. The author i s also grateful to the department of Mechanical Engineering for the use of the various f a c i l i t i e s . Thanks are due to the technicians of the machine and instrumentation shops for the construction of the models and advice received concerning the handling of the instrumentation. Patrick Chun, an undergratuate student in this department, spent l a s t summer on the project. I want to offer special thanks to him for his assistance. The project was f i n a n c i a l l y supported by the Natural Sciences and Engineering Research Council of Canada's grant to Dr.Modi. The author wants to express his appreciation for the research assistanship awarded from the grant. LIST OF SYMBOLS damper amplitude of o s c i l l a t i o n s screen open area ba f f l e area perpendicular to flow damping and c r i t i c a l damping c o e f f i c i e n t of the o v e r a l l system (including nutation damper), respectively inherent damping and c r i t i c a l damping c o e f f i c i e n t of the test f a c i l i t y (excluding damper), respectively aerodynamic drag c o e f f i c i e n t based on free stream v e l o c i t y and projected area normal to flow aerodynamic force c o e f f i c i e n t based on free stream ve l o c i t y and projected area normal to flow, Figure 1(b) number of chart d i v i s i o n used during c a l i b r a t i o n damper cross-section diameter or width b a l l diameter, Figure 21 perforated tube inside diameter, Figure 23 cross-section diameter of the structure, Figures 1(a) and 2 damper-ring mean diameter energy dissipated by the system (excluding damper), and o v e r a l l system (including damper) per cycle, respectively energy dissipated by the damper (excluding system) per cycle per unit mass of l i q u i d , (E -Es)/M, E ( j / f 2 d 2 , non-dimensional energy parameter based on damper cross-section diameter and system frequency E c j / f 2 a 2 , non-dimensional energy parameter based on damper amplitude and system frequency xi f n , f d natural and damped frequency of the system, respectively; f n » f d % f f w natural frequency of damping l i q u i d F aerodynamic force, Figure 1(b) Fs force applied to spring g acceleration due to gravity h l i q u i d height in damper h*, h' perforated tube and b a f f l e location in damper, respectively; Figures 23 and 24 I system i n e r t i a including damper; I = I a + I c + I d + I w I a , I c ,I d , I w arm, support, damper and dead weight i n e r t i a , respectively k s,k g,k spring, s t r a i n gauge transducer and system s t i f f n e s s , respectively, k = k s + ks k g/( k s+k g ), Figure 1-1 L length of the o s c i l l a t i n g arm of the s t a t i c stand test f a c i l i t y , Figure 1-4 1 spring-pivot point distance on the s t a t i c stand test f a c i l i t y , Figure 1-4 m number of cycles executed by the system with damper during a sp e c i f i e d change in amplitude m' number of cycles executed by the system without damper during a spe c i f i e d change in amplitude ms mass of the structure per unit length mt mass of tuned mass damper, Figure 4(a) M t o t a l system mass (including damper), M = Ma+ Mc+ Md+ M w Ma,Mc,Md,Mw arm, support, damper and dead weight mass, respectively M| mass of damping l i q u i d Ms mass of the structure, Figure 4(a) n number of b a l l s , screens, or baffl e s ri| number of horizontal layers xi i n 0 number of holes on perforated tube surface n w number of pieces of wood q perforated tube hole size r damper-ring inner radius R damper-ring outer radius S Strouhal number, f v d s / V t time U damping l i q u i d maximum ve l o c i t y during a cycle, U = af v volume of pieces of wood inside damper V free stream v e l o c i t y W, combined weight of damper, dead weight and damper support, W, = (M<j+Mw+Mc)g W2 weight of the o s c i l l a t i n g arm, W2 = Mag x spring displacement y l a t e r a l displacement, Figure 1(a) 0 rotational displacement ©o value of 0 at t = 0 e inner surface roughness of damper p density of damping l i q u i d p a free stream a i r density M v i s c o s i t y of damping l i q u i d a surface tension of damping l i q u i d with a i r 77 damping r a t i o , C/Cc ns inherent damping r a t i o (without damper), C s/C c s to n ,wd c i r c u l a r undamped and damped natural frequency of system , respectively; a>n = 2irfn , cj d = 27rfd <f> phase angle defined by i n i t i a l conditions of the system in rotation, 0 = t g " 1 Vi -T?2' /0cjn (0+7?o>n0) at t = 0 d i m e n s i o n l e s s p a r a m e t e r s a f f e c t i n g damper p e r f o r m a n c e , i = 1—25 e f f e c t i v e a n g l e o f a t t a c k d u r i n g v i b r a t i o n a n g l e o f a t t a c k i n t h e a b s e n c e o f l a t e r a l d i s p l a c e m e n t (y = 0), F i g u r e 1(b) l o g a r i t h m i c d e c r e m e n t 1 1. INTRODUCTION 1.1 Wind-Induced O s c i l l a t i o n s A variety of structures such as smokestacks, t a l l buildings, bridges and similar bluff bodies may exhibit large amplitude motion when subjected to the action of the natural wind. Such wind induced o s c i l l a t i o n s at r e l a t i v e l y low frequencies, t y p i c a l l y below 1Hz, have caused severe damage and, at times, even collapse of the structures has been reported. Vortex resonance and galloping are the two major forms of wind-induced i n s t a b i l i t i e s . Vortex resonance represents a forced vibration situation caused by the formation of the well known Karman Vortex Street when the frequency of the shedding vortices coincides with the natural frequency of the structure (Figure 1a). The use of the Strouhal number, which i s approximately constant for a given geometry and attitude over a range of Reynolds number, provides a good estimate of the frequency of the shedding vortices, and hence of the resonant wind condition for a given structure. Galloping i n s t a b i l i t i e s are self-induced o s c i l l a t i o n s caused by the motion of the structure i t s e l f . They take place when the system is aerodynamically unstable, causing an increase in the side force in the same dir e c t i o n as the disturbance. This i s i l l u s t r a t e d in Figure 1(b). The d i s t i n c t character of the two forms of i n s t a b i l i t i e s suggests that although vortex resonance i s always a p o s s i b i l i t y , not every structure would experience galloping. For instance, structures with c i r c u l a r section do not gallop. However, square or rectangular buildings, sleeted 2 Strouhal Number = S = f v d s / V . Resonance when f v= f n . For most blu f f structures (smokestacks, bridges, buildings, transmission l i n e s , etc.), f n < 1 Hz . Here: f v= frequency of shedding v o r t i c e s -f n = natural frequency of structure. S = Strouhal number. V = freestream wind'speed; d s = c h a r a c t e r i s t i c dimension. Figure 1 Wind induced i n s t a b i l i t i e s of bluff bodies: (a) vortex resonance Figure 1 Wind induced i n s t a b i l i t i e s of bluff bodies: (b) galloping 4 transmission l i n e s , etc., are found to gallop. Several quasi-steady analyses have been proposed over the years to predict the dynamical reponse during vortex resonance and galloping i n s t a b i l i t i e s . 1.2 Suppression of Wind-Induced O s c i l l a t i o n s Suppression of wind-induced o s c i l l a t i o n s has been the object of extensive studies. Broadly speaking, two d i f f e r e n t approaches have been used. The f i r s t one deals with the modification of the f l u i d mechanics associated with the section geometry responsible for the time-dependant exc i t a t i o n . It has mostly been considered for the suppression of vortex resonance. Various devices developed to t h i s end are summarized by Zdravkovich 1, and Every, King and Weaver 2, who present in-depth reviews of the work done in the f i e l d . The most popular procedure consists of h e l i c a l strakes fixed around chimneys. The concept was proposed by Scruton 3 in the 1950's who also gave optimal configurations in terms of p i t c h , profusion, and location (Figure 2). Strakes were found to af f e c t the boundary layer around the chimney, making i t more turbulent thus eliminating the regularity at which vortices are shed. Other e f f i c i e n t devices comprise of shrouds and s l a t s that a f f e c t the flow around structures and more p a r t i c u l a r l y the flow in the entrainment region downwind. Optimal configurations described by Wong and Cox" are also shown in Figure 2. Several other devices, such as f a i r i n g s , s p l i t t e r plates and flags (Figure 2) have been developed in order to s t a b i l i z e the wake behind the structures and hence to reduce Helical Strakes Shrouds Slats optimal pitch = 5d s protusion = 0.10 — 0.12ds location : top 33% of height drag penalty : » 1 .3 for 8x10* < Re < 2x106 ( i . e . 3 times higher in s u p e r c r i t i c a l range) Desirable Configurations gap : shroud-chimney = 0.12ds open area ratio = 20 — 36% location = top 25% of height drag penalty : C d » 0.9 for 8x10" < Re < 2x10 s ( i . e . 2 times higher in s u p e r c r i t i c a l range) s l a t width = d s/l1.5 gap : slat—chimney = d s/7 open area r a t i o = 40% drag penalty : C d » 1.05 to 1.1 for 1.5x10" < Re < 1.5X10 5 ( i . e . expected 2.5 times higher in s u p e r c r i t i c a l range) Figure 2 Several t y p i c a l devices aimed at modifying cn character of the flow to minimize i n s t a b i l i t y D r a g c o e f f i c i e n t : C d » 0.41 f o r c h o r d t o t h i c k n e s s r a t i o = 2.12, i n s u p e r c r i t i c a l r a n g e D r a g r e d u c t i o n : minimum C d x, 0.86 ( 6 9 % o f C d f o r p l a i n c y l i n d e r ) f o r s p l i t t e r p l a t e l e n g t h e q u a l t o c y l i n d e r d i a m e t e r , i n s u b c r i t i c a l r a n g e ( 1 0 " < Re < 5x10") D r a g r e d u c t i o n : C d f r o m 2.12 down t o 0.93 on o s c i l l a t i n g c y l i n d e r s F i g u r e 2 S e v e r a l t y p i c a l d e v i c e s a i m e d a t m o d i f y i n g c h a r a c t e r o f t h e f l o w t o m i n i m i z e i n s t a b i l i t y 7 the effects of vortex shedding. However, they exhibit less f l e x i b i l i t y with respect to the direction of the wind. A comparative study using wind tunnel tests was ca r r i e d out by Wong and Cox" which showed r e l a t i v e effectiveness of the various devices as presented in Figure 3, where amplitudes of motion are plotted against the system damping. Most of the devices thus far lead to a rather severe drag increase, by a factor of two (shrouds), or even three (strakes). Strake e f f i c i e n c y was demonstrated in real l i f e situations on steel smokestacks as reported by Hirsh and Rusheweyh5. There are conditions where the suppression of vortex resonance using aerodynamic means i s not possible as in the case of structures with very low damping or for rows of stacks, as discussed by Ruscheweyh6 and Zdravkovich 1. Aerodynamic means postpone vortex shedding rather than eliminate i t , which sometimes create problems as in the case of stacks or buildings in the wakes of others. Several mechanical devices are proposed in this s i t u a t i o n . The suppression of galloping i n s t a b i l i t i e s using aerodynamic devices has also been considered. Naudasher et a l . 