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Suppression of wind-induced torsional instability using partitioned nutation dampers Lim, Seng Boh 1996

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SUPPRESSION OF WIND-INDUCED TORSIONAL INSTABILITY USING PARTITIONED NUTATION DAMPERS Seng Boh Lim B.A.Sc. (Hons), The University of British Columbia, 1994 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF M A S T E R 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 C O L U M B I A September 1996 © Seng Boh Lim, 1996 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada DE-6 (2/88) ABSTRACT The thesis aims at the development of partitioned rectangular and toroidal dampers for suppressing wind-induced instabilities in torsion of bluff bodies like bridge-decks and bundles of transmission line conductors. To begin with, energy dissipation of the dampers as affected by the system frequency and liquid height, in the presence of partitioning, is assessed. This is followed by a qualitative flow visualization study of the surface waves to provide better appreciation of the dissipation mechanism. Finally, a set of wind tunnel tests with a square prism is undertaken to determine the effectiveness of the dampers in suppressing torsional galloping instability. Results suggest that the optimum partitioning corresponds to the compartment length to width ratio of 1.2 for the rectangular damper, while for the double toroid, it represents the diameter ratios of 1.125 and 2 for the outer and inner rings, respectively. In general, for the rectangular damper, roll motion led to a higher damping compared to the pitch degree of freedom. From flow visualization, it appeared that wave breaking as well as collision of waves promote energy dissipation. During the wind tunnel tests, both rectangular and toroidal dampers proved to be quite successful in suppressing galloping instability in torsion. The information can be used to advantage in the design of bridgedecks and high voltage transmission lines, which are often susceptible to this form of instability. i i T A B L E OF C O N T E N T S ABSTRACT ii LIST OF FIGURES vi LIST OF T A B L E S . . . xiii N O M E N C L A T U R E xiv A C K N O W L E D G E M E N T xvii 1. INTRODUCTION 1 1.1 Preliminary Remarks 1 1.2 A Brief Review of the Relevant Literature 2 1.3 Scope of the Investigation 9 2. TEST FACILITIES, INSTRUMENTATION, A N D M E T H O D O L O G Y 12 2.1 Preliminary Remarks . . 12 2.2 Free Vibrational Tests 14 2.3 Forced Vibration Tests 20 2.4 Damper Models 26 2.5 Data Acquisition and Reduction 28 3. STUDY WITH DAMPERS DISSIPATION 32 3.1 Preliminary Remarks 32 ii i 3.2 Rectangular Damper 35 3.2.1 Roll motion 35 3.2.2 Pitch motion 39 3.2.3 Coupled roll-pitch motion . . . 41 3.3 Double Toroidal Damper 60 4. FLOW VISUALIZATION 68 4.1 Preliminary Remarks 68 4.2 Free Vibration Tests 69 4.3 Forced Vibration Tests . . 69 5. WIND INDUCED INSTABILITY STUDY 82 5.1 Preliminary Remarks 82 5.2 Test-Arrangement 82 5.3 Structural Response with Nutation Dampers 88 6. CLOSING COMMENTS 94 6.1 Concluding Remarks . 94 6.2 Recommendations for Future Work 96 REFERENCES 98 APPENDIXT: ALGORITHM FOR THE ENCODING P R O G R A M ... . . 103 APPENDIX II: FREE VIBRATION TESTS . . . . 114 APPENDIX III: RELATION BETWEEN THE DAMPING RATIO A N D iv , REDUCED DAMPING LIST OF F I G U R E S 1-1 Types of wind-induced vibration suppression devices: (a) spoilers; (b) dampers. . . . 4 1-2 Schematic diagram of a toroidal damper 7 1- 3 Schematic diagram showing the plan of study 11 •2-1 Schematic representation of a nutation damper, supported by a structure, subjected to horizontal excitation. 13 2- 2 The fluid particle trajectories in two types of waves observed in a sloshing fluid: (a) progressive wave; (b) standing wave 15 2-3 The bridge-deck free vibration experimental setup: (a) plunging oscillation mode; (b) torsional oscillation mode (roll); (c) torsional oscillation in the transverse direction (pitch) 17 2-4 A photograph of the free vibration test-arrangement showing a rectangular bridge-deck damper with five compartments . 18 2-5 Free vibration data analysis: (a) flow chart illustrating procedure for determining r\; (b) digitized free vibration decay data; (c) typical time window used in the analysis 19 2-6 A schematic diagram of the forced vibration test set-up to obtain vi reduced liquid damping, T),.and added mass, M a , ratios. 22 2-7 The force balance used to measure the reduced liquid damping of nutation damper models under steady state excitation 23 2-8 Typical traces obtained during the steady state force excitation study: (a) displacement of the platform, x e ; (b) sloshing force, F s . . . . . . . . 25 2-9 Schematic diagrams showing basic nutation damper geometries studied: (a) rectangular damper; (b) toroidal damper; (c) circular cylindrical damper 27 2-10 Data acquisition system for forced vibration test-facility. . . 30 2- 11 Data acquisition system for free vibration test-facility 31 3- 1 Effect of the number of compartments and liquid height on the reduced damping of a rectangular damper free to undergo roll motion 36 3-2 Effect of an increase in the initial displacement on the roll damping characteristics of a partitioned rectangular damper 38 3-3 Variation of the reduced damping in roll, for the optimum five compartemnt case, as affected by the liquid frequency and initial excitation. 40 vii 3-4 Reduced damping in pitch as affected by the partitioning and liquid height 42 3-5 Reduced damping in pitch, r|p, versus normalized liquid height, h/T, and normalized frequency, CO/>r/cor, as affected by the initial displacement (e 0 AV) p = 0.009 and 0.017 43 3-6 Effect of the coupled pitch-roll motion on the reduced damping r\T as affected by the liquid height. Note, in absence of a pitch excitation, the system has pure roll response and the peak damping is higher . . . 45 3-7 Variation of the reduced damping in pitch as affected by the liquid height (i.e. Cu/ ) P) and initial distrubances. The system frequencies in pitch and roll are held fixed 46 3-8 Effect of variation in the pitch stiffness on r| r during the roll-pitch coupled motion of a rectangular partitioned damper 48 , 3-9 Variation of the T)r with the liquid height and pitch stiffness at a higher value of the initial disturbance 49 3-10 Reduced pitch damping in the coupled motion as affected by the liquid height and system pitch frequency. 50 3-11 Variation of the reduced damping in pitch with the liquid height and system pitch frequency at a higher initial disturbance 51 viii 3-12 Effect of the level of initial disturbances on the reduced damping in roll during the combined roll-pitch motion 53 3-13 Variation of the reduced damping with liquid height and initial distrubances at a higher pitch stiffness represented by cop = 1.37 Hz. . 54 3-14 A further increase in the pitch stiffness leaves the variation of T| r with the liquid height essentially the same in character except for a reducetion in its peak value due to increase in coupling with the pitch motion. . 55 3-15 Variation of the reduced damping with liquid height and initial excitation as affected by three different values of the system frequency: (a) (0P = 1.18 Hz 57 3-15 Variation of the reduced damping with liquid height and initial excitation as affected by three different values of the system frequency: (b) cop = 1.37 Hz 58 3-15 Variation of the reduced damping with liquid height and initial excitation as affected by three different values of the system frequency: (c) COp = 1.67 Hz 59 3-16 Geometry of circular, single toroid and double toroidal dampers. . . . 61 3-17 Reduced damping, T|, of a double toroid and a single toroid versus ix excitation frequency, u)e. 63 3-18 The effect of detuning of the liquid frequency of the inner and outer toroids on the performance of a double toroidal damper. Performance of the single toroidal damper is also included to assess the relative merit 65 3- 19 Effect of the tuning frequency parameter (Oi/co0 on the variation of the reduced damping with excitation frequency. . 66 4- 1 Visual study of the sloshing motion as affected by the number of partitions for a given amount of liquid: (a) two compartments; (b) five compartments; (c) ten compartments 70 4-2 Schematic diagram based on visual observation showing the free surface geometry as affected by the variation of the excitation frequency around the resonance: (a) standing wave; (b) propagating wave train; (c) propagating single pulse at resonance; (d) standing wave 71 4-3 Typical flow visualization photographs for a rectangular damper, with an aspect ratio of 1.64, showing: (a) wave train at (0//coe = 0.93; (b) single wave at resonance; (c) wave-breaking 73 4-4 Top and side views of the free surface oscillation modes in a toroidal damper as the forcing frequency, C0e, increases from below to beyond the sloshing resonance at a fixed oscillation amplitude, (Oe: x (a) standing wave; (b) wave trains; (c) two waves; (d) single propagating wave; (e) swirling single wave; (f) standing wave 74 4-5 Photographs showing surface wave characteristics for a tuned double toroida with (0/ = 0.58 Hz: (a) standing wave; (b) wave train. . . . . . . 76 4-5 Photographs showing surface wave characteristics for a tuned double toroid with (0/ = 0.58 Hz: (c) resonance showing collision of waves at the far end; (d) collision of waves at the near end during resonance 77 4-6 Progressive variation of the surface waves in the outer and inner toroids, during a mistuned condition, with increase in the excitation frequency: (a) wave train in the outer toroid, standing wave in the inner toroid; (b) resonance in the outer toroid, standing wave in the inner toroid. 78 4-6 Progressive variation of the surface waves in the outer and inner toroids, during a mistuned condition, with increase in the excitation frequency: (c) standing wave in the outer toroid, wave train in the inner toroid; (d) standing wave in the ourter toroid, two waves approaching collision at resonance in the inner toroid 79 4-6 Progressive variation of the surface waves in the outer and inner toroids, during a mistuned condition, with increase in the excitation frequency: (e) standing wave in the outer toroid, collision of waves at resonance in the inner toroid 80 xi 5-1 A schematic diagram of a high voltage transmission line showing a bundle of conductors undergoing torsional oscillations 83 5-2 A schematic diagram of the test-arrangement during wind tunnel experiments 85 5-3 Photographs showing springs, encoder and damper during wind tunnel tests. 86 5-4 Schematic diagram of the closed circuit laminar flow wind tunnel used in the test-program 87 5-5 Typical time histories of rotation signals, as given by the optical encoder, without and with a circular damper 89 5-6 Plots showing effectiveness of a circular cylindrical damper in regulating the torsional galloping instability 90 5-7 Remarkable effectiveness of a toroidal damper in regulating wind induced oscillations in galloping 92 5-8 Relative performance of three nutation damper geometries in controlling the galloping instability. . 93 III-1 One degree of freedom model of a structure installed with a nutation damper 117 xii LIST OF TABLES 2-1 Geometry of rectangular dampers used in the free vibration study. . . 26 2- 2 Toroidal dampers used in the forced vibration tests 26 3- 1 Liquid natural frequency as affected by the number of compartments for a given volume of liquid 37 3-2 Liquid frequency in roll as given by the inviscid small wave theory. . 39 3-3 The variation of liquid natural frequency in pitch with normalized liquid height 41 3-4 Roll and coupled roll-pitch frequencies as given by the small wave theory 52 3-5 Variation of C0/>p and u)/ ) rp with the liquid height as given by the small wave theory 56 xiii N O M E N C L A T U R E AR Aspect ratio, / ' AV for roll and W// ' for pitch c damping coefficient D j inner diameter of a double toroidal damper D m partition diameter of a double toroidal damper D 0 outer diameter of a double toroidal damper D diameter of a circular cylindrical damper d sectional width of a toroidal damper F d damping force F, inertia force F s sloshing force g gravitational acceleration H width of the square section wind tunnel model h liquid height mode shape of the surface wave Is system moment of inertia without damper 1/ system moment of inertia with damper Ji Bessel function of the first kind for i mode k spring constant xiv L length of the rectangular damper /' length of the individual compartment m number of decaying cycles M a added mass Mi liquid mass Ri, R 0 outside and inside radii of a troidal damper, respectively U free stream velocity U r reduced free stream velocity defined as U/(corH) V/ total liquid volume W width of the rectangular damper Xj i-th amplitude of a decaying response x e excitation displacement x velocity of a vibrating system Yj Bessel function of the second kind for the i mode Greek £e excitation amplitude e 0 initial displacement during free vibration test T| reduced damping coefficient, eq. (2.3) xv system reduced damping in pitch system reduced damping in roll %,p system reduced damping in pitch-roll coupled motion eigenvalue of the solution with the shallow water approximation phase difference between excitation and sloshing force e rotational displacement, degree excitation frequency ( 0 / , p liquid natural frequency in pitch CO/,r liquid natural frequency in roll W/,rp . liquid natural frequency in pitch-roll coupled motion COi liquid natural frequency in the inner ring of a double toroid CO.j liquid natural frequency for the i,j mode (Oo liquid natural frequency in the outer ring of a double