A N INVESTIGATION ON T H E SUPPRESSION OF F L O W I N D U C E D VIBRATIONS OF B L U F F BODIES Mae Lenora Seto B.A.Sc, University of British Columbia, 1987 M.A.Sc, University of British Columbia, 1990 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF T H E REQUIREMENTS FOR T H E D E G R E E OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies Department of Mechanical Engineering We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA March 1996 © Mae L. Seto, 1996 In presenting degree at the this thesis in partial University of fulfilment of of department this thesis for or by his or requirements British Columbia, I agree that the freely available for reference and study. I further copying the representatives. an advanced Library shall make it agree that permission for extensive scholarly purposes may be granted her for It is by the understood that head of copying my or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mec h a n iCaJ The University of British Columbia Vancouver, Canada frpri'!2H, Date DE-6 (2/88) Hnqi'neer" I'nej ABSTRACT Wind induced oscillations of bridges, tall buildings, smokestacks, transmission line conductors and similar bluff bodies have been of interest to scientists and engineers for a long time. Unchecked, the vortex resonance and galloping type of vibrations can cause damage to structures. Some examples include the orig- inal Tacoma Narrows Bridge, the dramatic cracking of industrial chimneys, and the destruction of building components as well as the structure itself. Apart from catastrophic destruction, this class of low frequency vibrations are known to cause undesireable working conditions leading to nausea, dizziness, disorientation and vertigo, particularly for those working at relatively greater heights as in tall buildings and air traffic control towers. Flow induced vibrations are of special relevance to engineers today because of the tendency to build taller structures and longer span bridges with ever lighter building materials. The thesis studies vortex induced and galloping type of instabilities associated with structural geometries of fundamental importance. Of particular interest is the effectiveness of energy dissipation to suppress the oscillations. To that end, a comprehensive study focuses on the design of nutation dampers and assesses their effectiveness in arresting wind induced instabilities. To begin with, a parametric study of the damper, in conjunction with frequency response tests, is used to identify important system variables contributing to significant energy dissipation. The results show that optimum combinations of the damper parameters such as the geometry, liquid height, surface seeding, and compartmenting can lead to an efficient damper, particularly if the operating conditions are conducive to wavebreaking. Among the dampers tested, the circular cylindrical geometry proved to be the most efficient. The addition of floating particles further improved the performance by around 30 %. Next, a numerical model, based on the nonlinear shallow water equations of motion, is developed to predict ii dissipation characteristics. The numerical results are also animated to provide better visual appreciation of the free surface wave dynamics. The agreement between numerical and experimental results is quite good considering the complex character of the flow. This is followed by construction of a fluid-structure interaction model through coupling of the nonlinear shallow water equations to a single degree of freedom structure undergoing vortex resonance. The agreement between wind tunnel experiments and the model is surprisingly good considering the nonlinear character of the fluid dynamics and structural interactions with it. The numerical algorithm should serve as a valuable tool in designing this class of dampers for practical applications. Finally, wind tunnel tests with two-dimensional models substantiate, rather dramatically, the effectiveness of the nutation dampers in arresting both vortex resonance and galloping types of instabilities. The dampers continue to be effective even for the case when the structure is located in the wake of other structures, the situation frequently encountered in practice. A visualization study, for flow within the damper and suppression of structural instabilities during the wind tunnel tests, complemented the experimental