7 have reviewed the various designs being tested, such as r i b s , recessed corners, and small plates. However, no conclusive assessment as to their effectiveness i s yet av a i l a b l e . The second approach i s to increase the damping of the structures. Several devices have been proposed to t h i s end and a few are under investigation. Tuned mass systems (Figure 4a) permit the main structural element to stay somewhat unaffected by the e x c i t i n g forces. Their 8 "D > CD "O •*-> 0.4-Q E < co Pe 0 .3 -o CO CD CL 0 .2 -CO CO 0 c o 'co en 0 . 1 -E b Plain cylinder 3-start strake system,full coverage Perforated shroud, 1/4 coverage 3-start strake system, 1/3 coverage Slat system, 1/2 coverage Slat system,full coverage Slat system with front and rear opening,2/3 coverage 0 20 30 40 Damping Coefficient , 2 m s 6/ p a d . F i g u r e 3 Maximum a m p l i t u d e o f o s c i l l a t i o n ( p e a k t o p e a k ) a g a i n s t t h e d a m p i n g c o e f f i c i e n t f o r c i r c u l a r c y l i n d e r f i t t e d w i t h s e v e r a l d a m p i n g d e v i c e s M s J v W 71 11 Tuned Mass Damper ( a ) Rubber covered Chain Stockbridge Damper I n t e r n a l d a m p i n g ; o u t o f p h a s e v i b r a t i o n s (b) Hydraulic Dashpot ( c ) Hanging-Chain Impact Damper I n c r e a s e i n d a m p i n g I n c r e a s e i n w e i g h t : ( d ) f a c t o r o f 5% Nutation Damper ( e ) i g u r e 4 S e v e r a l d e v i c e s u s e d i n p r a c t i c e t o p r o m o t e e n e r g y d i s s i p a t i o n 10 e f f i c i e n c y in suppressing wind-induced o s c i l l a t i o n s is discussed by Wardlaw and Cooper 8, as well as H i r s h 9 . The Stockbridge damper was developed in 1925 and is widely used on transmission lines to suppress galloping i n s t a b i l i t i e s 1 0 " 1 2 . A Stockbridge damper is made of two weights linked by a cable, the assembly being attached to the transmission l i n e (Figure 4b). Vibration control is acheived through internal damping in the cable strands as well as out-of-phase motion of the weights with respect to the transmission l i n e . Extensive a n a l y t i c a l and experimental studies have been reported on t h i s damper, in p a r t i c u l a r by Sturm 1 3 and Schafer 1*. Recently, Hagedorn 1 5 proposed a rather general procedure for c a l c u l a t i n g damped response of transmission l i n e s . Of course, mention should be made of the conventional hydraulic dashpots which have been used on guyed structures (Figure 4c). The concept of 'hanging chain impact damper' (Figure 4d) i s described in some d e t a i l by Reed 1 6. The use of v i s c o e l a s t i c systems i s another alternative discussed by Gasparini, Curry and Debchaudhury 1 7. More recently, H i r s h 9 proposed an active control system using sensor elements that trigger out-of-phase motion of counterweights to oppose o s c i l l a t i o n s . A nutation damper belongs to t h i s category of energy d i s s i p a t i n g devices. E s s e n t i a l l y , i t consists of a hollow ring p a r t i a l l y f i l l e d with l i q u i d (Figure 4e). Such dampers have been used with some success in minimizing l i b r a t i o n a l motion of s a t e l l i t e s where the period may be as long as 90 minutes to 24 h o u r s 1 8 " 2 " . As wind-induced o s c i l l a t i o n s of 11 structures occur at r e l a t i v e l y low frequencies, normally less than 1Hz, i t was thought appropriate to explore the potential of t h i s concept in attending to vibration problems of i n d u s t r i a l aerodynamics. 1.3 Scope of the Investigation A.preliminary study aimed at assessing effectiveness of a nutation damper in suppressing wind-induced i n s t a b i l i t i e s was c a r r i e d out by Modi et a l . 2 5 Results showed some promise in c o n t r o l l i n g vortex resonance and galloping o s c i l l a t i o n s during simulated conditions in wind tunnels. This thesis systematically studies the effect of various system parameters on the damper performance. Ultimate objective is to a r r i v e at configurations which are l i k e l y to be e f f e c t i v e in p r a c t i c e . The investigation i s mainly experimental as the energy d i s s i p a t i o n process i s quite complex and not readily amenable to analysis. Although several t h e o r a t i c a l models have been proposed over the y e a r s 1 3 " 2 3 • 2 6 , none is representative of the actual phenomenon. For instance, Brunner 2 6 treated the damper as having two dimensional viscous flow while A l f r i e n d et a l . 1 9 " 2 2 replaced damping l i q u i d by a r i g i d slug moving inside the ring. A n a l y t i c a l treatment of the problem thus remains a challenging task. A wide range of damper models are investigated using a s t a t i c stand f a c i l i t y which evaluates damper performance in the one-degree of freedom rotational motion. Major dimensionless numbers a f f e c t i n g the performance are many and their organized v a r i a t i o n leads to a vast body of 1 2 i n f o r m a t i o n . Hence, at t i m e s , only more s i g n i f i c a n t r e s u l t s u s e f u l i n e s t a b l i s h i n g t rends are recorded . Based on the a n a l y s i s of the d a t a , s i g n i f i c a n t system parameters a f f e c t i n g the energy d i s s i p a t i o n process in s l o s h i n g l i q u i d s are i d e n t i f i e d and recommendations made which would he lp evolve e f f i c i e n t n u t a t i o n dampers. 1 3 2. NUTATION DAMPERS 2.1 Damper Description A nutation damper is a torus shape container p a r t i a l l y f i l l e d with l i q u i d as i l l u s t r a t e d in Figure 5. The c i r c u l a r geometry ensures symmetry and renders i t s performance insensitive to the direction of o s c i l l a t i o n s . It can be positioned inside or around structures. The internal configuration, which consists of plain walls in i t s simplest form, may be modified by introducing b a f f l e s , screens, spheres, p a r t i t i o n s , etc., in order to promote the energy dis s i p a t i o n process. Nutation Damper D Figure 5 A schematic diagram showing the nutation damper and i t s application 14 Generally, the damper has a c i r c u l a r cross-section, however other shapes can also be considered. Its main features are : • simple construction, resulting in low cost; • f l e x i b i l i t y in terms of amount and type of l i q u i d to be used according to the level of damping required; • e f f i c i e n t energy d i s s i p a t i o n process resulting in a r e l a t i v e l y small weight of the damping l i q u i d compared to the size of the structure; • insensitive to wind d i r e c t i o n . 2 . 2 Parameters A nutation damper can be characterized by i t s dimensions, weight and other readily measurable independant variables which govern i t s performance. There are b a s i c a l l y two categories of parameters : those common to a l l nutation dampers and the ones s p e c i f i c to d i s t i n c t internal configurations aimed at improvement in the basic design. Common Parameters These involve the damper dimensions and the' l i q u i d c h a r a c t e r i s t i c s . Furthermore, variables involving the o s c i l l a t i o n c h a r a c t e r i s t i c s as well as the system i n e r t i a under which the damper responds are also included here. • Geometrical Dimensions : ring diameter D cross-section diameter d surface roughness e l i q u i d height h 1 5 • Physical Properties of Liquid • System Dynamical Properties • Natural Parameter : density p v i s c o s i t y n surface tension a : frequency f amplitude a t o t a l mass M inherent damping rjs : g r a v i t a t i o n a l acceleration g The l i s t contains only the variables considered most relevant to the damping c h a r a c t e r i s t i c s . Liquids, for instance, are characterized by only three main properties, i.e . density, v i s c o s i t y and surface tension in a i r . Specific Parameters The l i s t of parameters that follows refers to the various models tested in the experimental program (Table I, chapter 3, pp.27-28). They deal with p a r t i c u l a r modifications to the basic internal configuration aimed at increasing the damping due to sloshing motion of the l i q u i d . • Spheres (Balls) in the Flow • Screens • Inside Tube with Perforations number of b a l l s b a l l diameter number of screens open area tube diameter tube height hole size number of holes n n A * i h* q n 0 16 • Baffles (Rigid or Flexible) : baffle height h' baffle area b number of baffl e s n • Horizontal Layers number of layers • Pieces of Wood in Flow volume v number n w Each of these features showed a promising trend and e f f o r t s were made to optimize the parameters. 2.3 Dimensionless Parameters To evolve a rati o n a l test program, i t was desirable to arrive at a dimensionless group of numbers governing the energy di s s i p a t i o n process. There are twelve common parameters involving length (L), time (T), and mass (M) dimensions. Of the fourteen s p e c i f i c parameters, some are either dimensionless or involve only the length dimension. The repeated variables chosen in the dimensional analysis were the cross-section diameter d (L), the frequency of o s c i l l a t i o n s f ( T " 1 ) , and the l i q u i d density p (ML" 3). The associated dimensionless numbers are l i s t e d below : Common Parameters Twelve variables with three dimensions lead to nine dimensionless numbers: dimensionless amplitude of o s c i l l a t i o n s ir, = a/d dimensionless l i q u i d height ir2 = h/d dimensionless surface roughness ir3 = e/d 17 i n e r t i a s t r e s s / v i s c o u s s t r e s s it u = pd2l/n ( R e y n o l d s number) s h a p e f a c t o r it- = D/d l i q u i d f r a c t i o n , mass o f l i q u i d / t o t a l mass it- - pd 3/M i n h e r e n t d a m p i n g r a t i o it- = T J S i n e r t i a f o r c e / s u r f a c e t e n s i o n f o r c e it- = p d 3 f 2 / a (Weber number) i n e r t i a f o r c e / g r a v i t y f o r c e it- = f 2 d / g ( F r o u d e number) N o t e , t h e dampers e x p e r i e n c e t h e e f f e c t o f t h e s y s t e m i n e r t i a a n d d a m p i n g o n l y t h r o u g h t h e a m p l i t u d e o f o s c i l l a t i o n s , i . e . f o r a g i v e n a m p l i t u d e , dampers d i s s i p a t e t h e same amount o f e n e r g y ( E a , f ) , i n d e p e n d a n t o f t h e s y s t e m i n e r t i a a n d i n h e r e n t d a m p i n g ( T ? s ) . T h u s , i t i s p o s s i b l e t o e l i m i n a t e it- ( l i q u i d m a s s / s y s t e m mass r a t i o ) a n d it7 ( i n h e r e n t d a m p i n g r a t i o ) f r o m t h e a n a l y s i s , l e a v i n g s e v e n p a r a m e t e r s . D i m e n s i o n l e s s numbers c a n be m u l t i p l i e d t o f o r m new o n e s more r e p r e s e n t a t i v e o f t h e p h y s i c a l c h a r a c t e r o f t h e p r o b l e m . F o r e x a m p l e , iti.it- = it'- = ( a / d ) ( p d 2 f / n) = p d a f / M , it,2.it- = it'- = ( a / d ) 2 ( p d 3 f 2/o) = p d a 2 f 2 / a , TT , 2 . 