toroid cop natural frequency of the system in pitch (Or natural frequency of the system in roll Wr,p natural frequency of the system in pitch-roll coupled motion damping ratio defined by the logarithmic decrement, eq. (II. 1) system damping ratio with liquid damper system damping ratio without damper. xvi A C K N O W L E D G E M E N T I would like to thank Dr. V.J . Modi for his guidance and encouragement throughout the thesis program. I have gained from our many discussions and the example he has set as a world class research engineer. The work here has also benefitted from the suggestions of Dr. M . L. Seto. I would also like to acknowledge my mother, Mrs. N.C. Lim, who have always been supportive of my studies. This research project was supported by the Natural Science and Engineering Research Council of Canada, Grant No. A-2181. xvii This thesis is dedicated to the memory of my father, E.C. Lim (1938 - 1990). xviii 1 INTRODUCTION 1.1 Preliminary Remarks Wind induced oscillations of bluff bodies, such as tall buildings, bridges, smokestacks, cooling towers, iced transmission lines and many others, have been of concern to engineers for quite some time. Vortex resonance and galloping type of instabilities have frequently resulted in structural damage and, in extreme circumstances, collapse of the entire structure itself. Destruction of the Tacoma Narrow Bridges, in 1940, is a frequently quoted example illustrating this point. Besides the possibility of structural damage, the wind induced oscillations may also cause physical discomfort, nausea, vertigo, and feeling of disorientation as reported by occupants of tall buildings as well as air traffic controllers. The World Trade Center in New York is reported to sway upto 30 cm at its top in high winds. The fundamental natural frequency of most bluff bodies mentioned above is usually less than 1 Hz. Some of the internal organs, formed of biological tissues, also have relatively low level of natural frequencies. It is found that, in general, acceleration levels greater than 0.15g lead to human discomfort/ In the past, structural design engineers frequently used high factors of safety to account for uncertain wind load conditions. However, advances in the metallurgical science, computer aided design and emphasis on cost effective structures, have dramatically changed the approach. There is a strong trend . towards not only lighter and more flexible, but also longer and taller structures. Obviously, these tend to make the structures quite prone to wind induced instabilities. 1.2 A Brief Review of the Relevant Literature Wind-induced instabilities of concern here can be categorized into the forced oscillations of the vortex resonance type and the self-excited galloping. Large amplitude motion can result in either type of instabilities, particularly under low damping and favourable wind conditions. A bluff geometry object, when exposed to a fluid stream, experiences formation of vortices. Time dependent pressure forces are imposed on the body as vortices are shed with well-defined periodicity. When the frequency of shedding vortices matches the natural frequency of the body, the resonance condition is established and the body executes large amplitude oscillations This process has a self-limiting mechanism [1] . As the amplitude of the body increases beyond the critical value, the vortices begin to break up due to reduced coherence in the spanwise direction. As a result, the amplitude decreases to a value below the critical level. Self-excited oscillations depend on the motion of the structure. The aerodynamically unstable structure causes an increase in the excitation force in the same direction as the motion. Galloping and flutter type of instabilities belong to this category. As against the galloping, which refers to one degree of freedom oscillations, flutter is the term used by aeronautical engineers to describe more than one degree of freedom self-excited oscillations. In flutter, two or more generalized coordinates, say pitch and roll, are excited simultaneously. The frequency of a coupled response is always higher than the pure single degree of freedom motion. The forces involved here are much larger resulting in a very short build-up time and sudden onset of instability. This instability is known to 2 occur on ice coated transmission lines and bridges. Extensive studies have been reported on understanding the mechanism of such phenomena [2,3]. In galloping, the motion itself leads to a change in the effective angle of attack and the associated force in such a way that the energy is extracted from the fluid stream. The amplitude of the oscillation continues to grow until the rate at which the energy is extracted balances the rate of dissipation. Normally, vortex resonance occurs at a lower reduced wind speed while galloping manifests itself at higher values . However, this can change with damping and modes of vibration [4,5]. Over years, a variety of devices have been proposed to arrest these forms of instabilities. They may be classified as spoilers and dampers. Spoilers such as helical strakes, shrouds, slats, splitter plates, etc. attempt to interfere with the aerodynamics of the structure in such a way as to weaken the excitation force. Dampers, on the other hand, provide a mechanism for dissipation or transfer of energy. The conventional viscous dampers, stockbridge dampers used on transmission lines since 1925, tuned-mass dampers, and impact devices belong to this class (Figure 1-1). Zdravkovich [6] and Modi et al. [7] have reviewed the literature pertaining to these add-on devices and their varying degree of effectiveness in wind as well as marine applications. In 1950's, Scruton was the first one to propose application of helical strakes to suppress vibration of smokestacks. He also gave the -optimal configuration in terms of pitch, protrusion and location [8]. Strakes were found to trip the boundary layer to reduce coherence of shedding vortices along the span of the chimney. Other devices using the similar principles in controlling 3 Helical Strakes Shrouds Slats Figure 1.-1 Types of wind-induced vibration suppression devices: (a) spoilers; (b) dampers. 4 vortex resonance are strouds and slats. Their optimal configurations are described by Cox and Wong [9]. Fairings, splitter plates and flags have also been used to stabilize the wake and reduce the effects of vortex shedding. A l l these add-on devices, which interfere with the local aerodynamics, lead to an increase in the drag. Some are also dependent on the wind direction. In I960's, Reed investigated the applicability of impact dampers to lightmasts and antennas [10]. Dashpot dampers as shown in Figure 1-1 are usually installed on guyed structures [11]. Gasparini, Curry and Debchaudhury [12] have discussed the use of viscoelastic systems on smokestacks. Tuned mass dampers have found, relatively, wider acceptance in practice. Normally mounted within the structure, the system aerodynamics remains unaffected. It involves attachment of an auxiliary mass to the main structure through a spring. By tuning the auxiliary mass and the stiffness of the spring, the excitation energy is diverted from the main structure to the auxiliary mass [13]. In controlling the galloping instability of iced transmission lines, a type of tuned mass damper, called the Stockbridge damper, is frequently used. Effectiveness of the Stockbridge damper in controlling plunging oscillations is well documented [14, 15].. However, in general, suppression of torsional oscillations through the use of add-on devices have received little attention. Near earth satellites have successfully used the concept of tuned mass damper, i.e. transfer of excitation energy from one part of the structure to the other and eventually its dissipation, in controlling their librational motion (roll, yaw, pitch). In accordance with the Eulerian description of the general rotational motion in terms of spin, precession and nutation, the device is referred to as the 5 nutation damper. Here the auxiliary mass is replaced by liquid in a toroidal container (Figure 1-2). Nutation dampers have proved quite effective in suppressing low frequency librational motion of satellites with period ranging from 90 minutes to 24 hours! In the early designs, to guard against the microgravity environment, the damper was completely filled with liquid and energy dissipation was achieved through the wake of a freely moving sphere [16]. Of course, for application to ground-based structures, one can discard the sphere and take advantage of the sloshing motion at the free surface to promote energy dissipation. In mid-1960s, Modi et al. were the first ones to propose the use of toroidal nutation dampers for vibrational control of ground-based structures [17]. As the dampers are no longer stationed in micro-g condition, free surface was allowed to assist in energy dissipation. This change in concept of dissipation using the free surface opened a door to its possible application for suppression of different modes of vibration. A family of nutation dampers with different geometries such as toroidal, circular, rectangular, etc. slowly evolved, and their performance as affected by the system parameters was evaluated [18-23]. Modifications, like addition of screens, floating particles and internal baffles were also attempted to improve dissipation [21-23]. Made of rigid container, the water level and geometry of the damper determined the frequency response of the damper to the excitation. Under laboratory conditions, the damper showed promising results. Besides efficiency, it proved to be simple in design, economical and easy to maintain. Welt [21,22] as well as Seto [23] have presented excellent reviews on the subject of wind induced instabilities and their suppression through nutation 6 Figure 1-2 Schematic diagram of a toroidal damper. 7 dampers. Results of the parametric studies with the toroidal damper suggested that, in general, the damping improves with a decrease in the Reynold's number, frequency, and diameter ratio, d/D, of the toroidal damper. On the other hand, cross-sectional geometry of the toroid had little effect on the damper efficiency [21]. Furthermore, presence of screens and baffles seemed to deteriorate the damper performance as they tended to diminish the free surface motion. Results pointed to the fact that impact and breaking of waves together with viscous effects were major factors contributing to the energy dissipation. Subsequently, a detailed study by Seto confirmed this observation [23]. Extensive wind tunnel tests to assess effectiveness of the dampers in suppressing vortex resonance and galloping followed [18-20,22]. Results showed the toroidal damper to be quite successful in arresting both forms of instabilities during laminar and turbulent flows, with the structural model free to undergo two- or three-dimensional plunging motion. Seto [23] extended the study of Welt [21,22] to assess relative merit of three damper geometries: circular, rectangular, and toroidal. It is the most comprehensive study on the subject reported to date. Besides evaluation of the damper efficiency in dissipating energy and its substantiation through wind tunnel tests, Seto also provided a numerical finite, difference procedure, based on the shallow water waves with dissipation and dispersion corrections, accounting for conjugate character of the problem. Considering the complex nature of system and the relatively simple model used to characterize it, the analysis was successful in capturing the physics of the problem rather well, as suggested by excellent 8 correlation with the measured data. The flow visualization study, which complemented the experimental and numerical investigations, provided better appreciation as to the mechanism of energy dissipation [23-26]. The results showed the circular geometry damper to be the most efficient. Under optimum conditions of floating particles, the damper performance improved by as much as 30%. The study also indicated a possibility of improvement in the performance of the rectangular damper through partitions. A word about the application of the nutation dampers to real-life, prototype structures would be appropriate. The earliest installation dates back to 1986 atop the Yokohama Marine Tower in Japan. At 104 m, it is. the tallest lighthouse in the world [7]. Since then, around 15 buildings, airport control towers, microwave transmission masts, etc. around the world have used nutation dampers to suppress wind and earthquake induced oscillations [7,27]. Its application to ice-coated transmission line conductors [28] and offshore marine structures [29] has also been proposed. 