and numerical investigations. Both still photographs as well as a video were taken. Each phase of the study represents innovative contributions and the results obtained are of far-reaching consequence for a class of structures currently under design and those planned for the future. The thesis ends with some concluding comments and recommendations on rewarding avenues of research to pursue in the next phase of the work. iii T A B L E OF CONTENTS ABSTRACT ii LIST O F F I G U R E S viii LIST O F T A B L E S xvi NOMENCLATURE xvii ACKNOWLEDGEMENT . xxiv 1. I N T R O D U C T I O N 1.1 Preliminary Remarks 1 1.2 A Brief Review of the Relevant Literature 2 1.3 Scope of the Investigations 19 2. P A R A M E T R I C S T U D Y O F N U T A T I O N D A M P E R S 2.1 Preliminary Remarks .23 2.2 Test-facility, Instrumentation and Methodology 2.3 26 2.2.1 Free vibration tests 27 2.2.2 Forced excitation tests 30 2.2.3 Damper models 35 2.2.4 Data acquisition and analysis methodology . 39 2.2.5 Flow visualization 40 Damper Geometry and Dissipation 42 2.3.1 Preliminary remarks 42 2.3.2 The circular cylindrical damper 44 2.3.3 The toroidal damper 49 2.3.4 The rectangular damper 57 2.3.5 Effect of damper geometry 63 2.4 Effect of Partitioning a Damper iv 63 2.4.1 Partitioning the rectangular damper 64 2.4.2 Partitioning of the toroidal damper 71 2.5 Influence of Floating Particles 74 2.6 Summary of Results 86 3. W I N D I N D U C E D I N S T A B I L I T Y STUDIES 3.1 Preliminary Remarks 88 3.2 Wind Tunnel Test Facilities, Instrumentation and Methodology . . 88 3.3 Structural Response with Nutation Dampers 96 3.3.1 Isolated aerodynamic structure 96 3.3.2 Effect of floating particles 100 3.3.3 Structures affected by wake 100 3.4 Concluding Remarks 109 4. N U M E R I C A L A P P R O A C H T O S Y S T E M D Y N A M I C S W I T H NUTATION DAMPING 4.1 Preliminary Remarks 115 4.2 Model 115 4.3 Potential Flow Analysis 118 4.3.1 Equations of motion 118 4.3.2 Initial and boundary conditions 119 4.3.3 Solution of the potential flow equations 121 4.4 Integration of the Governing Equations along z-axis 123 4.4.1 Continuity equation 123 4.4.2 Momentum equations 123 4.5 Dissipation Model 126 4.5.1 Energy density function 4.5.2 Dissipation at the damper bottom and side walls 4.5.3 Dissipation within the body of the fluid v 127 . . . . 128 130 4.5.4 Dissipation due to free surface dynamics 131 4.5.5 Added mass and reduced liquid damping 135 4.5.6 Modified equations of motion 137 4.6 The Dispersion Relation 139 4.7 Numerical Solution for Sloshing Liquid Equations 141 4.7.1 Two-dimensional equations at resonance 141 4.8 Fluid-Structure Interaction Dynamics 146 4.9 Results and Discussion 149 5. CLOSING C O M M E N T S 5.1 Concluding Remarks 172 5.2 Recommendations for Future Work 174 REFERENCES 176 APPENDICES 188 I C A L I B R A T I O N O F I N S T R U M E N T A T I O N U S E D IN T H E T E S T PROGRAM LI Force Balance 189 1.2 L V D T Used to Measure Structural Displacement During W i n d Tunnel Tests II 190 LISTING OF P R O G R A M S D E V E L O P E D F O R T H E S T U D Y 11.1 logdec.m: Logarithmic Decrement from Digitized Data . . 11.2 getfreq.m: Amplitude, Frequency and Phase of the Sloshing Force 11.3 whtor.m: 11.4 solvtor.m: 191 192 Determination of Toroidal Geometry 196 Damper Geometries (Circular, Toroidal and Rectangular) for Given Liquid Frequency and Volume . . vi 199 Ill SHALLOW WATER MODEL FOR NUTATION DAMPER 111.1 Expression for dw/dt 201 111.2 Nondimensional Form of Nonlinear Shallow Water Wave Equations 202 111.3 Free Surface and Kinematic Boundary Conditions vii . . . . 205 LIST O F F I G U R E S 1-1 Various devices used to control vortex resonance and galloping types of instabilities. They provide: (a) modification in excitation force; (b) energy dissipation through damping 5 1-2 Parameters associated with a toroidal damper 8 1-3 Internal modifications of the nutation damper studied by Welt [34] to optimize damping 10 1- 4 Scope of the investigation 22 2- 1 Schematic representation of a tuned mass damper: (a) idealized configuration; (b) the nutation damper modelled as a tuned mass damper. F is the excitation force on the e primary body. 2-2 24 The fluid particle trajectories in two types of waves observed in a sloshing fluid: (a) progressive wave; (b) standing wave 2-3 The 25 bridge deck free vibration experimental set- up: (a) plunging