7 r 9 / 7 r 2 = it'- = ( a / d ) 2 ( f 2 d / g ) / ( h / d ) , = a 2 f 2 / g h . H e r e ' a f h a s t h e d i m e n s i o n o f v e l o c i t y . Thus it'n = pdU/n, it'- = p6U2/a, a n d it'- = U 2 / g h , w h i c h r e p r e s e n t t h e R e y n o l d s , t h e Weber, a n d t h e F r o u d e n u m bers, r e s p e c t i v e l y . H o wever, 7r„ a n d ir- a r e much more c o n v e n i e n t t o u s e a n d a r e e q u i v a l e n t 18 to ir{ and n%, respectively, when 7r-, is constant, which i s the case in most of the experiments. Sim i l a r l y , the influence of ir5 is discussed only when IT , and 7r2 are kept constant. Furthermore, TT8 and 7 r 9 are quite d i f f i c u l t to treat separately in the analysis and only their combined effect on damper performance i s studied. Specific Parameters Using 'd' as the nondimensionalizing parameter, the fourteen it numbers are obtained quite readily: • Spheres (Balls) in the Flow : number of b a l l s 7 r 1 0 = n dimensionless b a l l diameter ff,, = d b/d • Screens : number of screens n , 2 = n open area r a t i o i r , 3 = A/d 2 • Inside Tube with Perforations : dimensionless inside tube diameter jr,„ = d,/d dimensionless inside tube height 7r 1 5 = h*/d dimensionless hole size = q/d number of holes 7 r , 7 = n 0 • Baffles : dimensionless ba f f l e height i , B = h'/d ba f f l e area r a t i o = b/d 2 number of ba f f l e s 7 r 2 0 = n • Horizontal Layers : number of layers IT 2 1 = n| 19 • Pieces of Wood in Flow : volume r a t i o " 2 2 = v/d 3 number 7 r 2 3 = n w Note, T r 2 2 i s a measure of the volume of wood pieces used. The true volume r a t i o i s given by 7 ^ 2 = v / 7 r d 2D. The test program described later aims at assessing the influence of these 23 dimensionless numbers. 2.4 C r i t e r i a Damping Ratio Damping c h a r a c t e r i s t i c s of nutation dampers have been assessed through the use of appropriate c r i t e r i a . The dimensionless damping r a t i o 77 is a l o g i c a l and convenient parameter to use. It i s , however, dependant on the system i n e r t i a and inherent damping. Hence i t would be useful to evaluate the effectiveness of dampers using a common test f a c i l i t y and fixed i n e r t i a (higher the i n e r t i a , lower the value of TJ for a given damper). Now, 77 = C/Cc , where C c i s the c r i t i c a l damping of the system. In terms of amplitude decay, for linear one-degree of freedom systems, TJ = 8/[ (4TT2 + 82) 1 / 2m] , where: 6 = logarithmic decrement = l n ( x 1 / x m + 1 ) ; m = number of cycles; x n= amplitude at the nth cycle. For small TJ ( t y p i c a l l y ^ 0.20), t h i s can be approximated as TJ «. 5/2?rm. TJ i s a function of a l l the 20 independant parameters l i s t e d previously, hence, Tt 2 « = T} = f ( TT i , 7T 2 , TT 3 , 7T „ , 7T 5 , 7T g , Tt 7 , 7T 8 , . . . , 7T 2 3 ) . Energy Dissipation Rate Energy di s s i p a t i o n within a damping l i q u i d is an inherent property and hence does not depend on the structure on which the damper is i n s t a l l e d . The idea of l i q u i d e f f i c i e n c y can also be introduced by dividing the energy di s s i p a t i o n rate with the mass of the l i q u i d . It should be noted that the optimal damper performance l i e s in maximizing energy di s s i p a t i o n (or damping) while minimizing the l i q u i d weight, and therefore the highest e f f i c i e n c y also represents the l i g h t e s t damper. The energy dissipated per cycle per unit mass of l i q u i d can be expressed as, E d = (E-E s)/M|, where : E = t o t a l energy dissipated per cycle = [ ( k / 2 ) ( x 1 2 - x m + 1 2 ) ] / m ; k = s t i f f n e s s of a one degree of freedom system; E s= energy dissipated by the system alone (excluding dampers); M| = mass of the l i q u i d in damper; m = number of cycles. E s i s known through c a l i b r a t i o n of the test apparatus. For the analysis, E dshould be presented in a dimensionless form. With i t s dimension of L 2T~ 2, i t i s convenient to def ine, E d i f = E d / f 2 d 2 21 Multiplying by TT -,2 = (a/d) 2, in order to take the amplitude of o s c i l l a t i o n s into account gives, E a f = E d / f 2 a 2 . Thus, Ea,f i s the energy dissipated in l i q u i d per cycle per unit mass, amplitude and frequency. From the d e f i n i t i o n , i t follows that, Ea,f = E d / f 2 a 2 = (E-E S)/[M, (f 2 a 2 ) ] Now f 2 a 2 has the dimension of ( v e l o c i t y ) 2 ; hence M|(f 2a 2) i s a representation of the maximum kinetic energy in the l i q u i d . Thus E a,f represents the r a t i o of energy dissipated by the l i q u i d per cycle over the maximum kinetic energy present in the l i q u i d during the cycle. E a,f does not depend on the system i n e r t i a and damping, i . e . it 6 and it-,, hence " 2 5 = E a f = f ( 7 T i , 7T 2, 7T 3, It« , It 5 , It e , . . . , IT 2 3 ) . E a f — 7} Relationship The two c r i t e r i a mentioned before,i.e. 17 and E a f , are intimately related. From d e f i n i t i o n , Ea.f = [ ( ( k / 2 ) ( x 1 2 -x m + 1 2)/m} -E s ]/(M, f 2 a 2 ) , and TJ = (lnx ,/x m +, )/27rm , therefore, as shown in Appendix I I , Ea,f = (a ) ( k/M| f 2 ) { ( l - x m + 1 2 / x 1 2 ) / ( l n x l / x m t 1 ) } ( r ? - r ? s ) , where a i s a constant. Note k / f 2 represents the system i n e r t i a for a single-degree of freedom system. For a constant r a t i o ( x m + 1 / x , ) , E a,f = a'M/M| (77 - i 7 s) , a' = constant, which represents the absolute damping per unit mass of 22 l i q u i d , per unit frequency. It should be recognized that E a f accounts for the system i n e r t i a (M) and inherent damping ( T J S ) as mentioned e a r l i e r . Details of the system c h a r a c t e r i s t i c s k, M ,rj sare presented in Appendix I. 23 3 . T E S T P R O C E D U R E 3 . 1 T e s t F a c i l i t y N u t a t i o n d a m p e r s w e r e t e s t e d i n a o n e - d e g r e e o f f r e e d o m r o t a t i o n a l m o t i o n u s i n g t h e f a c i l i t y s h o w n i n F i g u r e 6 . E s s e n t i a l l y , i t c o n s i s t s o f a n a l u m i n u m r o d , 1 2 6 . 5 c m l o n g w i t h a 1 . 9 c m O . D . a n d a m a s s o f I 2 8 4 g m s , m o u n t e d o n a l u b r i c a t e d b e a r i n g , a n d a c r o s s - l i k e a l u m i n u m s u p p o r t w e i g h i n g 4 0 8 g m s o n w h i c h d a m p e r s w e r e p o s i t i o n e d . T h e f r e q u e n c y o f t h e s y s t e m c a n b e v a r i e d t h r o u g h t h e u s e o f a m o v e a b l e c o l l a r c o n n e c t i n g t w o s p r i n g s , o f d e s i r e d s t i f f n e s s , t o t h e a l u m i n u m r o d . A l i s t o f t h e v a r i o u s s p r i n g s t i f f n e s s e s u s e d i n t h e e x p e r i m e n t s i s g i v e n i n A p p e n d i x I . D i s p l a c e m e n t a n d f r e q u e n c y o f o s c i l l a t i o n s w e r e m e a s u r e d t h r o u g h a s t r a i n g a u g e a r r a n g e m e n t a c t i n g i n s e r i e s w i t h t h e s p r i n g s . T h e s t r a i n g a u g e , f o r m i n g o n e a r m o f t h e W h e a t s t o n e B r i d g e , s e n d s a s i g n a l t o a n a m p l i f i e r f r o m w h e r e i t i s d i r e c t e d t o a n o s c i l l o s c o p e a n d a c h a r t r e c o r d e r . T h e o v e r a l l s y s t e m i s m o u n t e d o n a h e a v y , r i g i d f r a m e . T h e f a c i l i t y w a s u s e d t o m e a s u r e a m p l i t u d e d e c a y s o f f r e e v i b r a t i o n s . O f c o u r s e , e v e n t u a l l y o n e w o u l d l i k e t o t e s t t h e e f f e c t i v e n e s s o f a d a m p e r i n a r r e s t i n g o r a t l e a s t m i n i m i z i n g v o r t e x - i n d u c e d a n d g a l l o p i n g o s c i l l a t i o n s o f s t r u c t u r e s t h r o u g h s i m u l a t e d w i n d t u n n e l t e s t s . H e n c e , i t w a s t h o u g h t a p p r o p r i a t e t o c o n d u c t d a m p i n g m e a s u r e m e n t s u s i n g a f a c i l i t y s i m i l a r t o t h e o n e l i k e l y t o b e u s e d d u r i n g w i n d t u n n e l t e s t s ( F i g u r e 7 ) . T h e m a i n d i f f e r e n c e l i e s i n N U T A T I O N D A M P E R \ S P R I N G M O V E A B L E C O L L A R RECORDER OR OSCILLOSCOPE BAM S T R A I N G A U G E A D J U S T A B L E S U P P O R T A N D H E I G H T TO C O N T R O L N A T U R A L F R E Q U E N C Y B E A R I N G /////)///////// gure 6 A schematic diagram of the s t a t i c stand t e s t f a c i l i t y 25 Figure 7 Static stand test f a c i l i t y used during the wind tunnel tests 26 the use of a i r bearings which contribute, r e l a t i v e l y , much lower damping. Due to a difference in length of the moment arms, distance travelled by the damper, for the same angular rotation, i s di f f e r e n t in the two test f a c i l i t i e s . The rotating parts consist of a square wooden block (balsa), 4.96cm wide (2in.), and 48.3cm long (I9in.), mounted on a hollow aluminum tube fixed to the shaft free to rotate in a i r bearings. A set of springs imposed the necessary s t i f f n e s s . As before, a s t r a i n gauge was used to measure displacements. A thin aluminum plate (instead of a cross) supported the dampers. 3.2 Models A family of dampers was designed to carry out the parametric study. Square as well as c i r c u l a r cross-sections were used together with a variety of dimensions, internal configurations, surface roughnesses and f l o a t i n g elements. The empty model mass varied from 26gms to I7l0gms and the l i q u i d capacity from 136ml to 826ml. Water was used in most of the experiments. The d e t a i l s of the models are given in Table I. Further d e t a i l s concerning damper internal configurations are given in Chapter 4 where results are di scussed. The square cross-section dampers, i . e . models 2 and 12 to 23 (Table I) have a removable top so that parametric changes can be effected e a s i l y in the internal configuration. They are also made of transparent plexiglas which permitted flow v i s u a l i z a t i o n . C i r c u l a r models 9, 10 Table I Damper models used in the test program Damper Dimen< sions D e s c r i p t i o n d (cm) D (cm) Internal C o nfiguration Cross-Section M a t e r i a l Weight (gms) Capacity (ml) 1 2.4 33.1 p l a i n c i r c u l a r rubber 914 422 2 4.5 15.9 p l a i n square p l e x i g l a s 1710 640 3 2.6 49.5 p l a i n c i r c u l a r rubber 1295 826 4 2.6 40.0 p l a i n c i r c u l a r rubber 1080 667 5 2.6 29.5 p l a i n c i r c u l a r rubber 810 492 6 2.6 20.0 p l a i n c i r c u l a r rubber 565 334 7 2.6 6.65 p l a i n c i r c u l a r p l a s t i c 26 167 8 1.9 28.9 p l a i n c i r c u l a r copper 842 235 9 1.55 23.6 p l a i n c i r c u l a r polyethylene 215 136 10 1.55 23.6 b a f f l e s c i r c u l a r polyethylene 217 136 11 1.55 23.6 perforated tube c i r c u l a r polyethylene 225 135 12 4.5 15.9 rough sandpaper square p l e x i g l a s 1711 640 13 4.5 15.9 smooth sandpaper square p l e x i g l a s 1711 640 Table I Damper models used in the test program Damper Dimensions D e s c r i p t i o n d (cm) D (cm) Internal C o n figuration Cross-Section M a t e r i a l Weight (gms) Capacity (ml) 14 4.5 15.9 b a l l s i n flow square p l e x i g l a s 1710 640 15 4.5 15.9 screens square p l e x i g l a s 1710 640 16 4.5 15.9 perforated tube square p l e x i g l a s 1740 635 17 4.5 15.9 b a f f l e s square p l e x i g l a s 1710 640 18 4.5 15.9 middle l a y e r square p l e x i g l a s 1771 589 19 4.5 15.9 pieces of wood i n flow square p l e x i g l a s 1710 640 20 2.4 33.1 b a l l s i n flow c i r c u l a r rubber 914 422 21 4.5 15.9 middle l a y e r and 2 tubes square p l e x i g l a s 1831 589 22 4.5 15.9 middle l a y e r and b a f f l e s square p i e x i g l a s 1781 573 23 4.5 15.9 middle l a y e r and pieces of wood square p l e x i g l a s 1787 567 24 4.5 15.9 tube and b a f f l e s square p i e x i g l a s 1750 619 29 and 11 are also transparent. They are made of f l e x i b l e plexiglas tubing formed into a ring with ends clamped together. Model 7 i s a moulded p l a s t i c hollow ring used as a toy and available commercially. The models were constructed in the Department's Machine Shop. Each model was provided with two small openings to f a c i l i t a t e introduction and drainage of the damping f l u i d as required. 