1.3 Scope of the Investigation With the passage of time, particularly during the past two decades, our understanding of the nutation damper, its performance and approach to design have progressed significantly. However, as is usually the case with any scientific inquiry, there are a number of issues which demand further attention. Dynamics of the Tacoma Narrows Bridge leading to its collapse clearly showed that a continuum structure may respond to wind excitation in a variety of single and coupled degrees of freedom. Unfortunately, wind induced instability of 9 structures in torsion has received very little attention in the literature. What will be the performance of a nutation damper during such structural response? Nutation dampers have proved to be effective during translational plunging motion of a structure. Can it also suppress torsional oscillations? We do not have information to answer such questions. With this as background, the present study focuses on the following fundamental aspects: (i) effect of partitioning on the performance of rectangular and toroidal dampers; (ii) influence of roll, pitch and coupled roll-pitch motion on the efficiency of a partitioned rectangular damper; (iii) effectiveness of the damper in suppressing torsional instability; (iv) visual study of the surface flow in the damper as well as dynamical response to get better appreciation of the phenomena. To begin with, optimum partitioning of a rectangular nutation damper and its performance in roll, pitch, and coupled roll-pitch motions is investigated using a specially designed versatile test-facility. This is a followed by assessment of a double toroidal damper in terms of energy dissipation for a given amount of liquid. Next, a flow visualization study of the surface waves, as affected by the excitation frequency and amplitude, is undertaken to gain better appreciation of the energy dissipation mechanism. Finally, effectiveness of the partitioned dampers in suppressing torsional instability of a square prism is assessed through carefully planned wind tunnel experiments. Figure 1-3 schematically summarizes the plan of study. Throughout the test-programme, the working fluid used in the damper is water. 10 OH C O c o 1/1 Cd • 2 , ; 5 o cd B C D o 00 C O I ax) 2. TEST FACILITIES, INSTRUMENTATION, AND METHODOLOGY 2.1 Preliminary Remarks It is clear that nutation dampers belong to a class of energy dissipative devices. A synthesis of desirable geometry and amount of liquid results in a damper that can efficiently extract and dissipate vibration energy from the attached structure. The sloshing liquid is the mechanism through which the energy is dissipated. Recently, some researchers have referred to the dampers as "Liquid Dampers", "Liquid Sloshing Dampers," "Tuned Liquid Dampers," or "Tuned Sloshing Liquid Dampers." Tuning a nutation damper to fit a particular situation consists in choosing the proper geometry and corresponding amount of liquid. This makes the damper particularly effective at a desired frequency. The objective is to adjust the damper frequency response to match the natural frequency of the structure so that, during peak response of the structure, the energy dissipation is maximum. Two factors play important role in the dissipation of energy: viscosity, which leads to boundary-layer at the walls and the free surface; and the surface waves. The latter involves nature of the waves (standing and different types of propagating waves), their interactions and collision with the damper walls. . The characteristics of these waves depend on the excitation frequency and amplitude, damper geometry, and physical properties of the liquid. Consider, for example, a nutation damper subjected to horizontal excitation as shown in Figure 2-1. 12 Nutation Damper \ Figure 2-1 Schematic representation of a nutation damper, supported by a structure, subjected to horizontal excitation. 13 Typical motions of liquid particles during progressive and standing waves are illustrated in Figure 2-2. In general, standing waves contribute little to energy dissipation. Primarily, the damping is through the shearing action between adjacent layers of liquid and collision of surface waves leading to their breaking, a nonlinear phenomenon representing spilling of waves in a dispersive medium. As pointed out before, the thesis focuses on the energy dissipation performance of two distinctly different classes of nutation dampers: rectangular and toroidal. In particular, the interest is on the effect of partitions on the damping efficiency. To that end, two distinct test-facilities were used: free vibration setup for rectangular dampers and forced vibration apparatus for toroidal dampers. This chapter describes the test-facilities and procedures used during acqusition and reduction of data. 2.2 Free Vibrational Tests The rectangular damper or a bridge-section is essentially a rigid Plexiglas box where partitions can be introduced as desired to change aspect ratio of the resulting compartments. The damper was supported by a universal joint made of a system of ball bearings to minimize friction. The versatile joint was so designed as to provide independent pitch, roll and yaw motions or their combinations. Each mode of rotation can be readily decoupled by locking the other two degrees of freedom. The bridge-section (damper) was supported from a rigid frame by a system of four identical springs attached to the damper at its four corners. The stiffness of the springs was appropriately chosen to impart the 14 ( a ) V a r i a t i o n o f P a r t i c l e M o t i o n w i t h D e p t h (b) U p p e r E n v e l o p e L o w e r E n v e l o p e L i n e a r D e c r e a s e i n P a r t i c l e M o t i o n w i t h D e p t h Figure 2-2 The fluid particle trajectories in two types of waves observed in a sloshing fluid: (a) progressive wave; (b) standing wave. 15 desired roll frequency to the sytem. In the same fashion, two similar springs connecting the box and the frame along the horizontal maximum momentum inertia axis, at the center of the bridge-section, controlled the frequency in the pitch. The two sets of springs provided motions about the principal axes (Figure 2-3). A photograph of the free vibration test-arrangement is shown in Figure 2-4. Friction and sensor-loading are important factors in designing free vibrational tests. To minimize these effects, a shaft encoder was installed at each joint to acquire the rotational information. The encoders were particularly selected for their small weight and low friction. Comprising of Light Emitting Diode (LED) and ball bearings, the encoder provides rotation in terms of a digital signal. The output signal is sent to a signal conditioning board for decoding. It can tackle four encoders simultaneously using a software based control. This is particularly important as different modes of rotational signals have to be recorded simultaneously. The source code for data acquistion and reduction algorithm is included in Appendix I. The resulting signals are collected and stored in a computer for later analysis. During a typical test with a rectangular damper, the compartment aspect ratio is identified first through the choice of an appropriate number of partitions. Now the right amount of liquid is introduced to obtain the desired liquid resonance frequency, the system frequency being known through the choice of the springs supporting the model. The deck is given a prescribed initial displacement and the vibration decay information is stored on the hard drive of a computer for later analysis. Figure 2-5 presents a flow chart of the test-16 Figure 2-3 The bridge-deck free vibration experimental setup: (a) plunging oscillation mode; (b) torsional oscillation mode (roll); (c) torsional oscillation in the transverse direction (pitch). 17 81 ao" c l-i n> to • P > i §• 5 o 3- p e r a S-CD 3 p~ CD S 3* 3 « T3 CD p H * -0 3 t± p CD g 3 | p CD i-i 3 r * r-f 3 C/3 CD cr 3 O CO procedure and a sample of the acquired data. The free vibration response is essentially a decaying sinusoid. The logarithmic decrement, a standard procedure for evaluating damping of a free vibrating system, is given by x. In 27im (2.1) where xi and x m correspond to amplitudes of the signal m cycles apart. To assess improvement in damping through sloshing liquid, Welt [21] has introduced the reduced damping parameter r\ defined as I (2.2) Here: system damping ratio with liquid damper; £ s = system damping ratio without damper; 1/ = system moment of inertia with damper; I s = system moment Of inertia without damper. 2.3 Forced Vibration Tests The free vibration test-facility, though useful, is not readily amenable to a systematic study of dampers over a broad range of excitation frequencies and amplitudes. On the other hand, from the vibration control point of view, system 20 response over a spectrum of frequency may be crucial to the damper design. In practical applications, availability of high damping at one frequency is often not adequate for a good design if the damper performance over a frequency spectrum around the design frequency is poor. Forced vibration dynamic tests readily provide this information. The test-system comprises of a Scotch-Yoke mechanism connected to a horizontal platform (Figure 2-6). A force balance on which a damper under test is mounted is rigidly connected to the platform (Figure 2-7). The balance was designed to test a wide variety of dampers. It consists of two cantilevered galvanized thin steel plates connected to a horizontal stainless steel platform, which supports the damper. Four strain gages, fixed to the plates near the root, formed a part of the Wheatstone bridge circuit. The materials were carefully selected to ensure a good compromise between stiffness and sensitivity. Natural frequency of the balance was designed to be 12 Hz, which is an order of magnitude higher than the excitation frequency range of interest. The strain gage balance measures the sloshing force acting on the damper wall. The signals from the four gages are fed to a bridge amplifier for signal conditioning and then recorded via the analog-to-digital (ATD) board. During the tests, the Scotch-Yoke steadily drives the platform harmonically in the horizontal plane at a constant frequency and amplitude. A Linear Variable Displacement Transducer (LVDT) is installed to monitor the time-history of the table oscillations. It is a mutual inductance sensor consisting of two concentric cylinders with a gap. The inner cylinder, referred to as core, is made of magnetic material and can move relative to two windings on the outer cylinder. An ac voltage is applied to one of 21 T 3 C 22 nutation clamper «« : •> £e cos coe t ure 2-7 The force balance used to measure the reduced liquid damping of nutation damper models under steady state excitation. 23 the coils called primary. As the magnetic core moves, it varies the mutual inductance between the coils which serves as a measure of the displacement. The conditioned L V D T signal is recorded together with the force balance data by the A T D board. This also provides phase difference between the excitation displacement and sloshing force. A low frequency, high torque motor was chosen as the driver for the Scotch-Yoke mechanism. A potentiometer is used to control the speed of the motor. Frequency of the table was monitored through a spectrum analyzer in real time. By carefully adjusting the current input to the motor, precise frequency control can be achieved. Figure 2-8 shows samples of the platform displacement and sloshing force signals. During a typical test, four parameters are measured: sloshing force (F s), excitation frequency (coe), excitation amplitude (ee), and phase between the excitation and the sloshing force (9 ) . These are used to define the reduced damping parameter (r|) and added mass ratio as (M a /M/) as suggested by Welt [22]: F F (2.3) M.e co2 / e e 1 cos 9 ; 2M,e co ' e - sin (p. 2 T The details are given in Appendix III. 24 CD E 00 I 3 25 2.4 Damper Models As indicated before, two different damper geometries were used during the parametric study: rectangular; and toroidal. A circular cylindrical damper was also used during the wind tunnel tests, as Seto [23] found it quite promising in arresting both vortex resonance and galloping instabilites in plunging motion (Figure 2-9). The dampers were constructed out of Plexiglas. Geometrical properties of the dampers are summarized in the following tables: Table 2-1 Geometry of rectangular dampers used in the free vibration study. RECTANGULAR DAMPER MODELS Model Length, L (cm) Width, W (cm) Aspect Ratio, 77W 1 91.4 10.2 0.6, 1.2, 3 and 6 .2 37.0 15.2 1.2 and 2.4 3* 10.0 10.0 1.0 * only for wind tunnel tests Table 2-2 Toroidal dampers used in the forced vibration tests. PARTITIONED TOROIDAL AND CIRCULAR DAMPER MODEL Model Outer Radius D 0 (cm) Middle Radius D m (cm) Inner Radius Dj (cm) Slenderness Ratio, Dm/Do Slenderness Ratio, Dj/Dm 4 30.0 20.0 - 0.67 -5 30.0 20.0 12.0 0.67 0.60 6 30.0 24.0 12.0 0.80 0.50 7* 10.0 7.2 - 0.72 -8* 10.0 • - - - -* only for wind tunnel tests 26 CO CL fYJT\ t Q o "D cd " O c cd > CO 03 c/5 CO - 5 to wo CO CO E o CO W) u. co CL c E CO *o E <- rt iS co 3 "C C T 3 o . 5 rt co b f l cd • S 3 o E ^ cd . -w- u. W ) co ed O . ^ £ cd cd rt "3 i <N CO t-l 3 27 Model 1 is a relatively large rectangular damper which can also simulate plunging and torosional oscillations of a bridge-deck (two-dimensional section)during free vibration tests. It can be subdivided into two to ten smaller chambers (compartments) through the use of one to nine dividers (partitions). Thus aspect ratio of the compartment can be adjusted to assess its effect on the damping level and frequency response. However, the model proved to be too big and heavy for the flow visualization study using the forced vibration table. It was also too big for wind tunnel tests. Hence two smaller models were constructed. Partitioned toroidal dampers were constructed by concentrically placing different sizes of rings in a 30 cm diameter circular tank. The rings were made of Plexiglas to minimize weight and hence the inertia loading. They are held in place by means of silicone gel. D i / D m and D m / D 0 represent bluffness ratios for the double toroidal arrangement. A toroid is considered slender for a bluffness ratio > 0.5. Model 4 is a single toroid damper with inner diameter set to zero. Models 5 and 6 are double toroidal dampers with different middle diameter. Toroidal dampers were used during the forced vibration tests. Smaller dampers numbered 3, 7 and 8 were specifically constructed for use during wind tunnel tests to assess their effectiveness in suppressing torsional instability. 