oscillation mode; (b) torsional oscillation mode (roll); (c) torsional oscillation in transverse direction (pitch) 2-4 28 A free vibration test facility for studying efficiency of nutation dampers 2-5 29 Free vibration data analysis: (a) flow chart illustrating procedure for determining r) i ; (b) digitized free vibration deTt cay data; (c) typical time window used in the analysis 2-6 31 A schematic diagram of the forced vibration test setup to obtain reduced liquid damping, 77,./, and added mass, M 32 a 2-7 The force balance used to measure the reduced liquid damping of nutation damper models under steady state excitation 34 viii 2-8 Example of data traces obtained during the steady state excitation study: (a) displacement of the platform, e ; (b) sloshing force, F 36 Schematic diagrams showing basic nutation damper geometries studied: (a) toroidal damper; (b) cylindrical damper; (c) rectangular damper 37 2-10 Data acquisition system for forced vibration test-facility 41 2-11 Photograph showing some of the nutation damper models used during the parametric study: (i) rectangular; (ii) toroidal; (iii) partitioned rectangular; (iv) circular cylindrical 43 Reduced liquid damping, r) i, for a circular nutation damper (Model # 7) at different liquid heights. Note, a distinct improvement in dissipation as h/R decreases in the excitation range f < 0.55 Hz 45 Circular cylindrical damper showing formation of a wave train in the direciton of excitation: u) /ui — 0.80 46 Top and side views of the free surface modes in a circular damper as the forcing frequency, u , increases from below to beyond the sloshing resonance. The damper is shown accelerating to the right: (a) standing wave; (b) propagating wave; (c) toroidal-type sloshing (resonance); (d) stronger propagating wave; (e) propagating waves with swirl; (f) standing wave 47 Variation of the reduced damping parameter with the excitation amplitude for a large value of D/d = 6. The liquid height and excitation frequency were held fixed at h/d = 0.5 and 6) = 1.15, respectively. 50 Effect of excitation frequency and liquid height on the reduced damping parameter. The excitation amplitude and the damper geometry parameter were held fixed . 52 e s 2-9 2-12 r e 2-13 e 2-14 e 2-15 e 2-16 2-17 Top and side views of the free surface oscillation modes in a toroidal damper as the forcing frequency, cu , increases from e ix below to beyond the sloshing resonance at a fixed oscillation amplitude, e : (a) standing wave; (b) wave trains; (c) two e waves; (d) single propagating wave; (e) swirling single wave, (f) standing wave 2-18 53 Typical flow visualization photographs for a toroidal damper showing: (a) two waves, on either side, approaching collision to the right, f = 0.6 Hz; (b) single wave at resonance, e f e 2-19 0.65 Hz 55 Effect of excitation frequency and liquid height on the dissipation for a rectangular damper 2-20 58 Wave train in a rectangular damper of aspect ratio L/W — 1.64 at u /ui = 0.95 59 e 2-21 Photographs showing time history of the propagating waves near the resonance and wavebreaking 2-22 60 Schematic diagrams based on visual observations 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) resonant sloshing; (d) propagating single pulse; (e) standing wave 2-23 62 The effect of compartmenting the rectangular nutation damper (Model # 10), for a given system frequency (5.78 rad/s) and oscillation amplitude (e /w = 0.17), on the damping ratio. 0 AR is the aspect ratio of an individual compartment 2-24 65 Performance of the optimally compartmented rectangular damper (Model # 10, AR = 1.2) at two natural system frequencies, f = 0.73 and 0.92 Hz, as affected by the liquid n height 2-25 67 Effect of initial excitation amplitude, e , and compartment 0 aspect ratio on the reduced damping of a rectangular damper (Model # 10). The experimental set-up is shown in the inset. The system frequency w was held at 5.78 rad/s with n the liquid height parameter h/W = 0.08 x 69 2-26 Performance of the rectangular damper (Model # 10) in pitch motion as affected by the compartment aspect ratio and excitaition amplitude: (a) h/W = 0.033; (b) h/W = 0.05; (c) h/W = 0.10 2-27 70 Geometry of the double toroidal damper as given by the inviscid small wave theory: fi = 0.58 Hz, V\= 730 ml of water, and D = 30 cm 72 0 2-28 Dimensions of the double toroid, single toroid (Model # 3) and