3.3 Instrumentation and Calibration Instrumentation used during the test program was simple and conventional. The signal from a Bridge Amplifier Meter (BAM, E l l i s Associates) was e s s e n t i a l l y sinusoidal without any s i g n i f i c a n t noise as shown in Figure 8. The signal was displayed on a storage oscilloscope (Tektronics 564, V e r t i c a l Amplifier Type 3A3, Time Base Type 2B67) and traced using a high speed recorder (Gulton Industries Inc., Type TR 722). Figure 8 Typical free o s c i l l a t i o n signal traced by a chart recorder at a frequency of 3Hz Calibra t i o n of the displacement s t r a i n gauge transducer was accomplished quite readily using a routine procedure. The system was imparted a known displacement and the 30 r e s u l t i n g t i m e / d i s p l a c e m e n t h i s t o r y r e g i s t e r e d o n t h e c h a r t r e c o r d e r . T h e r e a l t i m e a x i s w a s g i v e n b y t h e c h a r t s p e e d s u s e d , i . e . 1 m m / s e c o r 2 5 m m / s e c . T h e a c c u r a c y o f t h e c h a r t s p e e d s i n d i c a t e d o n t h e r e c o r d e r w a s c h e c k e d a n d t h e e r r o r w a s f o u n d t o b e n e g l i g i b l e ( 0 . 1 % ) . S i n c e a c o n v e n t i o n a l s t r a i n g a u g e g e n e r a t e s s i g n a l s a p p r o x i m a t e l y p r o p o r t i o n a l t o t h e f o r c e a p p l i e d , t h e v o l t a g e r e c e i v e d a t t h e c h a r t r e c o r d e r , a n d t h e r e f o r e t h e n u m b e r o f c h a r t d i v i s i o n s , s h o u l d b e p r o p o r t i o n a l t o t h e d i s p l a c e m e n t o f t h e l i n e a r s p r i n g ( F s = k x ) . C a l i b r a t i o n p l o t s s h o w i n g s p r i n g d i s p l a c e m e n t p e r c h a r t d i v i s i o n v e r s u s e f f e c t i v e s t i f f n e s s a r e s h o w n i n F i g u r e 9 f o r t w o d i f f e r e n t g a i n s e t t i n g s : l O m V / c h a r t d i v i s i o n a n d 2 0 m V / c h a r t d i v i s i o n . D a m p e r d i s p l a c e m e n t s a r e p r o p o r t i o n a l t o s p r i n g d i s p l a c e m e n t s a n d d e p e n d u p o n t h e p o s i t i o n o f t h e m o v e a b l e c o l l a r . I n m o s t o f t h e e x p e r i m e n t s , t h e c o l l a r w a s l o c a t e d 5 9 . 5 c m f r o m t h e p i v o t p o i n t . T h u s s p r i n g d i s p l a c e m e n t s w h e n m u l t i p l i e d b y a f a c t o r o f 2 . 1 2 6 g i v e t h e d a m p e r d i s p l a c e m e n t s . 3 . 4 E x p e r i m e n t a l P r o c e d u r e C h a r a c t e r i s t i c s o f n u t a t i o n d a m p e r s w e r e s t u d i e d t h r o u g h a s y s t e m a t i c v a r i a t i o n o f t h e t w e n t y - t h r e e d i m e n s i o n l e s s p a r a m e t e r s l i s t e d e a r l i e r . T h e e f f e c t o f e a c h p a r a m e t e r w a s s t u d i e d s e p a r a t e l y , k e e p i n g o t h e r s c o n s t a n t . T h i s w a s a c h e i v e d b y u s i n g d i f f e r e n t s p r i n g s t i f f n e s s e s , s y s t e m d e a d w e i g h t s , i n i t i a l a m p l i t u d e s a n d s c a l e m o d e l s . W a t e r w a s u s e d i n t h e m a j o r i t y o f t h e e x p e r i m e n t s f o r c o n v e n i e n c e , b u t o t h e r l i q u i d s s u c h a s e n g i n e o i l w e r e x/Ch ( — 1 .— » 500 1000 1500 2 0 0 0 k s k g / (k s +k g ) Figure 9 Calibration plots showing displacement against e f f e c t i v e spring s t i f f n e s s 32 needed to vary certain parameters. Amplitude decay of free vibrations was measured with a chart recorder, and damping c o e f f i c i e n t s as well as energy dis s i p a t i o n rates calculated. The experiments are categorized as : (i) those dealing with the 'common' parameters mentioned in section 2.2; and ( i i ) the ones dealing with ' s p e c i f i c ' parameters. Of the nine 'common' parameters, ir6 and ir7 can be eliminated as explained before (p. 17). Furthermore, TT8 and TT9 are analysed j o i n t l y . Thus, six sets of experiments were planned, each dealing with a common parameter. In the second category, experiments were grouped by the type of internal configuration as against the individual parameters. This added six sets of experiments. As the main objective was to identif y more promising damper configurations, those showing r e l a t i v e l y modest performance were not investigated extensively. Following i s a short summary of the experimental program. It may prove helpful in appreciating the results presented in Chapter 4. 'Common' Parameters • Ef f ect of 7T, = . a/d Damped response was recorded over a wide range of amplitudes during the 'same' test ( i . e . same damper, frequency, dead weight, l i q u i d height, e t c . ) . The experiment was repeated over a range of l i q u i d heights and frequencies of o s c i l l a t i o n s using dampers 1, 2 and 9. 33 • Effect of Tr2 = h/d Amplitude decay plots were recorded for d i f f e r e n t l i q u i d heights using the same damper, frequency, i n i t i a l amplitude, etc. The tests were repeated for dampers 1, 2 and 9 at d i f f e r e n t frequencies. • Effect of 7r 3 = e/d This set of tests aims at evaluating the effect of surface roughness through the use of d i f f e r e n t sandpapers. Dampers 2, 12 and 13 were used at frequencies of 2.27Hz and 0.22Hz. • Effect of TTd = pd 2f/ M Here, the tests involved changing of l i q u i d v i s c o s i t y . Water, as well as SAE 10W, 20W and 30W engine o i l s were used with damper 2 over a wide range of frequencies. • Effect of TT5 = D/d The tests used d i f f e r e n t models with the same cross-section diameter d but of d i f f e r e n t ring diameter D in order to study the eff e c t of D/d v a r i a t i o n . Dampers 3 to 7 were used at frequencies of 2.27Hz and 0.22Hz. • Effect of 7r8 = p d 3 f 2 / o and 7r9 = f 2d/g Two values of a i r - l i q u i d surface tension (engine o i l and water) and various frequencies of o s c i l l a t i o n s were used to study the eff e c t of TI 8 and TT9 . Damper 2 was the model used here. 34 • Additional Experiments (i) E ffect of pure rotation, as opposed to rotation and tra n s l a t i o n , was b r i e f l y studied and results are summarized in section 4.1(i). ( i i ) E ffect of damper material was looked at by using rubber coating on a plexiglas damper (damper. 2). This is discussed in sect ion 4.1(i i i ) . ( i i i ) Combined effect of nn , JTb and TT9 parameters was investigated by changing frequency only using dampers 1 and 2 p a r t i a l l y f i l l e d with water. This is discussed in section 4.1(vi ) . 'Specific' parameters • Tests here attempt to assess the effect of internal modifications in the o r i g i n a l damper. Six dif f e r e n t configurations as mentioned e a r l i e r (section 2.2) were tested. Dampers 10 to 20 were used at frequencies of 2.27Hz and 0.22Hz. Optimal parameters in each of the configurations were established. • Effect of combining d i f f e r e n t optimal configurations was studied at the same two frequencies of 2.27Hz and 0.22Hz, using dampers 21 to 24. A l l the experiments were c a r r i e d out using the test f a c i l i t y shown in Figure 6. The second test f a c i l i t y of Figure 7 proved useful in substantiating the re s u l t s . It was used only in a selected few cases ( f i r s t two experiments of the f i r s t category and four experiments in the second 35 category) . 36 4. RESULTS AND DISCUSSION The amount of information obtained through a systematic study of 'common' and ' s p e c i f i c ' parameters mentioned before is rather extensive. This presented a challenge in organizing the results in a concise yet informative manner. To impart continuity and f a c i l i t a t e understanding, influence of individual parameter is discussed separately. 4.1 Common Parameters (i) ff, = a/d Depending upon the value of parameter p d 2 f / M , the amplitude of o s c i l l a t i o n s appears to affe c t damping r a t i o T? as well as the energy d i s s i p a t i o n rate E af d i f f e r e n t l y (Figures 10, 11). In general, at high values of p d 2 f / V corresponding to f 2.08Hz and f = 2.53Hz, T? and Ea,f increase quickly with amplitude. This i s due to an increase in turbulence l e v e l with amplitude resulting in higher energy d i s s i p a t i o n . However, at lower values of p d 2 f / M (206 and 270) corresponding to f = 0.22Hz and f = 0.36Hz, r? and E af remain e s s e n t i a l l y constant (slight tendancy to increase or decrease, depending on other parameters involved such as h/d). It should be noted that at these low frequencies, the e f f e c t i v e Reynolds number i s low and the flow i s e s s e n t i a l l y laminar over the amplitude range investigated. Hence the energy dis s i p a t i o n process remains v i r t u a l l y unaffected. The results are reinforced by those of Figure 12 for model 9 obtained using the other f a c i l i t y . Although the frequency h/d=3/4 0 2.0 4.0 6.0 8.0 a/d 0 2.0 4.0 6.0 8.0 a/d CO Figure 10 Variation of the damping r a t i o with amplitude of o s c i l l a t i o n s as affected by the frequency and l i q u i d height. Note the e f f e c t of turbulence at higher Reynolds number 3 0 A 20] 10 Damper 1 D/d = 13.8 f = 2.53Hz p d 2 f / y = 1450 p d 3 f 2 / a = 1.21 f 2 d / g = 15.7x10 h/d=1/4 Eo,f 3 0 2 0 10H Damper 2 D/d = 4 . 5 f = 2.08Hz p d 2 f / * i = 2536 p d 3 f 2 / a = 2.53 f 2 d / g = I 5 . 4 x l 0 - 3 h/d=1/4 0 .4 0 .8 1.2 1.6 a/d 0 0 .4 0 .8 1.2 1.6 a/d 9 0 0 | 6 0 0 3 0 0 1/4 h/d=3/4? Damper 1* D/d = 13.8 f * 0.36Hz pd2t/n « 206 p d 3 f J / a = 0.02 f 2 d / g = 0 . 3 2 x l 0 - 3 3 0 0 J 2 0 0 1 0 0 1/4 Damper 2 D/d = 4 . 5 f - 0.22Hz pd2f/<i « 270 p d J f 2 / a * 0.03 f 2 d / g = 0.17x10-' h/d=3/4 0 2 .0 4 . 0 6 .0 8 .0 a/d ~2X) 4^0 6^0 & \ c T a/d Figure 11 Energy dissipation as affected by the damper amplitude and Reynolds number C O co 39 Damper 9 1.2-D/d = 15.2 f = 2.20Hz pd 2f/M = 525 p d 3 f 2 / o = 0.25 f 2 d / g = 7.6X10-3 0.8- 3/4 0.4-1/2 d, n * ' h/d -i 1 1 = 1/4 1.5 2.5 3.5 4.5 a/d Figure 12 Effect of rrt = a/d as recorded during an a i r bearing s t a t i c stand test 40 is high (2.20Hz), the small damper size leads to a low value of pd2i/u (525) resulting in only small variation in TJ. Similar trends were observed for p62f/u as high as 1000. In summary, damping remains constant with amplitude for low values of pd2t/u (< 1000) but increases with amplitude when turbulence sets i n . Typical chart records of amplitude decays are shown in Figure 13. Note, in later experiments, TJ and E a f are calculated using an arbit r a r y (convenient) amplitude r a t i o rather than evaluating i t over a cycle. Furthermore, the type of motion, i . e . pure rotation or translation plus rotation, a f f e c t s the energy d i s s i p a t i o n process. It was demonstrated, using the second test f a c i l i t y (Figure 7), that pure rotation contributes very l i t t l e to damping. Dampers were more e f f i c i e n t when i n s t a l l e d farther away from the axis. ( i i ) 7 r 2 = h/d Damping ra t i o TJ and energy dissipation rate E a f are highly dependant on the l i q u i d height. For c i r c u l a r cross-section dampers (models 1 and 9), optimum value of the l i q u i d height parameter leading to maximium TJ was found to be in the range 3/4 - 1. The parameters p d 3 f 2 / a and f 2d/g have only a s l i g h t effect on t h i s optimum value. However, E a f has a tendancy to be maximum for low values of h/d, t y p i c a l l y equal to 1/8. In other words, the increase in damping T? with h/d, as observed in Figures 14(a) and (b) , i s at the cost of an increase in the amount of l i q u i d . 