2.5 Data Acquisition and Reduction Two different systems were used for data acquisition. A strip chart recorder was employed for acquiring forced vibration test-data. It is a 12 bit, eight double-ended channel, analog to digital converter (Data Translation 2800, 28 Mass., U.S.A.) interfaced to an I B M Personal Computer (Figure 2-10). The other system is a PC to Incremental Encoder interface card (PC 7166, US Digital, W A , U.S.A.) , which is used for counting the digital pulses from the shaft encoders and delivering the data to the computer. The card comprises of four, 24-bit quadrature counters (LS 7166); four interface.channels; and multiplier function. One encoder appears with each rotational degree of freedom hence they are present during both free vibration and wind tunnel tests. The frequency range of interest in the present study is 0.1 to 1.0 Hz. To obtain reasonable resolution, a large number of samples was collected during each test. The sampling time was 1.5 minutes to ensure collection of data over sufficient number of cycles. The sampling frequency was at least an order of magnitude higher than the excitation frequency. Furthermore, each test was repeated at least five times. The results were stored in ASCII files for later processing (Figure 2-11). Routines in DOS and M A T L A B were written to analyze the data collected during free vibration, forced vibrational and wind tunnel tests. For the free vibrational tests, these include demultiplexing, spectral analysis, and determination of the logarithmic decrement as well as the reduced damping. In dynamic tests, the data were spectrally analyzed. The phase angle between the excitation and sloshing force was evaluated and the reduced damping calculated. Results from several repeated tests were statistically analyzed for standard deviation to establish their validity. 29 CD c O) o Cp ' o _CD O o o o -*—' CO "CO CO " D "D ee o H — Q . W o J o o o CD SZ c o X T3 O o O T — CQ CO CD H — bridg ampi mete o .rt CD •4—I c X) "> T 3 CD C J 1-CD >^  c« C o 3 cr CD rt rt rt Q C M CD U i 3 30 Set initial Displacement Initiate acquisition program Shaft Encoder Digital Signals Signal Conditioning by Counting Data Stored in P . C Figure 2-11 Data acquisition system for free vibration test-facility. 31 3. STUDY WITH DAMPERS DISSIPATION 3.1 Preliminary Remarks When a nutation damper is excited, surface waves are generated. Different wave modes appear depending on the frequency of excitation with other system parameters held fixed. Understanding of the wave character is crucial in design of the dampers as wave modes determine the damping characteristics. Hence, it is useful to have a simple tool to predict liquid frequency and the associated surface wave mode before constructing a damper for test. However, the theory governing the wave motion is highly nonlinear [30,31]. The exact closed-form solution for the wave motion and associated damping is not available. One may use a relatively simple small wave theory [5] to gain some understanding of the problem. This can also assist in planning of the experiments. It was particularly helpful during tests with double toroidal dampers as now frequency, for a given amount of liquid, can be determined only through iteration. Thus the small wave theory proved useful in planning tests with appropriate values of system pararmeters (damper geometry, liquid height, excitation frequency, etc.) which may result in significant energy dissipation. In this chapter, the damping effectiveness of a partitioned rectangular damper and double toroidal damper is examined through a series of parameteric tests. The rectangular damper, which can be considered as a two-dimensional section of a bridge-deck, was supported by a universal joint for free vibration tests as explained earlier. The toroidal damper was tested using a horizontal shaking table with a Scotch-Yoke type harmonic exciter as described in Section 32 2.3. The bridge-deck model, supported by the free vibrational facility, was tested in roll, pitch and roll-pitch coupled motion. Important parameters such as liquid height, excitation amplitude, number of partitions, etc. were varied systematically. The reduced damping parameter was calculated to assess the As can be expected and also confirmed by Seto [23], the peak energy dissipation takes place around the liquid fundamental frequency which, according to the small wave theory, is given by where g is the gravitational acceleration; h, the liquid height; and Ky is the eigenvalue associated with the mode i,j. The eigenvalue for the wave in a rectangular tank with dimensions a x b is given by Given the above relations and liquid height, approximate natural frequency can be found for a particular rectangular damper. The concept of partitioning to improve energy dissipation was also applied to toroidal dampers. A partition transformed a single toroid into a double toroid. To compare their relative performance, the liquid mass and frequency were kept fixed in designing the double toroid. The inviscid, approximate, small wave theory was again used to this end. The process is a bit involved now. The eigenvalue equation for the free surface motion is given by [23] damper performance. (3.1) (3.2) 33 Y . ( K . . ) J . 1 IJ 1 K . . •J R • J . ( K . . ) Y . i ij i K . R. ) i jj R = 0, (3.3) which leads to the resonant sloshing frequency for a toroidal damper as 7 K i i § (0 .=—^tanh 'J R h ) li R (3.4) Here: J„Yi = R, Ro h g Bessel functions of the first and second kind of order i , respectively; eigenvalue (wave number) for the jth transverse mode with the ith circumferential mode; inner radius; outer radius; liquid height; acceleration due to gravity. The volume of liquid in a toroid can be written as V . = 7 C R ' - R " h / l o (3.5) In the toroidal damper design ( O i j , V/ and R 0 are specified while K j j , Rj and h are the parameters to be determined. Of course, as observed from the experiments, the fundamental sloshing mode provides the best damping, hence (i,j) = (1,0). One is now faced with the solution of a set of equations (3.3) - (3.5) simultaneously in an iterative fashion. This was accomplished numerically using a computer. The same procedure was applied to a double toroidal damper with 34 each toroid treated separately. 3.2 Rectangular Damper Using the free vibration test-setup described earlier, experiments were conducted with the rectangular (i.e. bridge-deck) dampers undergoing roll, pitch and coupled roll-pitch motion. The amount of information obtained through variation of important parameters is substantial. Here some typical results are presented which help in establishing trends. 3.2.1 Roll motion Variation of the reduced damping parameter as affected by the liquid height and number of compartments is presented in Figure 3-1. Three different values of the aspect ratio (3, 1.2, 0.6) as affected by the number of partitions (1, 4, 9, respectively) are considered. The structural natural frequency in roll, cor, is held fixed at 7.51 rad/s, so is the initial displacement at the tip (e 0/W = 0.095). It is apparent that, for a given number of compartments, the reduced damping increases with the liquid height, reaches a peak value and then remains essentially constant (AR = 0.6) or diminishes (AR = 1.2, 3). Note, the peak value of % is significantly affected by the number of compartments, and in the present case it is as high as 2.8 for A R =1.2 (five compartments). The peak values occur in the vicinity of h/W = 0.096, when the liquid frequency is close to the system natural frequency in roll. The damping level is affected by the wavelength of the surface wave. In the five compartment case, the wavelength is close to twice the 35 3 . 0 h/W Figure 3-1 Effect of the number of compartments and liquid height on the reduced damping of a rectangular damper free to undergo roll motion. 36 compartment length / ' leading to a relatively steep gradient for the impacting wave. This results iri larger dissipation of energy. A flow visualization study confirmed this observation (Chapter 4). Seto has also reported similar trends with compartmenting of a rectangular damper [23]. As the system is nonlinear, one would expect its response to be sensitve to the initial displacement. Figure 3-2 studies the effect of increasing the initial excitation, (e 0/W) r, from 0.095 to 0.142. The trends are essentially the same as before and the five compartment case continues to be the optimum. However, now the peak r| r reaches the value of 3.8! This represents an increase in damping due to sloshing with reference to the no liquid case by around 280%! Of course, the extent of the sloshing motion, and hence the energy dissipation, would depend on the liquid natural frequency and its proximity to the resonance. To that end, liquid natural frequency in roll (CD/,r) was calculated using the small wave theory. The results are given below: Table 3-1 Liquid natural frequency as affected by the number of compartments for a given volume of liquid. Fundamental Liquid Frequency in the Rolling Mode Number of Chamber Aspect Ratio Frequency, rad/s h/W 2 3 1.82 0.097 5 1.2 7.10 0.097 10 0.6 16.90 0.097 Note the liquid frequency of 7.10 rad/s for h/W = 0.097 is quite close to the 37 4.0 Roll L - —»H 1 1 I (e0/W) r = 0.142 co =7.51rad/s 0.06 0.08 0.10 h/W 0.12 0.14 Figure 3-2 Effect of an increase in the initial displacement on the roll damping characteristics of a partitioned rectangular damper. 38 system natural frequency of 7.51 rad/s. This explains the success of the five-compartment damper in giving a large value of rj r- By finer tuning , one can attain even higher dissipation. To further emphasize the effect of liquid frequency, tests were conducted with the favourable five compartment case using additional values of liquid height (Figure 3-3). Two different inital displacements were considers: (e 0 /W) r = 0.0178 and 0.156. The abscissa shows both the nondimensional liquid height as well as the liquid frequency, obtained using the small wave theory (Table 3-2) and nondimensionalized with respect to cor. It is apparent that the peak values of the reduced damping parameter are close to the resonance conition in both the cases. Table 3-2 Liquid frequency in roll as given by the inviscid small wave theory. h/W 0.02 0.05 0.07 0.10 0.12 0.15 C0/,r H Z 0.57 0.81 0.99 1.13 1.25 1.36 3.2.2 Pitch motion During pitch, the damper executes rotational motion about its longitudinal axis. Obviously, the partitioning should have virtually no effect on the damping. Furthermore, as the damper width (W) is relatively small compared to the length (L), there is little opportunity for the surface wave to develop. The moment arm being small, liquid moves essentially as a rigid body (standing wave in contrast to propagating waves), which contributes little to the damping. This is shown in 39 0.63 0.83 1.03 CO, . / C O . IX r J I L_ 0.03 0.06 0.09 0.12 h/W 1.23 0.15 Figure 3-3 Variation of the reduced damping in roll, for the optimum five compartemnt case, as affected by the liquid frequency and initial excitation. 40 Figure 3-4. Note, the damping level is significantly lower than that observed during the roll motion. To be consistent, the aspect ratio is now defined as W//' . Figure 3-5 gives detailed view as to the effect of the liquid natural frequency in pitch (and the associated liquid height) for the damper with five compartments. As before, the small wave theory was used to obtain C0/)P, which is given in Table 3-3. Two initial excitation amplitudes in pitch are considered, (£ 0 /W) p = 0.009 and 0.017. It is of interest to observe that in both cases the peak damping seems to occur around C0/,p/a)p ~ 0.85, i.e. little below the resonance. Table 3-3 The variation of liquid natural frequency in pitch with normalized liquid height. h/W 0.03 0.06 0.09 0.12 0.15 0.18 CO/,p Hz 0.69 0.97 1.18 1.35 1.49 1.61 3.2.3 Coupled roll-pitch motion With some appreciation as to the damper performance in the individual roll and pitch degrees of freedom, the next logical step was to explore its effectiveness in the coupled roll-pitch motion. This is important as, in practice, bridge-decks and bundles of transmission line conductors have experienced such coupled motion. The coupled motion was excited by providing an initial distrubance in roll as well as pitch and monitoring the response in the individual degrees of freedom. Thus reduced damping contribution appears in both roll (r|r) and pitch 41 1.0 0.8 0.6 i i i i i - i — i — i — r -I • • » 1 1 t -» - -L- H 0.2 0.0 (e0/W)p = 0.284 co = 5.86 rad/s •w Compartments no. AR 2 0.33 5 0.83 -v—- 10 1.67 -I I I I I I I I I I L. 0.06 0.08 , , 0.10 h/r 0.12 -I I L. 0.14 Figure 3-4 Reduced damping in pitch as affected by the partitioning and liquid height 42 0.03 0.09 . n , 0.15 Figure 3-5 Reduced damping in pitch, r | p , versus normalized liquid height, h// ' , and normalized frequency, co/,r/cor, as affected by the initial displacement (e 0/W) p = 0.009 and 0.017. 