circular damper (Model #7) studied at // = 0.58 Hz, Vj = 730 ml 2-29 73 Effect of damper geometry and excitation frequency on the dissipation of: (a) single toroid (Model # 3); (b) double toroid (Figure 2-20a) ; (c) circular damper (Model # 7). A fixed volume of 730 mL of water was used 2-30 75 Representative plots showing the effect of particle geometry and concentration on the reduced damping of a toroidal damper (Model # 6). The liquid height was held fixed at h/d= 2-31 0.3 78 Representative plots showing the effect of oscillation amplitude and particle geometry on the reduced damping for a toroidal damper (Model # 6, h/d = 0.3, D/d = 4.5; u> = 1.0, e /d — 0.085). Suffix 'b' in the aspect ratio refers e e to particles with ends sealed or blocked 2-32 Representative plot showing the effect of particle concentration on (Model # 7). reduced damping for a circular damper Note, the liquid height is held fixed at h / R = 0.047 2-33 79 81 The effect of oscillation amplitude and seeding density on the damping efficiency of a rectangular damper (Model # 10) optimally compartmented (AR = 1.2) 2-34 82 The effect of seeding density, for liquid heights h/W = 0.10 and 0.15, on the performance of a rectangular damper (Model # 13, 1 compartment) xi 83 2-35 Comparative performance of a rectangular damper (Model # 13, h/W = 0.15), for the cases of plain water and fully seeded surface conditions, over a range of frequencies 3-1 85 W i n d tunnel test set-up to assess effectiveness of a nutation damper in suppressing vortex resonance and galloping. Three different damper geometries were used during the tests 3-2 89 Schematic diagram of the closed circuit laminar wind tunnel facility used in wind induced instability studies 3-3 91 Typical traces taken during wind tunnel tests showing variation of the structural model displacement with time: (a) without damper; (b) with a circular damper of mass ratio 2% 3-4 92 Spectrum of elastically mounted square cylinder displacement, Y(t) / H , under wind loading. The square cylinder is about to make the transition to galloping. The inset shows the time history of displacement for the square cylinder. Note the presence of strong modulations during transition from vortex resonance to galloping 3-5 94 The effectiveness of a circular nutation damper in arresting both vortex resonance and galloping instability of a square cylinder, -o- represents the system response with little damping, hence the square model gallops from the start. - A - presents the displacement data with system dissipation increased through introduction of eddy current dampers in order to separate vortex resonance from galloping, - o - shows the nutation damper suppressing both the vortex resonance and galloping over the entire range of the wind speed. 3-6 97 W i n d tunnel test results showing the effect of liquid mass ratio on the vibratory response of a square prism (Model # 1) with a circular cylindrical damper (Model # 9 ) . xii 98 3-7 Comparative performance of three different geometry dampers (Model # 6, 9, 14) showing their effect on the galloping response of a square cylinder (Model # 1) 3-8 99 Wind tunnel test results for a two-dimensional circular section model (Model # 3) in presence of a toroidal damper (Model #6) with particles 3-9 101 Effectiveness of the toroidal damper (Model # 6) in arresting galloping oscillations of a two-dimensional square cylinder (Model # 1). The effect of floating particles is also shown 3-10 102 Plan view of the interfering structural models arrangement: (a) inline square cylinders; (b) inline circular cylinders 3-11 103 Effect of wind speed on the response of a square cylinder (Model # 2) critically located in the wake of the other square cylinder (Model # 1): (a) undamped response with the wind speed lower than the resonance; (b) undamped beat-type response for the wind speed close to resonance; (c) damped response at resonance for the circular geometry damper (Model #9); (d) undamped response with the wind speed beyond the resonance value 3-12 105 Effect of the damper geometry on the resonant response of the inline square cylinder (Model # 1, 2): (a) mass ratio of 1 %; (b) mass ratio of 4%. Note the circular damper appears to be quite promising with a reduction in the amplitude by 98% 3-13 110 A comparison of damper performance at resonance for inline circular cylinders (Model # 3, 