41 = 2.53Hz, h/d = 1/4, damper 1 f = 1.29Hz, h/d = 3/4, damper 8 Figure 13 Typical amplitude decay plots for the system f i t t e d with a nutation damper Damper 1 a/d = 1.0 D/d = 13.8 1/4 1/2 3/4 (a) Damper 1 1 h/d a/d = 1.0 D/d = 13.8 . f = 0.36Hz V 1/4 1/2 3/4 1 (b) h/d 1.2 o.8 ^ 0.4 Damper 2 a/d = 1.0 D/d = 4 . 5 2.27Hz f = 1.79Hz 6.0-I 7/4 1/2 3/4 (c) Damper 2 1 h/d a/d = 1.0 D/d = 4 . 5 0.22Hz f = 0.42Hz 0 1/4 1/2 3/4 (d) Figure 14 Variation of the damping r a t i o with the l i q u i d height (h/d) at several d i f f e r e n t frequencies for c i r c u l a r (1) and square cross-section (2) dampers. Note the square section damper with lower frequency attains peak value at lower h/d h/d 43 The square cross-section damper (damper 2) showed similar trends at r e l a t i v e l y high values of pd2i/v, pd3t2/o and f 2d/g ( i . e . higher frequencies), where 77 reaches a maximum for h/d close to 1 (Figure 14c). However, at lower values of p d 3 f 2 / o and f 2d/g (lower frequencies), 77 and Ea,f are both optimal for h/d around 1/8 (Figures I4d, 15d). It should be mentioned that experiments with viscous engine o i l s ( i . e . very small pd2f/n) showed similar trends, suggesting that the optimal damping as a function of h/d is e s s e n t i a l l y unaffected by the p d 2 f / M parameter. In summary, there is a value of h/d for which the damping i s optimal. It depends on p d 3 f 2 / a , f 2d/g (frequency), D/d and cross-sectional shape (geometry). This value i s t y p i c a l l y in the range 3/4 - 1, although i t is shifted towards the lower end in some cases due to reduced contribution of the top surface at lower frequency and l i q u i d height. Energy dissipation rate i s generally maximum at low h/d, t y p i c a l l y < 1/8 when there i s high contact surface area per unit volume of water (Figure 15). ( i i i ) TT3 = e/d Models 12 and 13 with rougher bottom surfaces (fine and coarse sandpapers, type 3M garnet paper no. 220, Awt, and 40, Dwt, respectively) were tested for two conditions of pd2f/p., p d 3 f 2 / a and f 2d/g corresponding to f = 2.27Hz and 0.22Hz. The results are presented in Figures 16(a) and (b), respectively. Normally, one would expect higher roughness to raise the turbulence l e v e l . However, the influence of Figure 15 Effect of frequency on c r i t i c a l h/d corresponding to peak energy dissipation 45 a/d = 1 . 0 D/d = 4 . 5 damper 12 damper 2 ,e/dsgO h/d damper 13 damper 2 ,e/d%0 0 1/4 1/2 3/4 1 (b) h/d Figure 16 Effect of surface roughness on the damping r a t i o 46 e/d does not seem to be s i g n i f i c a n t here. Roughness seems to make variation of damping with l i q u i d height smoother, however, ov e r a l l damping c h a r a c t e r i s t i c s are not r e a l l y improved. In another experiment with damper 2 , a l l the inner walls were covered with rubber. No major difference in damping was recorded over that given by the o r i g i n a l plexiglas walls. This suggests that surface material, as well as surface roughness, do not affect damping c h a r a c t e r i s t i c s substantially. (iv) rr,, = pd 2f/M To assess the e f f e c t of Reynolds number, l i q u i d v i s c o s i t y was changed using water and various types of engine o i l s in damper 2 , for given frequencies. This helped iso l a t e the influence of surface tension and gravity at a given value of the Reynolds number. Results of Figure 17 show damping c o e f f i c i e n t 77 and energy d i s s i p a t i o n rate Ea,f to increase slowly as the Reynolds number decreases ( i . e . increasing v i s c o s i t y ) , except for p d 3 f 2 / o =0.61. Similar trends were observed for constant values of f 2d/g (Froude number). Reversal of trend for p d 3 f 2 / a = 0.61 at low values of pd 2f/M would suggest a change in damping and energy di s s i p a t i o n mechanism which i s now governed by the l i q u i d resonance as against v i s c o s i t y . In summary, damping increases smoothly with decreasing values of pd 2f/M (Reynolds number) as the viscous forces i o . 0 ; l 4.0-2.0-damper 2 a/d = 1.0 D/d =4.5 h/d = 3/4 o - C - - (^d 3 f 2/cf ^d3f2/<5-ed3f2/<r ed 3 f 2/d-£d3f2/tf 4.1 2.0 = 0.61 = 0.17 0.055 0 5 10 15 20 400 1200 2000 2800 £d 2f/|* ca,f 80-40-20 10 15 2 ^ 400 ' 1200 ' 2000 ' 2800 ^ 3*i7f Figure 17 Variation of the damping dissipation as functions of Weber numbers r a t i o and energy the Reynolds and 48 become more important. For instance, 77 and Ea,f r i s e by a factor of 6 and 15, respectively, as p&2t/n drops from 2300 to 5, for a constant p d 3 f 2 / a . No sudden t r a n s i t i o n due to changes in flow regime from laminar to turbulent was observed. (v) TT5 = D/d Figure 18 shows the effect of ring diameter on the di s s i p a t i o n process in high and low frequency regimes with the tube cross-sectional diameter held fixed. At a r e l a t i v e l y higher frequency (f = 2.27Hz), both the damping r a t i o and the energy d i s s i p a t i o n function decrease with an increase in the ring to tube diameter r a t i o (D/d < 10). Note, at higher D/d, the damping ratio increases due to larger volume of the l i q u i d , however, more accurate assessment of the e f f i c i e n c y i s reflected by E a f which remains e s s e n t i a l l y constant in this range (D/d > 1 0 ) . At low frequency (f = 0.22Hz) t h i s trend is reversed. 77 and E a f remain e s s e n t i a l l y constant over a range of D/d extending from 0 to 8 (sligh t variation in location of the c r i t i c a l point according to l i q u i d height h/d), and suddendly r i s e to much higher values at larger D/d. Again t h i s may be attributed to the l i q u i d resonance condition. (vi) 7T8 = p d 3 f 2 / o and TT9 = f 2d/g Keeping the Reynolds number (7r4 = pd 2f/y) constant permits the study of the combined effect of TT8 (Weber number) and 7r 9 (Froude number). The results can be plotted Figure 18 Effect of ring diameter on d i s s i p a t i o n c h a r a c t e r i s t i c s in high and low frequency regimes 50 in terms of ?T8 (Figures 19a and b), or TT9 (Figures 19c and d). The trends in both cases are the same : there is a c r i t i c a l region where rj and E a f are much higher as shown for three values of the Reynolds number. The c r i t i c a l range associated with the l i q u i d resonance is around 0.4-1.0 for the Weber number and (1.5-4.0) 10"3 for the Froude number. This dependance on TT8 and 7 r 9 , regardless of the viscous contribution, points out the importance of the free surface. The surface tension and gravity forces appearing in the Weber and Froude number indeed af f e c t the o s c i l l a t o r y motion of the free surface. However, contribution of the surface tension force i s expected to be i n s i g n i f i c a n t here. On the other hand, gravity represents the restoring force during sloshing motion, hence TT9 parameter should be predominant. The effect of Tr9 i s primarly confined to the c r i t i c a l region only, where the natural frequency of the sloshing l i q u i d (which is a function of gravity and therefore of TT 9) coincides with the excitation frequency leading to maximum E a j and T? (Appendix I I I ) . Additional sets of experiments were car r i e d out to measure the influence of frequency alone (for the same l i q u i d ) , which involves a v a r i a t i o n in pdzt/n, p d 3 f 2 / a and f 2d/g simultaneously. Dampers 1 , 2, 7 , 8 and 9 were used to this end (Figure 20). Since the influence of pd2f//x is very smooth, as shown in section 4.1 ( i v ) , and surface tension i s not an important parameter here, the e f f e c t of a change in frequency i s mostly r e f l e c t e d through the Froude number. For a given D/d r a t i o , corresponding to a given damper, 0 4.0 p d 3 ? 2 / ^ 0 4.0 ?d 3f 2/, •(Od2f/fA = 5 • f d 2 f / ^ = 2 0 >^d2f/p = 8 0 0 damper 2 1 0 0 H a/d = i.o D/d = 4.5 h/d = 3/4 8 1 2 1 6 f 2 d / g x 1 0 3 (c) f 2 d / g x l 0 * Plots showing the influence of the Weber and the Froude numbers on the dissipation parameters 10.0-4.0 2.0 • • d a m p e r 1, D / d = 1 3 . 1 • " d a m p e r 2, D / d = 4 . 5 A A d a m p e r 7, D / d = 2 . 6 5 - - • d a m p e r 8, D / d = 1 5 . 2 A A d a m p e r 9, D / d = 1 5 . 2 a/d = 1.0 h/d = 3/4 0.4 0 .8 1.2 1.6 2.0 2.4 2.8 3 .2 3 .6 4 . 0 4 .4 f ,Hz cn K) Figure 20 Effect of frequency and damper diameter r a t i o D/d on the dissipation functions 77 and E a f 53 there i s a region for which damping is s i g n i f i c a n t l y higher. This region is the low frequency area for high D/d ratios (e.g. D/d = 15.2), and is s h i f t e d to higher frequencies as D/d decreases (e.g. D/d = 4.5 and 2.65). This reinforces the results obtained in section 4.1 (v). Changes in damping in t h i s c r i t i c a l region are again sudden and drastic and therefore there i s a r e l a t i o n s h i p between the c r i t i c a l region location, D/d r a t i o and Froude number f 2d/g, corresponding presumably to the f i r s t natural frequency of the sloshing l i q u i d . Appendix III v e r i f i e s these premises. It i s interesting to note that damper 2 ( i . e . D/d = 4.5) shows a second peak at f = 2.10Hz, which corresponds to the second natural frequency of the l i q u i d . 4.2 Specific Parameters (i) Balls in Flow This brings us to the effect of the various internal configurations described e a r l i e r . The f i r s t configuration involved introduction of small spheres r o l l i n g in the l i q u i d f i e l d during nutational motion. This results in additional energy d i s s i p a t i o n mechanisms through sphere drag and impact of the b a l l s against the walls of the dampers. Effect of two parameters were investigated : the number of b a l l s and the b a l l size db/d. Although dampers 14 and 20 were used in the test program, for conciseness only the results for damper 14 are presented in Figure 21, for several values of h/d r a t i o and frequency. It i s clear that increasing n improves damping of the 54 damper 14 0.6-0.4-0.2-damper 14 a/d = 1.0 D/d =4.5 h/d = 3/4 f = 1.35Hz pd 2f/p. = 1643 p d 3 f 2 / a = 1.06 f 2d/g = 6.5xl0' 3 4.0+ 2.0 0.8i db/d=0.31 0.42Hz damper 14 n = 1 0.1 0.2 0.3 db/d (a) a/d = 1.0 D/d = 4.5 h/d = 1/8 f = 2.2 7Hz ~2 1.2H 0.8 0.4 d b/d=0.31, n = 10 0 l 7 4 1/2 3/4 (c) 4 6 (b) 8 10 n — damper 14 —- damper 2 a/d = 1.0 D/d = 4 . 