43 (rip). To better appreciate couples interactions between the two degrees freedom involved, the response results for the roll and pitch motions are presented separately. As the five-compartment configuration resulted in the maximum damping, it is used during the roll-pitch coupled test-program. Figure 3-6 shows the effect of increase in the liquid height on the reduced damping in roll (r)r) for two different initial excitation conditions: pure roll excitation and coupled roll-pitch excitation. In the first case, the system responds only in roll although it is free to undergo coupled motion. The damping increases with the liquid height, reaches a peak value close to resonance and then diminishes. With the coupled initial distrubance, both the roll and pitch degrees of freedom are excited and there is mutual transfer of energy. The surface wave direction is changed and its impact on the wall, which is responsible for the energy dissipation, is slightly less efficient. This results in a reduction in the peak value of the roll damping, r| r. To have better appreciation of the phenomenon, the stiffness in pitch as well as initial conditions were changed systematically. Figure 3-7 presents variation of the damping in pitch with the liquid height, for two sets of initial conditions exciting the pure pitch as well as coupled roll-pitch motions. This is similar to the case considered earlier in Figure 3-6 where the pure roll motion was also excited and the focus was on the damping in roll. The trends are as anticipated: inital increase in r j p with the liquid height reaching a peak value at C0/)P/C0p ~ 0.89 followed by a decrease in pitch damping as CO/)P increases further. Note an increase in the peak value of the damping during the coupled motion. As mentioned earlier, this is associated with the 44 i 0.03 0.06 0.09 0.12 0.15 h/W Figure 3-6 Effect of the coupled pitch-roll motion on the reduced damping r| r as affected by the liquid height. Note, in absence of a pitch excitation, the system has pure roll response and the peak damping is higher 45 Figure 3-7 Variation of the reduced damping in pitch as affected by the liquid height (i.e. C0/,p) and initial distrubances. The system frequencies in pitch and roll are held fixed. 46 energy exchange between the two coupled degrees of freedom. It also explains drop in the maximum value of %, observed during the coupled motion, in Figure 3-6. There is a tendency to have another smaller secondary peak at a higher frequency but it is of little importance. As the primary peak is already flat, it provides favourable dissipation over a wide spectrum of liquid frequencies. The same was true for the roll damping results presented in Figure 3-6. This suggests robust character of the liquid damper, a desirable feature in its design. Figure 3-8 shows variation in the damping performance in roll with the liquid height for three different values of pitch stiffness, cop = 1.24, 1.37 and 1.67 Hz. The system frequency in roll and initial distrubance are held fixed during these variations. Results suggest that, as before, T | r attains a peak value at a characteristic liquid height. However, the optimum damper performance seems to occur at a lower liquid height (i.e. lower co/>r), particularly at the lower value of the pitch stiffness. This suggests that, for the system parameters considered, the increased pitch stiffness reduces the roll-pitch coupling. The similar trend persists even at a higher initial disturbance (Figure 3-9). Corresponding results for the reduced damping in pitch (r)p) are presented in Figures 3-10 and 3-11. The variation of r | p with liquid height and the system pitch stiffness are shown in Figure 3-10. During each test, the system was subjected to the same initial disturbance. Similar to the case of the roll damping, the peak value of r|p occurs at a lower liquid frequency (lower h//') as the system's pitch stiffness diminishes. The trend persists even at a higher disturbance (Figure 3-11) as in the case of the roll damping (Figures 3-8, 3-9). 47 12.0 h 6.0 i i i i 0.03 cop, H z 1.24 1.37 1.67 C0= 1.18Hz (80/W)p= 0.017 (E 0 /W)= 0.156 007 , n A , 0.11 h/W 0.15 Figure 3-8 Effect of variation in the pitch stiffness on % during the roll-pitch coupled motion of a rectangular partitioned damper. 48 18.0 12.0 "Hr i i i i 0.03 -i 1 1 r 0.07 CO= 1.18Hz (£ 0 /W) p = 0.027 (£ 0 /W) = 0.274 Roll Pitch h/W 0.11 J I I L. 0.15 Figure 3-9 Variation of the r i r with the liquid height and pitch stiffness at a higher value of the initial disturbance. 49 1.6 M, n 1 - 1 —r —i 1 1 r "V H z 1.24 1.37 1.67 CD= 1.18Hz ( 8 0 / W ) p = 0.017 (eoA/V) = 0.156 Figure 3-10 Reduced pitch damping in the coupled motion as affected by the liquid height and system pitch frequency. 50 -i 1 1 r - i 1 1 r - -i 1 1 r i i i Figure 3-11 Variation of the reduced damping in pitch with the liquid height and system pitch frequency at a higher initial disturbance. 51 As pointed out before, the system being nonlinear, the energy dissipation character may be susceptible to the level of the initial distrubance. To assess this aspect, the system was subjected to three different progressively increasing coupled disturbances. As the liquid frequency is dependent on the height, the variation of T| r was plotted with liquid height as well as the normalized liquid frequency. The liquid frequency (0/j as well as the coupled roll-pitch frequency u)/,rp were obtained using the small wave theory (Table 3-4). The results are presented in Figure 3-12. Table 3-4 Roll and coupled roll-pitch frequencies as given by the small wave theory. h/W 0.02 0.05 0.07 0.10 0.12 0.15 (0/,r Hz 0.57 0.81 0.99 1.13 1.25 1.36 CO/,rp Hz 0.90 1.26 1.52 1.73 1.90 2.04 It is apparent that, as before, a,larger initial disturbance leads to a higher peak damping value. At a lower initial displcement, the damping curve looks quite similar to that for the single degree of freedom case (Figure 3-3). However, as the displacement increases, the peak damping occurs at a lower liquid frequency in roll. The similar trend continues at higher pitch stiffness as shown in Figures 3-13 and 3-14. Note, the character of the damping response remains essentially the same as before. As expected, due to increased coupling with the pitch motion, 52 gure 3-12 Effect of the level of initial disturbances on the reduced damping in roll during the combined roll-pitch motion. 53 Figure 3-13 Variation of the reduced damping with liquid height and initial distrubances at a higher pitch stiffness represented by co p =1.37Hz. 54 0.03 0.06 0.09. „ ' 0.12 0.15 Figure 3-14 A further increase in the pitch stiffness leaves the variation of r) r with the liquid height essentially the same in character except for a reducetion in its peak value due to increase in coupling with the pitch motion. 55 the peak value of the reduced damping parameter in roll is now slightly lower. Similar frequency shift was also observed in the pitch damping results. The effect of initial coupled distrubance on the pitch damping (r\p) is studied in Figure 3-15 for three different values of the system frequency in pitch, C0p = 1.24 Hz, 1.37 Hz and 1.67 Hz. The variation in the damping is presented as affected by the liquid height as well as the corresponding liquid frequency in the pitch motion (C0/, p) obtained using the small wave theory (Table 3-5). The coupled roll-pitch frequencies are also indicated in the table. Table 3-5 Variation of C0/iP and co/ ) rp with the liquid height as given by the small wave theory. hir 0.03 0.06 0.09 0.12 0.15 0.18 co/,p Hz 0.69 0.97 1.18 1.35 1.49 1.61 CO/,rp Hz 0.90 1.26 1.52 1.73 1.90 2.04 For the lowest system frequency in pitch of co p = 1.24 Hz, the effect of an increase in the initial disturbance level is to increase the peak value of T | p , which now occurs at a lower liquid height, i.e. reduced C0/5p. Also of interest is the broad character of the peaks suggesting favourable performance of the damper over wide spectrum of liquid frequencies, i.e. its robustness as mentioned before. In general, the character of the damping response remains similar even at higher values of the system frequency in pitch (co p) as shown in Figures 3-15(b) and 3-15(c). Of course, as can be expected, there are local differences, however, a shift 56 Figure 3-15 Variation of the reduced damping with liquid height and initial excitation as affected by three different values of the system frequency: (a) cop = 1.18 Hz . 57 t i , _ , , , i , ,—3 0.03 0.09 0.15 Figure 3-15 Variation of the reduced damping with liquid height and initial excitation as affected by three different values of the system frequency: (b) (0p = 1.37 Hz. 58 ( £ 0 /W ) p (£ Q /W) r 0.009 0.078 0.017 0.156 0.027 0.274 t i i _ _ , i 1 1 1 1—3 0.03 0.09 0.15 h/r C0p= 1.67Hz C0r = 1.18Hz Figure 3-15 Variation of the reduced damping with liquid height and initial excitation as affected by three different values of the system frequency: (c) (0p = 1.67 Hz. 59 towards the lower C0/ i P for peak T | p at a higher distrubance is distinct. 3.3 Double Toroidal Damper The partitioning of a rectangular damper proved quite successful in improving the damper dissipation for a given volume of liquid. The earlier study by Seto [23] showed the circular cylindrical geometry damper to be remarkably efficient. This raises a natural question: Is it possible to improve the performance of a circular geometry damper by partitioning to convert it into a single or a double toroidal damper? To answer this question, a set of carefully planned experiments were conducted based on some available information: (i) As the excitation frequency is fixed by the wind speed, the resonant sloshing frequency should be the same in each toroid. (ii) The earlier study by Welt [22] showed that, in general, slender toroids lead to higher damping. (iii) The invisicid small wave theory [30], though approximate, is sufficiently accurate and can assist in the design of circular cylindrical partitioned dampers. With this as background, it was decided to focus on a double toroidal damper, i.e. one partition (Figure 3-16). The single toroid configuration served as reference. The following parameters were treated as fixed: outer diameter of the toroid, D 0 = 2R 0 = 30 cm; inner diameter of the toroid, Di = 2Rj =12 cm; total liquid volume V/ = 1000 ml. Liquid resonant frequency in each toroid was usually required to be 0.58 Hz, 60 61 however, at times it was purposely mistuned to assess its effect on the damping level over a spectrum of excitation frequency. The diameter D m of the partition that would satisfy the above conditions was obtained iteratively using the small wave theory as explained earlier. It is important to recognize that the solution for Dm is not unique. It depends on the proportional division of the liquid volume between the outer and inner toroids. Thus the combinations of system parameters associated with the damper design are literally infinite. Here only a few sample designs were tested to have some appreciation as to the effect of partitioning on the performance of a toroidal damper. Figure 3-17 presents variation of the reduced damping r\ as a function of the excitation frequency, e.g. the vortex shedding frequency which, when close to the resonance, is responsible for large amplitude oscillations of buildings, bridges, transmission line conductors, etc. The liquid frequency in the outer and inner toroids is the same (0.58 Hz), i.e. u)i/co0 = 1. The partition diameter D m is 24 cm which corresponds to the liquid volume in the outer toroid to be 500 ml (50%). Thus D m / D 0 = 0.8 and D i / D m = 0.5. The results show subtle differences in the response of the two dampers. The peak r\ for the double toroid is slightly higher, however the gain is not significant. In fact, the single toroid appears attractive as it performs favourably, although at a slightly lower value of the peak r\, over a spectrum of the excitation frequency in the range of 0.55 to 0.70 Hz. It suggests an ability to respond to a range of wind velocity and degree of robustness, i.e. uncertainty in the damper design parameters, or leakage. This is in contrast to the double toroid case which showed a relatively narrow peak but slightly higher value of the maximum r). 62 -l 1 1 r D /D D./Dni m o J m ~O80 0.50 Single Toroid cop ( H z ) 1 . 0 Figure 3-17 Reduced damping, Tj, of a double toroid and a single toroid versus excitation frequency, coe. 63 It is likely to perform better, if tuned properly, in the narrow frequency band near the resonance (coe - 0.54 - 0.58 Hz). The effect of a deliberate detuning of the toroid's liquid frequency, i.e. making the liquid frequency of the inner and outer toroids different, is presented in Figure 3-18. The ratio of the frequency is now 1.33 as against 1 in the previous case. The partition diameter is now 20 cm with the volume of the liquid in the outer toroid of 700 ml. The difference in the performance is rather remarkable. Although the peak value of rj for the double toroid is now lower compared to the single toroid case, it is able to provide a measure of damping over a wide range of excitation frequency. Thus the damper may be used to advantage in countering buffeting type of oscillations casued by turbulence and gusty wind conditions, where the forcing function has a broad frequency spectrum. Next, with the partition diameter held fixed at 20 cm, the effect of the distribution of liquid between the two toroids (with the total volume held fixed at 1000 ml) was assessed. Of course, this will also affect the liquid frequency in the two toroids, i.e. the tuning parameter C0i/co 0. Two values were considered: C0i/C0o =1.13 and 1.84 corresponding to the liquid volume in the outer toroid of 800 ml and 600 ml, respectively. The trends are rather clear. An increase in the value of the tuning parameter lowers the peak value of T| but extends the response over a wider range of the excitation frequency. Note, for u)i/co0 = 1.84, there are two distinct peaks, each in the vicinity of the outer and inner toroid's liquid frequency (co0 = 0.55 Hz, C0i = 1.01 Hz). Thus the double toroid configuration 64 Figure 3-18 The effect of detuning of the liquid frequency of the inner and outer toroids on the performance of a double toroidal damper. Performance of the single toroidal damper is also included to assess the relative merit. 65 66 presents a scope for providing a family of dampers suitable for a wide variety of applications. 67 4. F L O W V I S U A L I Z A T I O N 4.1 Preliminary Remarks To obtain better appreciation of the energy dissipation process through sloshing, a visual study of the liquid modes as affected by the damper geometry, liquid height, excitation frequency, etc. was carried out. The tests conducted and the facilities used were as follows: (i) free vibration tests with the bridge-deck or partitioned rectangular damper showing the effect of the number of compartments; (ii) forced vibration tests, using the harmonic excitation facility, on a rectangular damper with an aspect ratio of 1.64; (iii) forced vibration tests for the tuned double toroidal case; (iv) forced vibration tests on a double toroid intentionally mistuned. During the free vibration tests, the time to damp was quite short, and the control on the system parameters was limited. However, the forced vibration study proved to be extremely useful as it permitted controlled variation of the excitation frequency using the Scotch-Yoke mechanism (Figure 2-6). The fundamental as well as subharmonics and higher harmonics were observed and recorded using a still camera and a S-VHS video camcorder. In particular, the video showed, rather dramatically, progressive evolution of the waves, their merging, dissolution, regeneration and collision. The amount of visual information obtained through photographs and videos is substantial. Only a few typical photographs are presented here to have some appreciation of the complex free surface dynamics. 68 4 .2 Free Vibration Tests The free vibration tests with a rectangular bridge-deck damper were carried out for two, five and ten compartment conditions. In each case, the amount of liquid was kept fixed. The system was free to undergo roll motion. It should be recalled that the effect of the number of partitions is to change the compartment's aspect ratio and hence the liquid's natural frequency (CO/). When CO/ coincides with the structural natural frequency in roll (cor), one would expect maximum sloshing motion and hence higher damping. This should occur for the five compartment case. Figure 4-1 clearly shows relatively large sloshing motion of water, purposely coloured to help visualization, for the five compartment case. 4.3 Forced Vibration Tests Rectangular Damper Here the tests were carried out on a rectangular damper with an aspect ratio of 1.64. This value was chosen to help compare with the flow visualization results reported by Seto [23]. Figure 4-2 presents sketches of the free surface dynamics based on visual observations as the excitation frequency is slowly increased from below to above the resonance value. Note, the modal character changes from the plane standing wave undergoing rocking motion (very little dissipation, Figure 4-2a) followed by a propagating wave train (Figure 4-2b), and a single wave with the larger amplitude at resonance leading to wave-breaking (Figure 4-2c). In the post resonance phase, the rocking mode at a higher frequency reappears (Figure 4-2d). 69 Figure 4-1 Visual study of the sloshing motion as affected by the number of partitions for a given amount of liquid: (a) two compartments; (b) five compartments; (c) ten compartments. 70 (b) propagating wave train (c) propagating single pulse (d) standing wave line of nodes ~+ • e e cos co e t Figure 4-2 Schematic diagram based on visual observation showing the free surface geometry as affected by the variation of the excitation frequency around the resonance: (a) standing wave; (b) propagating wave train; (c) propagating single pulse at resonance; (d) standing wave. 71 Typical photographs of the free vibration modes are presented in Figure 4-3. It shows the formation of the wave train at cO//coe = 0.93 (Figure 4-3a). The following two photographs depict the resonance condition, i.e. co/= co e . In Figure 4-3(b), the single wave after impacting the far wall is returning. It has covered, approximately, half the length of the damper. In Figure 4-3(c), the wave has reached the front wall, impacted, and there is wave-breaking. Tuned Double Toroid The double toroid used in the flow visualization study was the same as that employed during the damping investigation reported in Chapter 3. Thus its dimensions were: Di = 12 cm.; D 0 = 30 cm.; and D m = 20 cm. Tuned condition refers to the case where the liquid frequency (CO/) in the outer and inner toroid is the same. In the present case (0/ = 0.58 Hz. If precisely tuned, the surface wave dynamics in both the toroids should be the same. The development of typical surface modes with an increase in the excitation frequency are approximately sketched in Figure 4-4. At a very low frequency, the free surface pivots like a plane about the line of nodes, extending from 90° to 270° points, and the motion is described by a standing wave. As the excitation frequency increases, two progressive wave trains appear simultaneously, propagating in opposite directions circumferentially, colliding at 0° and 180° positions (Figure 4-4b). As the frequency approaches the resonance, the wavelets in the train merge, finally leading to a single wave at resonance (Figures 4-4c,d). Now the dissipation of energy through collision of waves reaches a maximum. Beyond the sloshing resonance, the two colliding waves merge into one and 72 73 line of nodes 90° wave (a) standing wave (b) wave trains (c) two waves line of nodes (d) single propagating wave (e) swirling single wave (f) standing wave (maximum dissipation) £ e cos coet Figure 4-4 Top and side views of the free surface oscillation modes in a toroidal damper as the forcing frequency, coe, increases from below to beyond the sloshing resonance at a fixed oscillation amplitude, £ e : (a) standing wave; (b) wave trains; (c) two waves; (d) single propagating wave; (e) swirling single wave; (f) standing wave. 74 display swirling motion with negligible damping (Figure 4-4f). A representative set of photographs captures several phases of the surface dynamics as shown in Figure 4-5. Starting with a standing wave about the horizontal line of nodes (Figure 4-5a), an increase in excitation frequency results in two wave-trains circumferentially travelling in opposite directions (Figure 4-5b). They collide at diametrically opposite ends of the double toroid and lead only to a small amount of damping as the collision is mild. Intense collision occur at resonance (Figure 4-5 c,d) with large dissipation of energy. Mistuned Double Toroid Next the double toroidal damper was purposely mistuned, i.e. the liquid frequency in the outer and inner toroids were made different through a change in the distribution of the liquid volume. The liquid frequency co0/(Oi was 0.75. Obviously, the surface dynamics in the two toroid is different at a given excitation frequency. Photographs in Figure 4-6 attempt to show this as the excitation frequency is gradually increased. At a very low frequency, a standing wave first appears in the outer toroid followed by a wave train as shown in Figure 4-6(a). Note, even with a wave train in the outer toroid, the inner toroid displays a small amplitude standing wave. With an increase in the frequency, collision of two single circumferental waves appear at resonance in the outer toroid; while the inner toroid still' exhibits the standing wave (Figure 4-6b). As the excitation frequency increases further, the standing wave reappears in the outer toroid while a wave train appears in the inner toroid (Figure 4-6c). The standing wave continues to persist in the outer toroid as the inner toroid appraches resonance (Figure 4-6 d, e). It is this 75 Figure 4-5 Photographs showing surface wave characteristics for a tuned double toroid with CO/ = 0.58 Hz: (a) standing wave; (b) wave train. 76 Figure 4-5 Photographs showing surface wave characteristics for a tuned double toroid with CO/ = 0.58 Hz: (c) resonance showing collision of waves at the far end; (d) collision of waves at the near end during resonance. 77 Figure 4-6 Progressive variation of the surface waves in the outer and inner toroids, during a mistuned condition, with increase in the excitation frequency: (a) wave train in the outer toroid, standing wave in the inner toroid; (b) resonance in the outer toroid, standing wave in the inner toroid. 78 Figure 4-6 Progressive variation of the surface waves in the outer and inner toroids, during a mistuned condition, with increase in the excitation frequency: (c) standing wave in the outer toroid, wave train in the inner toroid; (d) standing wave in the ourter toroid, two waves approaching collision at resonance in the inner toroid. 79 Figure 4-6 Progressive variation of the surface waves in the outer and inner toroids, during a mistuned condition, with increase in the excitation frequency: (e) standing wave in the outer toroid, collision of waves at resonance in the inner toroid. 80 interplay between the outer and inner toroid surface wave dynamics that resulted in the broadband frequency response (Chapter 3, Section 3-3). 81 5. WIND INDUCED INSTABILITY STUDY 5.1 Preliminary Remarks With some appreciation as to the damper performance, the next logical step was to assess effectiveness of the nutation dampers in controlling wind-induced instability. To that end, a set of carefully planned experiments was carried out using the department's laminar flow wind tunnel. As pointed out before, the focus here is on the torsional instability control, which has received relatively less attention [33-36]. The classical Stockbridge damper has proved to be quite successful in suppressing plunging oscillations of high voltage transmission line conductors [14, 15, 37-39], however, it is virtually ineffective in controlling torsional oscillations of a bundle of conductors (Figure 5-1). On the other hand, the nutation damper can be readily arranged in an appropriate fashion to control plunging, torsion as well as coupled oscillations. Rectangular, toroidal and circular dampers were used to this end. 5.2 Test-Arrangement The model under test consists of a two-dimensional cylinder with square cross-section having dimensions 0.10 x 0.10 x 0.91 m. Square prisms are known to undergo torsional instability [4, 40] in galloping. The model attempts to simulate torsional instability in galloping encountered with a variety of bluff bodies, including bridges and bundles of transmission line conductors, as pointed out before. It was constructed from balsa-wood and supported by a steel shaft, passing through its center along the longitudinal axis. The steel shaft, and 82 Bundle of Transmission Line Conductors Figure 5-1 A schematic diagram of a high voltage transmission line showing a bundle of conductors undergoing torsional oscillations. 83 hence the model, are supproted by a pair of low friction ball-bearings, permitting free rotational motion about the horizontal longitudinal axis. The axis is oriented normal to the wind direction. An optical shaft encoder (US Digital Corp.), mounted at one end, is supported by the signal conditioning circuit to provide time-history of the rotational motion. It has a sensitivity of 0.25°. A system of springs, attached to the shaft through a bracket, gave a desired torsional stiffness to the model. The shaft also carries a small platform to support a damper under test. The encoder, bracket, springs, and damper are all located outside the wind tunnel test-section, and hence do not affect the flow. Figure 5-2 shows schematically the model and test-arrangement. Photographs in Figure 5-3 indicate the location of the torsional encoder, and a damper during tests. The dynamic tests were carried out in a closed circuit laminar flow wind tunnel shown in Figure 5-4. The tunnel has a test section of 0.69 x 0.91 x 2.44m. This is a low-speed, low-turbulence, return type wind tunnel where air speed can be varied from 1.5-50 m/s with a turbulence level less than 0.1%. The pressure differential across the contraction section of *J:1 ratio is measured with a Betz micromanometer with an accuracy of 0.2 mm of water. The rectangular test-section, 0.69 x 0.91m, is provided with 45° coiner fillets which vary from 13.25 x 13.25cm to 12.10 x 12.10 cm to partly compensate for the boundary-layer growth. The spatial variation of mean velocity in the test-section is less than 0.25%. The wind tunnel is powered by a 15 hp direct current motor driving a commercial flow fan with the Ward-Leonard System for speed control. During a typical test, the model response in galloping was first studied in 84 Figure 5-3 Photographs showing springs, encoder and damper during wind tunnel tests. 86 absence of a damping device, i.e. with only the inherent structural damping present. As expected, this resulted in a large amplitude limit cycle type of self-excited oscillations (galloping) beyond a critical wind speed. The amplitude of oscillations progressively increased with an increase in the wind speed and at times exceeded the limit of the apparatus. This was followed by the test with a nutation damper under consideration to assess its effectiveness with different amount of liquid. Visual record of the system response was also made using a video camera. ' 5 . 