5 f = 2.27Hz pd2t/u = 2762 p d 3 f 2 / o = 3.02 f 2d/g = 18.4X10" 3 h/d Figure 21 Representative results showing the influence of spheres in the f l u i d f i e l d on the damping r a t i o 5 5 system only s l i g h t l y (Figure 21b). Up to ten b a l l s with db/d = 0.46 and 0.31 were introduced in the flow of damper 20 and 14, respectively. The v a r i a t i o n in 7? with the l i q u i d height was mostly small except for two peaks. Optimal b a l l size was found to be around d^/d * 0.20, for h/d = 3/4, with damper 14 (Figure 21a). However, i t seems to be a function of other parameters such as D/d, f, etc. The performance of damper 14 with ten b a l l s of d^/d = 0.31 was tested over the entire range of h/d and compared with damper 2 (no b a l l , but same geometry), as shown in Figure 21(c). The damping i s increased over the entire range, with the peak value changing from 1.14% to 1.42%. Ea_f i s not shown here and is purposely omitted in the rest of the discussion, since the interest i s in performance of dampers with modified geometry as compared to the plai n damper 2 containing the same amount of l i q u i d . The comparison extends over the entire range of h/d in order to provide a more complete picture of the r e l a t i v e performance. (i i) Screens In another set of experiments, the damper was f i t t e d with screens of d i f f e r e n t wire diameter and open area, in order to promote turbulence. Two simple screens of open area r a t i o A = 0.77, each made of two wires bonded together into a cross shape, were f i r s t tested to evaluate merit of the concept. Results did not show s i g n i f i c a n t improvement in the peak value, although they were promising since the l o c a l values increased for l i q u i d heights of h/d = 1/4 to 56 3/4 (Figure 22a). A screen of A = 0.70 was then t r i e d and a similar trend was observed (Figure 22b). The number of screens was now varied. Damping ba s i c a l l y increases with increasing n, although performance depends on screen position with respect to damper direction of motion, as shown in Figure 22(c). Note, the performance is optimal when screens are positioned perpendicular to the d i r e c t i o n of motion. Under this condition, the damping increased to 2.25% with two screens, at a r a t i o of h/d = 15/16 and a frequency of 2.27Hz (iru = 2762). Systematic tests were not carried out to optimize the open area, but indications are that a value greater than 0.70 is undesirable. It should be noted that, for f = 0.22Hz, screens have no or even detrimental e f f e c t s . In other words, i t is helpful to increase the turbulence l e v e l of a flow already turbulent, which i s the case at pd 2f/n = 2762. But in laminar flows, at pd 2f/y = 270 for instance, the effect of screens was n e g l i g i b l e or even detrimental (not shown). ( i i i ) Inside Perforated Tube The idea of an inside tube i s to increase l i q u i d - w a l l interactions. A c i r c u l a r tube of 1.1cm diameter (d,/d = 0.31) and without holes was f i r s t positioned at the bottom of damper 16. Although magnitude of the damping ratio did not change by a s i g n i f i c a n t amount, i t remained e s s e n t i a l l y uniform over a wide range of h/d, from 1/4 to 3/4, as shown in Figures 23(a), (b) for the condition of ?ra, 7r 8 and TT9 57 damper 15 a/d =1.0 D/d = 4.5 f = 2.27Hz pd2t/u = 2762 p d 3 f 2 / a = 3.02 f 2d/g = 18.4X10-3 1.2 0.8 0.41 — damper 15 damper 2 A/d 2 =0 0 1/4 1/2 3/4 ( a ) T),% 1.21 0.8 0.4 -damper 15 damper 2 A/d 2=0.70 , n=1 1/4 1/2 3/4 (b) damper 15 h/d = 15/16 A/d 2 = 0 .70 1 h/d F i g u r e 22 T y p i c a l r e s u l t s s h o w i n g t h e e f f e c t o f s c r e e n s on t h e d a m p i n g r a t i o 58 damper 16 dj/d =0.31 1.2 0.8 0.4 h7d=q/d = n o=0 1/4 1/2 3/4 1 (a) 6.0 4.0 2.0 h/d=q/d=n o=0 0 1/4 1/2 3/4 1 (b) damper 16 -damper 2 a/d = 1.0 D/d =4.5 f = 2.27Hz p d 2 f / M = 2762 p d 3 f 2 / o = 3.02 f 2d/g = 18.4X10~3 h/d -damper 16 damper 2 a/d = 1.0 D/d =4.5 f = 0.22Hz pd2f/*x = 270 p d 3 f 2 / o = 0.03 f 2d/g = 0 . 1 7 X 1 0 ' 3 h/d Figure 23 Plots showing the influence of inside perforated tube on the damping .ratio: (a) high frequency case with inside tube without holes; (b) low frequency case with inside tube without holes 59 corresponding to f = 2.27Hz, 0.22Hz. Next, the number of holes n 0, the tube height h*/d and the hole size q/d were optimized. The inside tube, with thirty-two holes of q/d = 0.17, located at the top of the damper (h*/d = 1), was found to provide optimal damping as shown in Figures 23(c), (d), (e) and ( f ) . It should be noted that lower placement- of the inside tube (h*/d ^  1/2) improves damping for low values of h/d whereas i t s higher location (h*/d > 1/2) improves damping for larger h/d. Optimal arrangement led to a s i g n i f i c a n t improvement in the damping ra t i o at a higher frequency f = 2.27Hz, with r? increasing from 1.14% to 2.2%. However, the concept i s not p a r t i c u l a r l y e f f e c t i v e at a lower frequency (f = 0.22Hz). (iv) Baffles Baffles add pertubations to flow as well as provide more contact surfaces for the l i q u i d in motion which should contribute to the energy d i s s i p a t i o n process. Eight r i g i d p lexiglas baffles r i g i d l y fixed to the walls were f i r s t t r i e d in damper 17. The projected surface area of each b a f f l e was 25% of the tube cross-sectional area (b/d 2 = 0.25) and i t s location was varied. Figures 24(a) and (b) c l e a r l y show that the optimal ba f f l e height i s 1/2 for both low and high frequency conditions. Several schemes for positioning of baff l e s were investigated. The configuration involving alternate location of baffles on inside and outside walls was found to be more e f f i c i e n t . Here, the damping r a t i o jumped to 1.63% at f = 2.27Hz, for optimal 2.4 1.6 0.81 damper 16 a/d = 1.0 D/d = 4.5 f = 2.27Hz pd2t/u = 2762 pd 3 f 2 / o = 3.02 f 2d/g = 18.4x10 . h/d =1/4 * h*/d=1/2 a h*/d=3/4 • h*/d=1 q/d=0.11 , nQ=32 1/4 1/2 3/4 (c) damper 16 a/d =1.0 D/d = 4.5 f = 0.22Hz pd 2f/*i = 270 p d 3 f 2 / o = 0.03 f 2d/g = 0.17x10"3 h/d • h/d =1/4 • rf/d=1 q/d = 0.11 , n G=32 •--• <s 1/4 1/2 3/4 (d) h/d 2.01 1.5 1.01 q/d=0.17 2.41 1.6 0.81 - h/d=l/2 h/d=0 damper 16 a/d = 1.0 D/d =4.5 f = 2.27Hz pd 2f//i = 2762 p d 3 f 2 / o = 3.02 f 2d/g = I8.4xl0- 3 3/4 15/16 (e) damper 16 a/d = 1.0 D/d =4.5 f = 2.27Hz p d 2 f / M = 2762 p d 3 f 2 / o = 3.02 f 2d/g = 18.4X10" 3 0 1/4 1/2 3/4 (f) Figure 23 Plots showing the influence of inside perforated tube on the damping r a t i o : (c) tube location , high frequency; (d) tube location, low frequency; (e) number of holes; (f) hole size at high f requency h/d • q/d=0.43 A q/d=0.29 • q/d=0.17 h*/d=1 , n Q=32 h/d Ch O 61 damper 17 b/d2=0.25 a/d D/d 1.0 4.5 f = 2.27Hz h'/d = 1/2 rubber,.^ hinged plexiglas / \\ \./" cardboard 1/4 1/2 3/4 1 h/d (b) f = 0.22Hz h'/d = 1/2 \ hinged / ;\plexiglas t \ V cardboard / >' rubber—f 1/4 1/2 3/4 1 h/d (d) 1/4 1/2 3/4 1 h/d ( c ) F i g u r e 24 E f f e c t o f r i g i d a n d f l e x i b l e b a f f l e s on t h e e n e r g y d i s s i p a t i o n p r o c e s s . N o t e t h e s t a g g e r e d a r r a n g e m e n t o f b a f f l e s on i n s i d e and o u t s i d e w a l l s a t h i g h f r e q u e n c y r e p r e s e n t s a f a v o u r a b l e a r r a n g e m e n t . B a f f l e s a r e n o t p a r t i c u l a r l y e f f e c t i v e a t l o w f r e q u e n c y 62 h/d. Furthermore, an improvement was seen over the entire range of h/d. More importantly, the damping increased at lower frequency as well (f = 0.22Hz) which was not the case with screens, inside perforated tube, or spheres in flow. It should be noted that the number of baff l e s and the baffle size were not studied systematically, but indications are that eight baffles of half the tube height and width are l i k e l y to be most e f f i c i e n t . Tests were also conducted with several f l e x i b l e baffles to assess the potential of exploiting energy dissipation through the structural damping of the baf f l e material i t s e l f . F l e x i b l e rubber b a f f l e s , hard paper and hinged plexiglas baffles were t r i e d but the results showed l i t t l e change from the no b a f f l e case (Figure 24c, d). Thus f l e x i b l e b a f f l e s do not seem to affect substantially the energy dis s i p a t i o n mechanism involving w a l l - l i q u i d and l i q u i d - l i q u i d interactions. (v) Horizontal Layers A horizontal plexiglas floor was i n s t a l l e d into damper model 18 to increase l i q u i d - w a l l interactions. Results were not encouraging at high frequency, as demonstrated by Figure 25(a), but were very good at low frequency (Figure 25b). Improvement in the damping r a t i o was seen over the entire range of h/d and optimal damping c o e f f i c i e n t was increased from 4.0% to 7.5%. This suggests that liquid-bottom wall interactions are very important at low frequencies of o s c i l l a t i o n s . However, the i r contribution to energy 63 0 1/4 1/2 3/4 1 (a) 0 1/4 1/2 3/4 1 (b) damper 18 damper 2 a/d = 1.0 D/d = 4.5 f = 2.27Hz p d 2 f / M = 2762 p d 3 f 2 / a = 3.02 f 2 d / g = 18.4X10"3 h/d •damper 18 damper 2 a/d = 1.0 D/d =4.5 f = 0.22hz pd 2f/jx = 270 p d 3 f 2 / a = 0.03 f 2 d / g = 0.17x10"3 h/d Figure 25 P a r t i t i o n i n g of the damper to provide two layers of l i q u i d s i g n i f i c a n t l y improves the damping at low frequency 64 dis s i p a t i o n diminishes as the frequency increases. The number of layers was not increased any further, however one can anticipate similar trend. (vi) Pieces of Wood in Flow Damping results with three f l o a t i n g rectangular pieces of wood, 3.5cmx1.9cmx1.1 cm in size , are shown in Figure 26. Si g n i f i c a n t improvement in d i s s i p a t i o n character i s seen at high frequency (2.27Hz), where drag and f r i c t i o n generated by the pieces of wood increase damping for a l l conditions of h/d, with the peak r? raised from 1.14% to 2.27%. At lower frequency (f = 0.22Hz), the concept does not seem to be e f f e c t i v e as the damper performance, with or without pieces of wood, i s about the same. Additional experiments were carri e d out to find the optimum number of pieces. For the given v/d 3 = 0.17, n w = 3 was found to be the best. Additional tests are necessary to explore the concept further. To help assess comparative performance, the damping results for more promising configurations are summarized in Figures 27 and 28. In the high frequency range, perforated inside tube device and f l o a t i n g pieces of wood exhibit highest peak values of rj, while baf f l e s show more consistent improvement over the entire range of h/d. As discussed e a r l i e r , the two layer damper i s not competitive at high frequency, but i s most successful in the low frequency range (f = 0.22Hz.). As can be expected, the energy dis s i p a t i o n rate E a f shows peak values in the low range of h/d where 65 damper 19 v/d3=o.17 n w = 3 2.27 0 1/4 1/2 3/4 damper 19 damper 2 a/d = 1.0 D/d =4.5 f = 2.27Hz pd'f/y = 2762 p d 3 f 2 / o = 3.02 f 2d/g = 18.4xl0" 3 h/d 6.0-1 4.0 2.01 — damper 19 damper 2 a/d = 1.0 D/d =4.5 f = 0.22Hz p d 2 f/ M = 270 p d 3 f 2 / o = 0.03 f 2d/g = 0 . 