3 Structural Response with Nutation Dampers Typical time histories of signals given by the optical torsional transducer without and with a circular damper are given in Figure 5-5. Note, the signal for the case without damper is quite amplitude modulated. With the circular damper, there is a dramatic decrease in the amplitude and the model appears to execute seemingly random small oscillations about the equilibrium position. Hence, the torsional response from now on is plotted as the rms value. The effectiveness of a circular damper in controlling the galloping instability is illustrated quite vividly in Figure 5-6. In absence of the damper, the galloping instability sets in at a critical reduced wind speed of 6.55. As the wind speed is increased gradually beyond this value, the amplitude of oscillations increases until the physical limit of the apprartus is reached (= ±10°). Next, a circular cylindrical damper was installed. The tests were carried out for two liquid heights, with the damper mass approximately equal to 2% of the structure. The damper was tuned to the structural natural frequency in torsion. 88 68 Figure 5-6 Plots showing effectiveness of a circular cylindrical damper in regulating the torsional galloping instability. 90 This was accomplished using the small wave theory mentioned earlier. It is apparent that the damper is quite effective reducing the amplitude by about an order of magnitude. For example, at reduced wind speed of 10.30, the damper is successful in decreasing the vibration amplitude by about 85% (h/D = 0.111). Figure 5-7 gives similar results for a toroidal damper. The tests were carried out for three liquid heights. The response remains essentially the same for the three cases. As before, the damper is quite successful in arresting the galloping instability. Next, to assess relative merit of the damper configuration, three different geometries were considered: circular, toroidal, and square having the side length W. Each was tuned to the structural frequency in torsion. Figure 5-8 compares their relative effectiveness. It is apparent that all the dampers perform quite well. The square damper appears to have slightly lower efficiency as the aspect ratio is 1.0 instead of the optimum value of 1.2 as discussed in Chapter 3. Efficiency of the circular and toroidal dampers is essentially the same, however, the latter seems to perform slightly better at reduced wind speeds higher than 8.84. However, these minor differences in the damper performance are within the error associated with the test and data acquisition system which is estimated at around 8%. Hence, one can say that the dampers showed essentially the same level of efficiency. A video taken during the wind tunnel tests shows rather effectively, spectacular performance of the three dampers in arresting the structural oscillations, as well as surface wave conditions in the dampers. 91 92 I 1 1 I I I ' 1 ' 1 I I I I I I L 7 8 T T 9 1 0 Figure 5-8 Relative performance of three nutation damper geometries in controlling the galloping instability. 93 6. C L O S I N G C O M M E N T S 6.1 Concluding Remarks The thesis represents a rather fundamental study aimed at improvement in the energy dissipation of rectangular and toroidal dampers. This is achieved through a comprehensive test-program with three distinct phases: (a) parametric study using free and forced vibration test-facilities to arrive at the optimum configurations of partitioned rectangular and toroidal dampers; (b) flow visualization of the surface wave dynamics to have better physical appreciation of the conditions favourable to energy dissipation; (c) wind tunnel tests to substantiate effectiveness of the dampers in suppressing galloping type of self-excited oscillations experienced by bridge-decks and bundles of transmission line conductors. Based on the investigation, some of the more important results may be summarized as follows: (i) For a given amount of liquid, the damper performance can be improved through optimal partitioning in the direction of the wave motion. For a rectangular damper, this corresponds to the compartment aspect ratio of 1.2. As can be expected, optimal partitioning condition is related to the surface wave condition that leads to peak energy dissipation. In general, partitioning of a toroidal damper into a double toroid leads to a broadband response with a small reduction in the damping level. (ii) With the optimum configuration (i.e. the aspect ratio of 1.2), the 94 rectangular damper has a substantially higher level of damping in roll compared to that in pitch. As can be expected, due to improved sloshing, the peak damping occurs in the region where the liquid natural frequency is close to the system natural frequency. In general, increase in the initial disturbance leads to higher value of the peak damping. (iii) The effect of the coupled roll-pitch motion is to slightly reduce the peak damping in roll of the optimum rectangular damper, together with an increase in the pitch damping by a small amount. This is directly related to the changed direction of the surface waves at the time of impact with the damper walls. Level of the initial disturbance affects the damping response in the same manner as in the case of the uncoupled dynamics, however, now the peak damping occurs at a slightly lower liquid frequency. On the other hand, an increase in the system's pitch frequency causes the peak damping in roll and pitch to occur at a higher value of liquid height. This is attributed to the highly nonlinear character of the sloshing dynamics. (iv) Because of its broadband response to the excitation frequency, a toroidal damper, through suitable partitioning, can be used to advantage in regulating response of structures to turbulent winds and gusty conditions leading to beffeting. (v) Flow visualization study provided better qualitative appreciation of the free surface dynamics leading to the efficient dissipation of energy. In general, higher damping is associated with wave-breaking through impact with the damper walls for rectangular dampers, and that between the waves travelling in opposite directions for the toroidal case. 95 (vi) The wind tunnel tests substantiated effectiveness of the nutation dampers in suppressing wind-induced galloping instability in torsion. The results should prove useful in the safe design of bridges, transmission lines and other structures susceptible to this form of instability. 6.2 Recommendations for Future Work The pursuit of knowledge always presents new challenging problems as one's insight into the subject deepens. Although considerable progress has been made in understanding and design of nutation dampers, there are several avenues for further investigation which are likely to be rewarding: • Systematic tests (free, forced and flow visualization; uncoupled as well as coupled) with a rectangular damper having the optimum aspect ratio of 1.2 should be conducted to gain detailed fundamental information about this efficient configuration. • Efforts should be made to improve damping performance of the above configuration through modification of wall geometry, introduction of speed bumps, momentum injection through pendulum, etc. to promote wave-breaking. • Double toroidal damper having a large slenderness ratio appears encouraging. Its performance should be checked through a carefully planned test-program. • Wind tunnel test-arrangement should be modified to allow large amplitude as well as coupled motions. This wil l permit assessment of nutation dampers' effectiveness in suppressing plunging-torsion and roll-pitch 96 coupled motions often encountered in practice. Development of a numerical approach, based on the shallow water small wave model, as applied to cylindrical and toroidal dampers would represent an important contribution to the field. It should provide design engineers with a convenient, efficient and economical tool to design this class of dampers. 97 R E F E R E N C E S [1] Williamson, C.H.K., and Roshko, A. , "Vortex Formation in the Wake of an Oscillating Cylinder," Journal of Fluids and Structures, vol. 2, 1988, pp. 355-381 [2] Strum, R.G., "Vibration of Cables and Dampers," Electrical Engineer, Vol . 55, 1936, pp. 637-688. [3] Scanlan, R.H. , and Velozzi, J.W., "Catastrophic and Annoying Responses of Long-Span Bridges to Wind Action", Annals of the New York Academy of Sciences, Vol . 352, 1980, pp. 247 - 264. [4] Blevins, R., Flow Induced Vibration, Van Nostrand Reinhold, New York, Second Edition, 1990, pp. 54-82. [5] Blevins, R., Formulas for Natural Frequency and Mode Shape: Sloshing in Tanks, Basins and Harbours , Van Nostrand Reinhold, New York, 1979, pp. 364-385. [6] Zdravkovich, M . M . , "Review and Classification of Various Aerodynamic and Hydrodynamic Means for Suppressing Vortex Shedding," Journal of Wind Engineering and Industrial Aerodynamics, Vol . 7, 1981, pp. 145-189. [7] Modi, V.J . , Welt, F., and Seto, M.L . , "Control of Wind-Induced Instabilities through Application of Nutation Dampers: A Brief Review," Engineering Structures, Vol . 17, No.9, 1995, pp. 626-638. [8] Scruton, C , and Walshe, D.E.J. , " A Means for Avoiding Wind-Excited Oscillations of Structures with Circular or Nearly Circular Cross-Section," National Physical Laboratory, U.K. , Report 335, 1957. [9] Wong, R.H., and Cox, R.N., "The Suppression of Vortex Induced Oscillations of Circular Cylinders by Aerodynamic Devices," Proceedings of the 4th Colloquium on Industrial Aerodynamics, Editors: Kramer, C , Gerhardt, H.J., Ruscheweyh, H. , and Hirsch, G., Published by Fluid Mechanics Laboratory, Department of Aeronautics, Fachhochschule, Aachen, Germany, 1980, Part 2, pp. 185-204. [10] Reed, W.H., "Hanging Chain Impact Dampers: A Simple Method for Damping 98 Tall Flexible Structures," Proceedings of the International Research Seminar on Wind Effects on Structures , Ottawa, September 1967, University of Toronto Press, Vol.2, pp. 287-320. [11] Den Hartog, J.P., Mechanical Vibrations, McGraw Hil l , New York, 1956, pp. 305-309. [12] Gasparini, D.A. , Curry, L .W. , and Debchaudhury, A. , "Passive Viscoelastic Systems for Increasing the Damping of Buildings", Proceedings of the Fourth Colloquium on Industrial Aerodynamics, Editors: C. Kramer, H.J. Gerhardt, H. Ruscheweyh, and G. 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"Apparatus for Demonstrating Dynamics of Sloshing Liquids", Bulletin of Mechanical Engineering Education, Vol. 5, 1966, pp. 65 - 70. [18] Modi, V . J . , Sun, J.L.C., Shupe, L.S., and Solyomvari, A.S. , "Suppression of Wind Induced Instabilities using Nutation Dampers," Proceedings of the First Asian Congress of Fluid Mechanics, Bangalore, India, Editor: K.S. Yajnik, Vol . A , 1980, pp. A50-1 to A50-9; also Proceedings of the Indian Academy of 99 Science, Vol . 4, Part 4, 1981, pp. 461 - 470. [19] Modi, V.J . , and Welt, F., "Nutation Dampers and Suppression of Wind Induced Instabilities," ASME Winter Annual Meeting, Symposium on Flow Induced Oscillations, New Orleans, Louisiana, December 1984, Paper No. 84/045. [20] Modi, V.J . , and Welt, F., "On the Vibration Control Using Nutation Dampers," Proceedings of the International Conference on Flow Induced Vibrations, London, England, 1987, Editor: R. King, B H R A Publisher, London, England, pp. 369-376. 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[33] Modi, V.J. , Slater, J.E., and Yokomizo, T., "Wind Induced Instability and Near-wake of a Structural Angle Section", Proceedings of ASME International Symposium on Flow-Induced Vibration and Noise, ASME Winter Annual Metting, Anaheim, California, U.S.A., November 1992, Editors: M.P. Paidoussis, C. Dalton, and D.S. Weaver, A S M E Publisher, New York, FED Vol . 138, V.6, pp. 171 - 192. [34] Sakamoto, H . , Haniu, H . and Obata, Y . , "Fluctuating Forces Acting on Two Square Prisms in a Tandem Arrangement", Journal of Wind Engineering and Industrial Aerodynamics, Vol . 26, 1987, pp. 85 - 103. [35] Parkinson, G.V. , "Phenomena and Modelling of Flow-Induced Vibrations of Bluff Bodies", Progress in Aerospace Science, Vol . 26, 1989, pp. 169 - 224. 101 [36] Parkinson, G.V., and Smith, J.D., "The Square Prism as an Aeroelastic Non-Linear Oscillator", Quarterly Journal of Applied Mathematics, Vol 17, No. 2, 1986, pp. 225 - 239. [37] Richardson, A.S . , Martucelli, J.S., and Price, W.S., "Research Study on Galloping of Electrical Power Transmission Lines," Proceedings of the First International Conference on Wind effects on Buildings and Structures, Teddington, England, 1965, H .M. Stationery Office, Vol . II, pp. 612-686. [38] Dhotarad, M.S., Ganesan, R., and Rao, V .B .A. , "Transmission Line Vibrations," Journal, of Sound and Vibration, Vol . 60, 1978, pp. 217-237. [39] Rowbottom, M.D. , "The Effect of an Added Mass on the Galloping of an Overhead Line," Journal of Sound and Vibration, Vol . 63, 1979, pp. 310-313. [40] Slater, J. E., "Aerodynamic Instability of a Structural Angle Section," Ph. D. Thesis, University of B.C., Vancouver, B.C., Canada, 1969. 102 APPENDIX I: ALGORITHM FOR THE ENCODING PROGRAM Aqusition Program for Shaft Ecoder Three seperate files, sp.cpp, softpot.h, and softpotl.cpp, written in Borland C++ are needed in compiling the aquisition program, sp.cpp is the main program which calls on the routines and files contained in softpot.h and softpotl.cpp. In particular, softpotl.cpp contains all the required commands and codes. File sp.cpp /*This program contains only the main program of softpot.exe /*In order to compile it properly, softpotl.cpp and softpot.h should /*be included in the project. /*This program is written by Gary Seng-boh Lim /*Last update on March 19, 1996 at University of B.C. #include <stdio.h> #include <stdlib.h> #include <conio.h> #include <malloc.h> #include "softpot.h" main() { char input,inputc; float testval; double s_freq; int pos_add; int pos_end; int ts, fs; void *pos; (posp *)pos = (posp *) malloc(4*6000*sizeof(int)); if (pos == NULL) { printf("\n allocation error - aborting ..."); exit(l); } */ */ */ */ */ 103 pos_add =(int) pos; s_freq = sysclock(); init(); clrscr(); printf("\nPress C or c to start collecting data"); input = getche(); if ((input == 'c') II (input == 'C')) { printf("\nEnter the sampling time (in sec):"); scanf("%d",&ts); fseek(stdin, 01,0); printf("\nEnter the sampling frequency (in Hz):"); scanf("%d", &fs); fseek(stdin,01,0); printf("\nYou have entered %d sec and %d Hz", ts, fs); printf("\nCollecting..."); printf( Initial Addre=%d",pos); pos_end = reading(ts, fs,pos,s_freq); printf("\nThe positions are %d and %d\n",pos_add, pos_end); write_to_file(pos_add, pos_end,pos); } inputc=(char) choice(); printf("\nYour choice is %c\n",inputc); }while ((inputc == 'y') II (inputc == 'Y')); printf("\nBye-bye"); return(O); } 1 0 4 File softpot.h /*This is a help file to be include in the project file for compiling sp.cpp */ /*This program is written by Gary Seng-boh Lim */ /*Last update on March 19, 1996 at University of B.C. */ #define DEF_BASE 0X300 /* default base address of interface board */ #define DATA1 0 /* data register of LS7166 */ #define CONTROL1 1 /* control register of LS7166 */ #define DATA2 2 /* data register of LS7166 */ #define CONTROL2 3 /* control register of LS7166 */ #define DATA3 4 /* data register of LS7166 */ #define CONTROL3 5 /* control register of LS7166 */ #define DATA4 6 /* data register of LS7166 */ #define CONTROL4 7 /* control register of LS7166 */ #define L A T C H 8 /* a read causes all LS7166s to latch counter */ /* LS7166 commands */ #define MASTER_RESET 0X20 /* master reset command */ #define INPUTJSETUP 0X68 /* command to setup counter input */ #define QUAD_X1 0XC1 /* command to setup quadrature multiplier to 1 */ #define QUAD_X2 0XC2 /* command to setup quadrature multiplier to 2 */ #define QUAD_X4 0XC3 /* command to setup quadrature multiplier to 4 */ #define ADDR_RESET 0X01 /* command to reset address pointer */ #define LATCH_CNTR 0X02 /* command to latch counter */ #define CNTR_RESET 0X04 /* command to reset counter */ #define PRESET_CTR 0X08 /* transfer preset to counter */ typedef struct pot{int potl; int pot2; int pot3; int pot4; } pot_pos, *posp; char choice(void); void write_to_file(int ba, int end_addr,struct pot *wloc); long read_position(int controlnum, int datanum); void init(void); void hard_latch(void); double sysclock(void); int reading (int t, int freq,struct pot *loc, double s_freq); 105 File softpotl.cpp /*This is a program contains all subroutines required to run sp.cpp /*This is written by Gary Seng-boh Lim /*Last update on March 19, 1996 at University of B.C. #include <stdio.h> #include <conio.h> #include <dos.h> #include <stdlib.h> #include <time.h> /*#include <7166.c>*/ */ */ */ /* addresses */ #define DEF B A S E 0X300 /* default base address of interface board */ #define DATA1 0 /* data register of LS7166 */ #define CONTROL1 1 /* control register of LS7166 */ #define DATA2 2 /* data register of LS7166 */ #define CONTROL2 3 /* control register of LS7166 */ #define DATA3 4 /* data register of LS7166 */ #define CONTROL3 5 /* control register of LS7166 */ #define DATA4 6 /* data register of LS7166 */ #define CONTROL4 7 /* control register of LS7166 */ #define L A T C H latch counter */ 8 /* a read causes all LS7166s to /* LS7166 commands */ #define MASTER_RESET #define INPUT_SETUP #define Q U A D . X l quadrature multiplier to 1 */ #define QUAD_X2 quadrature multiplier to 2 */ #define QUAD_X4 quadrature multiplier to 4 */ #define ADDR_RESET #define L A T C H . C N T R #define CNTR_RESET #define PRESET_CTR 0X20 /* master reset command */ 0X68 /* command to setup counter input */ 0XC1/•* command to setup 0 X C 2 / * command to setup 0 X C 3 / * command to setup 0X01 /* command to reset address pointer */ 0X02 /* command to latch counter */ 0X04 /* command to reset counter */ 0X08 /* transfer preset to counter */ 106 int base = DEF_BASE; double sjfreq; struct pot{ long potl; long pot2; long' pot3; long pot4; . } pot_pos; struct pot *pos; /*input procedure*/ char choice() { char ans; printf("\nDo u want to continue? (y or n) \n"); ans = getche(); return(ans); } /*Procedure of writing a file*/ void write_to_file(int ba,int end_addr) { FILE *data_file; /*defining the file name*/ char filename[12]; /*filename variable*/ int control, addr_track; control = 0; pos = (pot *) ba; Presetting pointer position*/ printf("Writing got %d at address=%d\n",pos->potl, ba); printf("now the pointer is at %d\n",(int)pos); /*pos = (int) data_in; initializing the address tracker*/ do { printf("Please enter the filename:"); scanf("%s",filename); fseek(stdin,0L,0); printf("\n"); 107 printff'You entered %s\n",filename); if ((data_file=fopen(filename,"r")) != NULL){ printf("your filename is invalid\n"); fclose(data_file); control = 0;} else { data_file=fopen(filename,"w"); control = 1;} }while(control ==0); printf("\nThe address is %d and the end is %d\n",(int)pos,end_addr); printf("\nthe first value is %d\n",pos->potl); while ((int)pos != end_addr) { fprintf(data_file,"%ld %ld %ld %ld\n",pos->potl, pos->pot2, pos->pot3, pos->pot4); pos++; /*addr_track = (int) data_in;*/ printf("this is the next address %d\n",(int)pos); } ; fclose(data_file); } /""Initiation Program for aquisition*/ void init(void) { /* initialize the LS7166s */ outportb(base + CONTROL1, MASTER_RESET); outportb(base + CONTROL 1, INPUT_SETUP); outportb(base + CONTROL1, QUAD_X4); outportb(base + CONTROL1, CNTR_RESET); outportb(base + CONTROL2, MASTER_RESET); outportb(base + CONTROL2, INPUT_SETUP); outportb(base + CONTROL2, QUAD_X4); outportb(base + CONTROL2, CNTR.RESET); outportb(base + CONTROL3, MASTER_RESET); outportb(base + CONTROL3, INPUT_SETUP); outportb(base + CONTROL3, QUAD_X4); outportb(base + CONTROL3, CNTR_RESET); outportb(base + CONTROL4, MASTERJRESET); 108 outportb(base + C0NTR0L4 , INPUT_SETUP); outportb(base + C0NTR0L4 , QUAD_X4); outportb(base + C0NTR0L4 , CNTR_RESET); } void hard_latch(void) { /* latch position with low pulse on pin 3 */ outportb(base + L A T C H , 0); /* toggle pin 3 of LS 7166 */ } /*reading routine*/ long read_position(int controlnum, int datanum) { /* read position of encoder */ union pos_tag { long 1; struct byte_tag {char bO; char b l ; char b2; char b3;} byte; } posl; /* long position;*/ outportb(base + controlnum, ADDR_RESET); /* reset address pointer */ posl.byte.bO = inportb(base + datanum); posl.byte.bl = inportb(base + datanum); posl.byte.b2 = inportb(base + datanum); /* position = (long)inportb(base + datanum); /* least significant byte */ /* position += (long)inportb(base + datanum) « 8; position += (long)inportb(base + datanum) « 1 6 ; /* most significant byte */ if (pos 1 .byte.b2 < 0) pos 1 .byte.b3 = -1; else posl.byte.b3 = 0; return pos 1.1; /* return position;*/ } /*checking the system clock*/ double sysclock(void) { time_t start,end; double arg_t; /^artificial time*/ double sys_freq; /*system frequency*/ clrscr(); printf("\nThe system is starting\n"); 109 start = time(NULL); for (arg_t = 0; arg_t< 100000; arg_t++); end = time (NULL); sys_freq=arg_t/difftime(end,start); printf("\nThe system frequency is % f (CPS)\n",sys_freq); return(sys_freq); /*subroutine reading values from pots and put it into a variable*/ void reading(int t, int freq) { /*Define variables*/ int ts; /*sampling time constant*/ unsigned int i , end; long temp; end = t*s_freq; printf("\nThe end value is %d",end); delay(5000); ts = 1000/freq; printf("\nreading"); /*Loop of taking reading*/ while ( i <= end) { hard_latch(); temp= read_position(CONTROL 1 ,D AT A1); printf("\ntemp value is %ld\n",temp); pos->potl = temp; printf("value = %ld\n",pos->potl); /* printf(".");*/ /* pos.potl++;*/ temp = read_position(CONTROL2,DATA2); pos->pot2 = temp; printf("value2 = %lx\n",pos->pot2); /* printf(".");*/ /* pos.pot2++;*/ temp = read_position(CONTROL3,DATA3); pos->pot3 = temp; printf("."); /* pos.pot3++;*/ 110 temp = read_position(CONTROL4,DATA4); pos->pot4 = temp; printf('V'); /* pos.pot4++;*/ printf("Val=%d\n",pos->potl); pos++; delay(ts); i=i+l+ts; /* printf("\n%d\n",i);*/ } } 1 1 1 File Double Toroidal Exp' t Design This is a spreadsheet program written in Microsoft Excel 5.0 for calculating the natural frequencies of a double toroidal damper with different liquid heights in the two ring sections. The total volume is kept at 1000.00 cm3. Rl/cm R2/cm R3/cm g(m/s2) V/cmA3 R2/R1 R3/R2 R3/R1 6.00 10.00 15.00 9.81 1000.00 1.67 1.50 2.50 Lamda2 Lamda3 Lamda h h' Lf Lf2 0.87 0.89 0.76 1.68 2.55 -0.363 -0.339 V2/cmA3 h2/cm f2/Hz Df2<0 h/d V3 h3/cm f3/Hz Df3 <0 h/d 0.00 0.00 0.00 N/A 0.00 1000.00 2.55 0.71 -0.27 0.51 50.00 0.25 0.36 -2.78 0.06 950.00 2.42 0.69 -0.31 0.48 100.00 0.50 0.51 -1.71 0.12 900.00 2.29 0.67 -0.36 0.46 150.00 0.75 0.62 -1.19 0.19 850.00 . 2.16 0.66 -0.41 0.43 200.00 0.99 0.72 -0.85 0.25 800.00 2.04 0.64 -0.46 0.41 250.00 1.24 0.80 -0.60 0.31 750.00 1.91 0.62 -0.52 0.38 300.00 1.49 0.88 -0.40 0.37 700.00 1.78 0.60 -0.58 0.36 350.00 1.74 0.95 -0.24 0.44 650.00 1.66 0.57 -0.64 0.33 400.00 1.99 1.01 -0.10 0.50 600.00 1.53 0.55 -0.72 0.31 450.00 2.24 1.08 0.03 0.56 . 550.00 1.40 0.53 -0.80 0.28 500.00 2.49 1.13 0.14 0.62 500.00 1.27 0.50 -0.88 0.25 550.00 2.74 1.19 0.24 0.68 450.00 1.15 0.48 -0.99 0.23 600.00 2.98 1.24 0.34 0.75 400.00 1.02 0.45 -1.10 0.20 650.00 3.23 1.29 0.42 0.81 350.00 0.89 0.42 -1.24 0.18 700.00 3.48 1.34 0.50 0.87 300.00 0.76 0.39 -1.40 0.15 750.00 3.73 1.39 0.58 0.93 250.00 0.64 0.36 -1.61 0.13 800.00 3.98 1.44 0.65 0.99 200.00 0.51 0.32 -1.88 0.10 850.00 4.23 1.48 0.72 1.06 150.00 0.38 0.28 -2.26 0.08 900.00 4.48 1.52 0.78 1.12 100.00 0.25 0.22 -2.88 0.05 950.00 4.72 1.56 0.84 1.18 50.00 0.13 0.16 -4.23 0.03 1000.00 4.97 1.60 0.90 1.24 0.00 0.00 . 0.00 N/A 0.00 Control f/Hz Df R1&R3 0.82 -0.39 R2&R3 0.71 -0.27 The important parameters are f, h/d and Df as they indicate the natural frequency of the damper, the non-dimensional water height and the shallow water criterion. V is the total volume of the damper. 112 V2 is the volume of liquid in the inner ring while V3 is for outer ring. R1,R2,R3 are the radii Lamda, Lamda2, Lamda3 are the wavelength, h, h2, and h3 are the liquid height. f,f2, and f3 are the natural frequency. Df,Df2,and Df3 are the shallow water parameters; they have to be <0. 113 A P P E N D I X II: F R E E V I B R A T I O N TESTS In the experiment, the rectangular nutation damper was tested with a two-degree-of-freedom free vibrational experimental setup. To assess the damping effectiveness, the well-known Logarithmic Decrement approach was used. This involves determination of the decay in amplitude over a specified number of cycles. The damping ratio, is defined as, 1 X i C = — — In —, (II.1) s 27tm X , m + l where X\ and X m + i are the amplitudes of the first peak and the m+l peaks, respectively. The damping ratios of the system with and without damper were found using the above relation. To evaluate the improvement in damping, a relatve damping ratio is defined as, r r i c = j-, (II.2) where and £ s are the damping ratios of the system with and without dampers, respectively. The corresponding reduced damping coefficient as defined by Welt [21] can now be written as, n = r , ^ , (ii.3) s 114 where 1/ and I s are the moment of inertia of the liquid and system, respectively. 115 APPENDIX III: RELATION BETWEEN THE DAMPING RATIO AND REDUCED DAMPING For a simple dashport damper, the related damping force, Fd, can be modelled by F d =cx, (111.1) where c is the damping coefficient and x is the velocity. Assuming c to be constant over a period of time, T, and for a given frequency, it can be defined as the energy dissipation per cycle of oscillation by integrating eq. (Hl. l) with respect to displacement dx, t+T J F^x dt t d c = X t + T . (III.2) J" xdx Now, consider a single degree of freedom spring dashport system to model a structure installed with a nutation damper as shown in Figure III-1. Damping coefficient, the mass of the structure, and the liquid mass of the damper are represented by c, M s , and M / , respectively. Let the system be subjected to a harmonic excitation, x e = e e sin(coet), (III.3) where e e and coe are the excitation amplitude and frequency, respectively. 116 c , N u t a t i o n D a m p i n g C o e f f i c i e n t M, , Liquid Mass of the Damper M , Mass of the Structure S t a t i o n a r y R e f e r e n c e Figure III-1 One degree of freedom model of a structure installed with a nutation damper. Inside the damper, the sloshing liquid will create an inertia force, Fj, which can be written as, Fj = M / C 0 e 2 £ e sin(coet). (III.4) It can be modified to model the sloshing force, F s , by adding a phase difference, cp, between the force and the excitation, F s = M/CO e 2£ 0 sin(coet - cp). (III.5) Because of this phase difference, F s can be used to reduce or even cancel the excitation force. In other words, F s can be seen as the damping force, Fd. The effective damping coefficient c can be found by substituting eq. (III.5) and eq. (III.3) into eq. (III.2) and evaluating the integrals, IF c = £ CO e e sin cp. (III.6) 117 The equivalent reduced damping coefficient is defined by Welt [22] as, . (HI.7) 2co M, e ' which now becomes ri = 2M,e co' I e e sin cp. (III.8) One can represent the equivalent added mass contributed by the sloshing force as [22] ^ I C O S ID (III.9) M = a F - M . e co s ' e e cos cp 2 E CO e e i.e. in the nondimensional form as . 2 " M F - M . e co s ' e e M M.e co / e e cos (p. (III.10) In this thesis, a harmonically excited shaking table in conjunction with the above dynamic model was employed to evaluate the effectiveness of a nutation damper. 118 

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