1 7 X 1 0 " 3 0 1/4 1/2 3/4 1 h/d F igure 26 E f f e c t of f l o a t i n g wooden p i e c e s on dampinq c h a r a c t e r i s t i c s 2.4 2.0 1.6 1.2 • • Plain Damper (2) * * Inside Tube (16) • • Baffles (17) • D j w o Layers (18) A A Wood Pieces (19) r),% a/d =1.0 D/d = 4.5 f = 2.27Hz pd2t/u = 2762 p d 3 f 2 / a = 3.02 f 2d/g = 18.4x10-3 1/4 1/2 3/4 1 h/d 12 10H 8 6 4 2 a/d = 1.0 D/d =4.5 f = 0.22Hz pd2t/u = 270 p d 3 f 2 / a = 0.03 f 2d/g = 0.l7xi0- 3 0 1/4 1/2 3/4 1 h/d Figure 27 Comparative performance of several more promising damper configurations cn :o,t 12-10-8-6-4-2--• Plain Damper (2) * Inside Tube (16) • -Baffles (17) • a Two Layers (18) A A Wood Pieces (19) a/d = 1.0 D/d = 4.5 f = 2.27Hz pd2t/u = 2762 p d 3 f 2 / a = 3.02 f 2d/g = 18.4X10" 3 •a,f 600-500-400-300-200-100-1/4 1/2 3/4 1 h/d a/d = 1.0 D/d =4.5 f • 0.22Hz p d 2 f / M = 270 p d 3 f 2 / 0 = 0.03 f 2d/g = 0.17x10-3 * A £t • 1/4 1/2 3/4 1 h/d CTl Figure 28 Relative energy d i s s i p a t i o n c h a r a c t e r i s t i c s of four more favourable damper geometries 68 dampers are most e f f i c i e n t (Figure 28) whereas Figure 27 emphasized high values of h/d where damping ratios are maximum. However, both figures lead to similar conclusions. Figure 29 assesses comparative performance of dampers tested on the s t a t i c stand test f a c i l i t y and the a i r bearing system i l l u s t r a t e d in Figure 7. It i s interesting to notice that, although conditions of D/d and p d 2 f/ M , p d 3 f 2 / o , f 2d/g (involving frequency) are quite d i f f e r e n t , the same trends are present : dampers with perforated inside tube show highest peak performance, whereas those whith b a f f l e s display best consistency over the entire range of h/d. ( v i i ) Combined Devices In previous, experiments, d e f i n i t e improvements in di s s i p a t i o n character were observed with inside perforated tube, b a f f l e s , spheres, etc., at either f = 2.27Hz or f = 0.22Hz, or both. It was, therefore, thought that combining the devices may further improve performance in the low and high frequency ranges. The following combinations were t r i e d : • one perforated tube and baff l e s (damper 24); • one horizontal p a r t i t i o n with two perforated tubes (damper 21); • one horizontal p a r t i t i o n with b a f f l e s (damper 22); • one horizontal p a r t i t i o n with pieces of wood (damper 23). Surprisingly, combination of devices did not r e a l l y improve the peak performance, probably 0.6-0.4 0.2H 0.6H 0.4 0.2 • Plain Damper (9) • Baffles (10) * Inside Tube (11) a/d = 1.0 D/d = 15.2 f = 2.20Hz pd2t/u = 525 p d 3 f 2 / a = 0.25 f 2d/g = 7.6x10 1/4 1/2 3/4 1 h/d -a,f 4.0 3.0 2.0 1.0 a/d = 1.0 D/d = 15.2 f = 3.32Hz pd2t/u = 792 p d 3 f 2 / a = 0.56 f 2d/g = 17.4x10-3 4.0 3.0-2.0-1.0 1/4 1/2 3/4 1 h/d a/d = i.O D/d = 15.2 f = 2.20Hz p62t/u = 525 p d 3 f 2 / o = 0.25 f 2d/g = 7.6x10-3 1/4 1/2 3/4 1 h/d a/d = i.O D/d = 15.2 f = 3.32Hz pd2t/u = 792 p d 3 f 2 / a = 0.56 f 2d/g = 17.4x10-3 0 1/4 1/2 3/4 1 h/d Figure 29 Plots showing comparative performance of the more successful configurations when tested using the a i r bearing f a c i l i t y 70 because of the complex energy dissipation mechanisms, not yet understood (Figure 3 0 ) . However, combined device dampers having a horizontal layer behaved well in both frequency ranges and therefore are more v e r s a t i l e than single device dampers, in spite of lower peak values. 2.4-2.0-1.6 1.2-0.8-0.4-A • — • d a m p e r 2 —•damper 21 -adamper 2 2 --^damper 2 3 --Adamper 2 4 a/d = 1.0 D/d = 4.5 f = 2.27Hz pd2t/u = 2762 pd3t2/o = 3.02 f 2 d / g = 1 8 . 4 x 1 0 - 3 12 10 a / d = 1 . 0 D/d = 4.5 f = 0 . 2 2 H z p d 2 f / M = 270 p d 3 f 2 / o = 0 . 0 3 f 2 d / g = 0 . 1 7 X 1 0 " 3 0 1/4 1/2 3/4 1 h/d 0 1/4 1/2 3/4 1 h/d Figure 30 Plots showing comparative performance of -J combined devices in high and low frequency ~* regimes: 2 (plain damper); 21 (layers with tubes); 22 (layers with b a f f l e s ) ; 23 (layers with wood pieces); 24 (tube with baf f l e s ) 72 5. CLOSING COMMENTS 5.1 Conclusions The experimental program has attempted to assess the influence of more s i g n i f i c a n t parameters on the energy dis s i p a t i o n process. The amount of information obtained is rather extensive and would be useful in the design of e f f i c i e n t dampers for a wide range of applications. Based on the tests results following general conclusions can be made : (i) In general, at higher Reynolds numbers, an increase in amplitude of motion results in higher energy di s s i p a t i o n rate due to the corresponding increase in the turbulence l e v e l . Although higher turbulence l e v e l i s b e n e f i c i a l , viscous forces are more e f f i c i e n t in d i s s i p a t i n g energy. Thus lower values of p d 2 f / V i s desirable. Recognizing that f cannot be controlled and variation in d would affect several other dimensionless parameters, the easiest way to r e a l i z e t h i s condition is through the use of high kinematic v i s c o s i t y f l u i d . ( i i ) Damping i s primarily dependant on h/d ( l i q u i d height), D/d (damper geometry), and f 2d/g (frequency of o s c i l l a t i o n s ) parameters. There i s a relationship between optimal damping and these three parameters that corresponds to Qthe situation when the l i q u i d frequency reaches resonance. The information should prove useful in the damper design when the structural frequency of o s c i l l a t i o n i s known. For 73 instance, low structural frequencies imply use of dampers with high D/d r a t i o for a given h/d. ( i i i ) The damper material and i t s surface roughness do not have s i g n i f i c a n t influence on the damper c h a r a c t e r i s t i c s . (iv) As damping is primarily due to viscous action, velocity gradient appears to be the key parameter. Hence linear motion of the damper aff e c t s dissipation substantially compared to pure rotational motion about i t s diameter. (v) S i g n i f i c a n t increase in damping can be acheived by appropriate modifications of the damper internal geometry. Baffles, perforated tube, pieces of wood and multiple layers can increase damping r a t i o by about 100%. Each individual device provides optimal damping for s p e c i f i c conditions of damper parameters (f, h/d, e t c . ) . Although combination of the multiple layer damper with other configurations (baf f l e s , perforated tube, pieces of wood) s l i g h t l y reduced the peak damping, i t may be preferred due to improved performance under a l l conditions (f, h/d, e t c . ) . (vi) In general, low l i q u i d heights (h/d ^ 1/8) lead to favourable E a f , which suggests that torus dampers with wide cross-section and low wall height would be most e f f i c i e n t . On the other hand, a torus damper with c i r c u l a r cross-section is l i k e l y to be more readily available. For a given damper volume, to dissipate a desired amount of energy, i t 74 may become necessary to use larger quantity of f l u i d even at a cost of dissipation e f f i c i e n c y E a f . In such situations dampers of c i r c u l a r cross-section with h/d = 3/4 - 7/8 may prove to be desirable. ( v i i ) The design of an optimal damper would therefore include the following features: • A viscous l i q u i d p a r t i a l l y f i l l i n g the damper to h/d in the range 3/4 - 7/8. • An internal configuration consisting of: two (or more) horizontal layers with each chamber carrying a perforated tube of optimal c h a r a c t e r i s t i c s described in Chapter 4. A l t e r n a t i v e l y , i t may carry eight b a f f l e s , or three rectangular wooden blocks of optimum proportions. • . The structural frequency represents a key input information. When known, D/d should be chosen (for selected h/d) so that l i q u i d resonance i s attained for optimal damping. 5.2 Recommendations for Future Work (i) The present study dealt with c h a r a c t e r i s t i c s of nutation dampers under the condition of free o s c i l l a t i o n s . The next l o g i c a l step would be to assess their effectiveness during forced vibrations simulating real l i f e conditions of vortex resonance and galloping. Although i t i s anticipated that forced vibrations would not a l t e r the damper c h a r a c t e r i t i c s s u b s t a n t i a l l y , i t s experimental v e r i f i c a t i o n i s e s s e n t i a l , 7 5 both in laminar as well as turbulent shear flows. Of course, turbulence effects are l i k e l y to be less c r i t i c a l in the vortex resonance case. Furthermore, i t would be useful to conduct tests in plunging, torsion and coupled conditions to establish damper effectiveness over a range of dynamic conditions. ( i i ) It would be p r o f i t a b l e to assess the damper effectiveness for structures located in the wake of other structures. This is p a r t i c u l a r l y s i g n i f i c a n t because several conventional devices for wind-induced o s c i l l a t i o n suppression have proved i n e f f e c t i v e in this s i t u a t i o n . ( i i i ) Tests with prototype dampers may prove to be d i f f i c u l t and expensive, however, they would provide the most convincing substantiation of the concept. In this respect, prototype tests with transmission l i n e s may prove to be r e l a t i v e l y less d i f f i c u l t to acheive. (iv) Application of the concept to damping of off-shore structures under wind, waves and/or ocean current excitations i s a f i e l d that remains v i r t u a l l y untouched. 76 BIBLIOGRAPHY 1. Zdravkovich, M.M., "Review and Assessment of Various Aero and Hydrodynamic Means for Suppressing Vortex Shedding", Proceedings of the 4th Colloquium on  Industrial Aerodynamics , Aachen, June 18-20, 1980, Edited by C.Kramer, H.J.Gerhardt, H.Ruscheweyh and G. Hirsh, F l u i d Mechanics Laboratory, Department of Aeronautics, Fachhachschule, Aachen, Part 2, pp.29-47. 2. Every, M.J., King, R., and Weaver, D.S., "Vortex-Excited Vibrations of Cylinders and Cables and their Suppression", Ocean Engineering , Vol.9, No.2, 1982, pp.135-157 . 3. Scruton, C , and Walshe, D.E.J., "A Means for Avoiding Wind-Excited O s c i l l a t i o n s of Structures with Circular or Nearly Circular Cross-Section", National Physical  Laboratory , Report 335, 1957; also: B r i t i s h Patent No.907, 851. 4. Wong, H.Y., and Cox, R.N., "The Suppression of Vortex Induced O s c i l l a t i o n s on C i r c u l a r Cylinders by Aerodynamic Devices", Proceedings of the 3rd Colloquium  on Industrial Aerodynamics , Aachen, June 14-16, 1978, Edited by C.Kramer, and H.J.Gerhardt, F l u i d Mechanics Laboratory, Department of Aeronautics, Fachhochschule, Aachen, Part 2, pp.185-204. 5. Hirsh, G., and Ruscheweyh, H., " F u l l Scale Measurements on Chimney Stacks", Journal of Industrial  Aerodynamics , Vol.1, 1975/1976, pp.341-347. 6. Ruscheweyh, H., "Straked In-Line Steel Stacks with Low Mass Damping Parameter", Proceedings of the 4th  Colloquium on Industrial Aerodynamics , Aachen, June 18-20, 1980, Edited by C.Kramer, H.J.Gerhardt, H. Rusheweyh, and G.Hirsh, F l u i d Mechanics Laboratory, Department of Aeronautics, Fachhochschule, Aachen, Part 2, pp.195-204. 7. Naudascher, E., Weske, J.R., and Fey, B., "Exploratory Study on Damping Galloping Vibrations", Proceedings of  the 4th Colloquium on Industrial Aerodynamics , Aachen, June 18-20, 1980, Edited by C.Kramer, H.J.Gerhardt, H.Ruscheweyh, and G.Hirsh, F l u i d Mechanics Laboratory, Department of Aeronautics , Fachhochschule, Aachen, Part 2, pp.283-293. 8. Wardlaw, R.L., and Cooper, K.R., "Dynamic Vibration Absorbers for Suppressing Wind-Induced O s c i l l a t i o n s " , Proceedings of the 3rd Colloquium on Industrial  Aerodynamics , Aachen, June 14-16, 1978, Edited by C.Kramer, and H.J.Gerhardt, F l u i d Mechanics Laboratory, Department of Aeronautics, Fachhochschule, Aachen, Part 2, pp.205-220. 77 9. Hirsh, G., "Control of Wind-Induced Vibrations of C i v i l Engineering Structures", Proceedings of the 4th  Colloquium on Industrial Aerodynamics , Aachen, June 18-20, 1980, Edited by C.Kramer, H.J.Gerhardt, H.Ruscheweyh, and G.Hirsh, F l u i d Mechanics Laboratory, Department of Aeronautics , Fachhochschule, Aachen, Part 2, pp.237-256. 10. Den Hartog, J.P., "Self-Induced O s c i l l a t i o n s " , Mechanical Vibrations , McGraw-Hill, New York, 1956, chapter 7, pp.282-329. 11. Lawson, T.V., "Wind Loading: A e r o e l a s t i c i t y " , Wind  Effects on Buildings , Applied Science Publisher LTD, London 1980, chapter 6, pp.97-166. 12. Sachs, P., "Suppression of Vortex Exc i t a t i o n " , Wind  Forces in Engineering , Pergamon Press, 1972, chapter 5, pp.144-146. 13. Sturm R.G., "Vibration of Cables and Dampers", E l e c t r i c a l Engineering , Vol.55, No.4, A p r i l 1936, pp.637-688. 14. Schafer, B., "Zur Entstehung und Unterdruckung winderregter Schwingungen an Freileitungen", Vom  Fachbereich Mechanick der Technischen Hochschule  Darmstadt , Dissertation, 1980. 15. Hagedorn, P., "On the computation of Damped Wind-Induced Excitation of Overhead Transmission Lines", Journal of Sound And Vibration , 1982, pp.253-271. 16. Reed, W.H., "Hanging-Chain Impact Dampers: a Simple Method for Damping T a l l F l e x i b l e Structures", Proceedings of the International Research Seminar , Ottawa, September 11-15, 1967, Wind Eff e c t s on Structures, University of Toronto Press, Volume II, 1968, pp.287-320. 17. Gasparini, D.A., Curry, L.W., and Debchaudhury, A., "Passive V i s c o e l a s t i c Systems for Increasing the Damping of Buildings", Proceedings of the 4th  Colloquium on Industrial Aerodynamics , Aachen, June 18-20, 1980, Edited by C.Kramer, H.J.Gerhardt, H.Ruscheweyh, and G.Hirsh, F l u i d Mechanics Laboratory, Department of Aeronautics , Fachhochschule, Aachen, Part 2, pp.257-266. 18. Schneider, C.C., and L i k i n s , P.W., "Nutation Dampers versus Precession Dampers for Axisymmetric Spinning Spacecraft", Journal of Spacecrafts and Rockets , Vol.10, No.3, March 1973, pp.218-222. 78 19. A l f r i e n d , K.T., " P a r t i a l l y F i l l e d Viscous Ring Nutation Damper", Journal of Spacecrafts and Rockets , Vol.11, No.3, July 1974, pp.456-462. 20. A l f r i e n d , K.T., "Gravity Effects in the Testing of Nutation Dampers", Proceedings of the CNES-ESA  Conference on Attitude Control of Space Vehicles- Technological and Dynamical Problems Associated with  the Presence of Liquids , Toulouse, October 10-12, 1977, ESA SP-129, pp.193-197. 21. A l f r i e n d , K.T., "Methods for Evaluating Gravity Effects in the Testing of Nutation Dampers", Journal of  Spacecrafts and Rockets , Vol.12, No.6, June 1975, pp.359-362. 22. A l f r i e n d , K.T., and Spencer, T.M., "Comparison of F i l l e d and Partly F i l l e d Nutation Dampers", AAS/AIAA  Astrodynamics S p e c i a l i s t Conference , Lake Tahoe, Nevada, August 3-5, 1981, Paper No.81-141. 23. Anzai, T., Ikeuchi, M., Igarashi, K., and Okanuma, T., "Active Nutation Damping System of Engineering Test S a t e l l i t e - I V " , XXIInd IAF Congress , Rome, 1981, Paper No.IAF-81-350. 24. Malik, N.K., and Goel, P.S., "Passive Nutation Dampers for Spinning S a t e l l i t e s " , Indian Space Research  Organization , Bangalore, January 1977, Technical Note ISRO-ISAC-TN-06-77. 25. Modi, V.J., Sun, J.L.C., Shupe, L.S., and Solyomvari, A.S., "Suppression of Wind-Induced I n s t a b i l i t i e s Using Nutation Dampers", Proceedings of the Indian Academy of  Sc iences , Vol.4, Part 4, December 1981, pp.461-470. 26. Brunner, A., "Amortisseurs d ' o s c i l l a t i o n s hydrauliques pour cheminees", les i n s t a b i l i t e s en hydraulique et en  mecanique des fluides , compte rendu des huitiemes journees de 1'hydraulique, L i l l e , Juin 1964, Tome 10, edition La Houille Blanche, Grenoble, France, pp.258-263. 27. Blevins, R.D., "Fluid Systems", Formulas for Natural  Frequency and Mode Shape , Van Nostrand Reinhold Company, 1979, pp.364-385. 28. Banning, D.A., Hengeveld, L.D., and Modi, V.J., "Apparatus for Demonstrating Dynamics of Sloshing Liquids", B u l l e t i n of Mechanical Engineering  Education , Pergamon Press, Vol.5, 1966, pp.65-70. 79 APPENDIX I - SYSTEM CHARACTERISTICS 1 System Mass and Inertia (i) Mass The t o t a l system mass (M) consists of: Ma = I284gms; Mc = 408gms; M<j and Mw. Thus M = 1 692gms + Md + M w . (i i) Inertia The i n e r t i a is calculated with respect to the pivot point. It includes: I a = M aL 2/l2 + M a(L/2) 2 = M aL 2/3 = 1.284x1.2652/3 (L = 1.265m) = 0.685 Kg-m2 ; I c= M CL 2 (considered point mass here) = 0.408x1.2652 = 0.653Kg-m2 ; I = (M d/2)[R 2+(5r 2/4)] + M dL 2 ; where: R = outer radius = (D+d)/2, r = inner radius = (D-d)/2. In most cases : I d « M d L 2 . Iw= MWL 2 . Thus I = t o t a l i n e r t i a = 1.34Kg-m2 + M d[(R 2/2) + (5r 2/8) + 1.60] + 1.60MW. For the alternate a i r bearing system : M = 611gms + Md + M w ; I = O.lOlKg-m2 + M d[R 2 + (5r 2/8) + 0.249] + 0.249MW. 2 System S t i f f n e s s The system has the one-degree of freedom in rotation. It i s held by a spring on one side and a similar spring and 8 0 a s t r a i n gauge in series on the other, with damping. Schematically i t i s shown below where: k s = spring s t i f f n e s s ; kg= s t i f f n e s s of the s t r a i n gauge transducer; C = system damping; M = t o t a l mass; I = t o t a l i n e r t i a about the hinge. Figure 1-1 System representation The equivalent system s t i f f n e s s can be expressed as follows : k = k s + [ ( k s kg ) / ( k s + k g ) ] . Below i s a table of spring and system stiff n e s s e s used. It should be noted that the same st r a i n gauge transducer was used throughout the test program, with k g = 14,700N/m, which does not affect the system s t i f f n e s s s i g n i f i c a n t l y ( k « 2k s). 81 Table II Spring and system stif f n e s s e s Spring. ks(N/m) k (N/m) 1 1891 .0 3566.0 2 1476.0 2818.0 3 671 .6 1313.8 4 247. 1 490.2 5 112.2 244 .4 6 71.6 142.8 3 System Damping There are three sources of damping in the system without nutation dampers: (i) F r i c t i o n : f r i c t i o n of the moveable parts, mainly in the bearings, leads to damping. Spring-rotating arm and spring-frame junctions may also contribute a small amount of energy d i s s i p a t i o n . ( i i ) Aerodynamic damping: o s c i l l a t i n g system generates drag contributing to energy losses. ( i i i ) Hysteresis damping: springs dissipate energy during their tension-compression stress cycles. Other parts may be assumed to be completely r i g i d , and therefore do not contribute to energy l o s s . In general, these contributions to an o v e r a l l system damping vary with i n e r t i a , spring type (material, c o i l 82 diameter, wire diameter, number of turns), frequency, etc. However, the effect of frequency appears to be dominant, as shown in Figure 1-2. The second test f a c i l i t y f i t t e d with a i r bearings showed similar trends. It should be noted that rj s (system inherent damping, does not include nutation damper) is an average value over a range of decaying amplitudes. TJ s has the tendancy to be rather constant with amplitude, showing an exponential decay (Figure 1-3), which indicates that f r i c t i o n (Coulomb damping) i s not the dominant source of damping. 4 System Dynamics The forces acting on the system are shown in Figure 1-4: s t a t i c e q u i l i b r i u m Figure 1-4 Force diagram 0.800H 0.600J 1.896 3.030 3.807 3.835 3.943 0.400H 0.200H 1.0 2.0 3.0 4.0 f . Hz Figure 1-2 Variation of system damping ra t i o (without nutation damper) with excitation frequency over a range of values of system mass. Note the effect of mass is r e l a t i v e l y small 84 f = 0.55Hz, M =2.760Kg, S p r i n g 5 F i g u r e 1-3 T y p i c a l p l o t s s h o w i n g e s s e n t i a l l y e x p o n e n t i a l d e c a y o f a m p l i t u d e w i t h t i m e o v e r a r a n g e o f v a l u e s o f s y s t e m p a r a m e t e r s 85 Taking moment about '0', 10 + C0L + klsin©lcos© - w,Lsin0 - W 2Lsin0/2 = 0 . For small 0, I© + C0L +kl 20 - W,L0 - W2L0/2 = 0 , (l) where W, = weight of support, damper and dead weight = (Mc +Md +Mw)g , and W2 = weight of o s c i l l a t i n g arm = Mag . Equation (1) can be rewritten as, I© + C0L + ( k l 2 - W,L - W2L/2)0 = 0 . (2) I, k, W, and W2 are system constants, however, C i s normally a function of 0, ©, 0. Since C is small, one can use i t s average value over a specified time i n t e r v a l . Hence the system can be treated as l i n e a r , with the solution, 0 = C 1e _ 1' w" tcos(w dt ~ <P) , where: at t = 0 , 0 = 0 O and 0 = 0 , giving C, = constant , The system natural frequency, f n = cjn/2n, can be calculated from equation (2), wn= V U l 2 - W,L - W 2L/2)/l' Hence the damped frequency i s expressed as: f d= (1 /2TT) V( 1 - 7 j 2 ) ( k l 2 - w,L - W 2 L/2 ) / l " The expression gave an accurate value of f d , quite close to the measured value. These equations apply to s t a t i c stand test f a c i l i t y of Figure 6 with b a l l bearings. Similar equations can be obtained for the f a c i l i t y of Figure 7 using a i r bearings. 86 APPENDIX II - ENERGY DISSIPATION VERSUS DAMPING Energy di s s i p a t i o n rate E af and damping c o e f f i c i e n t 77 are the two parameters used to assess damper performance. They are not independant. The relationship between the two can be found as follows. As obtained before (section 2 . 4 ) : Ea,f = [ { ( k / 2 ) ( X l 2 - x m + 1 2)/m} - E s]/(M,f 2a 2) ; (1) and 77 = ln(x ,/x m +, )/27rm (m = number of cycles) . ( 2 ) Let E s be the energy dissipated per cycle by the system without a nutation damper (in the same amplitude range), then E s = ( k / 2 ) ( x , 2 - x m' + 1 2)/m' , where xm'+ , = x m + 1 and m' = number of cycles . Now the logarithmic decrement for the system can be written as 7 7 S = ln(x,/x m ; , ) / 2 r n n ' ; m' = ln(x 1/x m' t , ) / 2 T T 7 7 S . Thus E s = ( k / 2 ) ( x , 2 - xm'+, 2 ) 2 7 T 7 7 S/In (x ,/xmV T ) = (k/2)(x, 2 - x m + , 2 )27T7? s/ln(x 1/x m + , ) . From equation ( 2 ) , m = In (x i / x m + , )/2irrj . Substituting for m and E s into equation ( 1 ) gives, Ea,f= [ { ( k / 2 ) ( x , 2 - x M + 1 2 ) 2 7 T 7 7 / l n ( x 1 / x m + 1 ) } - {(k / 2)(x, 2 - x M + L 2 ) 2 7 T 7 7 s / l n ( x 1 / x m + 1 ) } ] / ( M | f 2 a 2 ) = [ T T U X , 2 - x m + , 2 )/ln(x , / x m t , ) ] [ ( T J -7 7 s)/(M,f 2a 2)] ; Ea,f= «rk[(1 - x m + , 2 A i 2 ) / l n ( x 1 / x m + , ) ] [ (77 -7} S)/(M! f 2 ) ] [ x 2 / a 2 ] . During pure rotational motion, spring displacement (x,) and damper displacement (a) are related, a/x = constant (= 2.126 for the test f a c i l i t y of Figure 6 and 2.763 for Figure 7 Therefore Trx^/a 2 = constant. Thus, E a | f= a[k/M,f 2][(l - x m + , 2/x , 2 ) /In ( x , / x m + , ) ] (rj - T? As an i l l u s t r a t i o n , for x,/x m + 1 = 5, Ea,f= 0.41 5[ k/(M| f 2 ) ] (T? - r?s ) , for a/x = 2.126. 88 APPENDIX III ~ DETERMINATION OF LIQUID NATURAL FREQUENCY During the parametric study, optimal damping appeared to occur at the f i r s t natural frequency of the l i q u i d . This was thought to be responsible for the various values of h/d, D/d and f 2d/g corresponding to damping peaks. The l i q u i d natural frequencies are functions of l i q u i d height, damper geometry as well as l i q u i d properties and would therefore involve several dimensionless numbers mentioned e a r l i e r . It should be noted that a formula for the natural frequencies of sloshing l i q u i d in torus shape containers exists when potential flow can be assumed 2 7: f w = (X/2rrD) (gh) 1 / 2 for shallow l i q u i d , = ( 1/2IT) (2Xg/D) 1 / 2 for deep l i q u i d . In order to v e r i f y these observations, a test f a c i l i t y was used to determine water natural frequencies inside the various dampers. The test arrangement is shown in Figures 111-1,2. E s s e n t i a l l y , i t i s a Scotch-Yoke mechanism providing simple harmonic motion of controlled frequency and amplitude. More d e t a i l s are given in reference 28. The f i r s t natural frequency of l i q u i d for damper models 2 and 8 at various values of h/d are tabulated below: Figure 1 11 - 1 Test arrangement used to determine natural frequency of damping l i q u i d s 90 Table III Variation of l i q u i d natural frequency with l i q u i d height for dampers 2 and 8 Damper h/d f w 2 1/8 0.60 1/4 0.70 1/2 0.83 3/4 1 .00 7/8 1 .07 8 1/8 0.97 1/4 1 .07 1/2 1 .40 3/4 1 .60 Figures 111-3, 111-4 show t y p i c a l f i r s t and second modes, respect i v e l y . Figure III-4 Water in damper 2 executing 2nd mode; h/d = 3/4, f = 2.15Hz 

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