4) under critical conditions of wind speed and separation: (a) liquid mass ratio is 1%; (b) liquid mass ratio is 4%. The circular damper (Model # 9) continues to be the most efficient 4-1 112 Variable definitions used in deriving the equations of motion for the rectangular damper xiii 117 4-2 Modelling nonlinear sloshing in a rectangular nutation damper: (a) variable definition; (b) discretization over the solution domain 4-3 143 Flow chart illustrating procedure to derive the nonlinear shallow water sloshing equations for the nutation damper. The number in the parenthesis refer to the equation resulting from executing the step indicated in the box 4-4 145 Schematic diagram showing the test arrangement for the fluid-structure interaction model 4-5 147 Solution algorithm for implementing nonlinear shallow water sloshing in a rectangular nutation damper. 4-6 150 Time history of the free surface profile for a rectangular damper subjected to an excitation frequency Q) — 0.90: e (a) numerical results based on shallow water model; (b) animation of the numerical data. Note the standing wave character of the free surface 4-7 151 Free surface motion over a period at u> = 0.95: (a) numere ical results based on shallow water model; (b) animation of the numerical data. The distinctive feature is the presence of a wave train 4-8 153 Free surface dynamics at resonance: (a) numerical results based on shallow water model; (b) animation of the numerical data. The train with two waves is apparent. Note larger wave amplitude at the wall (r = 0.5) 4-9 155 Time histories of wave height at five spanwise locations of the rectangular damper at resonance 4-10 157 The linear shallow water solution for a rectangular nutation damper at resonance clearly showing its limitation. Note, it is unable to give the propagating wave solution. The sloshing force also shows discrepancy 4-11 158 The nonlinear shallow water numerical solution for a rectangular nutation damper oscillated at a frequency u) = e 1.05: (a) numerical results based on shallow water model; xiv (b) animation of the numerical data. Note the train with a single wave and higher amplitude at the wall. The sloshing force is also higher 4-12 161 Experimental results for a rectangular cross-section box damper showing the effect of liquid height and excitation frequency on the damping ratio. Note V refers to the numerical results corresponding to resonance. Correlation with experimental results is good 4-13 163 A comparison between the numerical model, its animation and flow visualization results for a rectangular damper free surface showing: (a) wave train; and (b) single wave 4-14 165 Typical plots illustrating interaction between the damper fluid and structure during the vortex resonance: (a) fluid wave height at the damper wall; (b) displacement of the structure, x t, with the damper. The undamped peak value 3 of x t was found to be 1.0. 3 Modulated character of the x signal agrees well with the experimental observations a reported in Chapter 3 4-15 168 Comparison between experimental measurements and numerically obtained structural displacement x t/H, 3 ratio Mi/m , 3 near vortex resonance (u> = 1.05): (a) M\jm e 0.01; (b) Mi/m 3 5-1 with mass a = 0.04. The agreement is acceptable. = 169 Proposed novel damper geometries which may lead to increase dissipation: (a) curved partitions; (b) speed bumps; (c) pendulum. 1-1 175 Calibration of the force balance used in shaking table tests: static calibration with the system at rest undergoing bending when subjected to a known stress 1-2 189 Calibration of the L V D T used to measure displacement of the shaking table and structural displacement during wind tunnel tests 190 xv LIST OF T A B L E S 2-1 Nutation damper models used in the parametric study 2-2 Toroidal damper design for given liquid frequency and volume 2-3 38 56 Important parameters in the study of floating particles to enhance dissipation 76 2-4 Floating particles used during the test-program 76 2- 5 Liquid height and corresponding resonant sloshing frequencies used in the test-program 76 3- 1 Structural models used during the wind tunnel tests 93 3- 2 Dampers used in wind tunnel tests 93 4- 1 Nondimensionalization of the shallow water equations of motion The cases studied numerically for steady excitation with 138 the model initially at rest 149 4-2 xvi NOMENCLATURE A wave amplitude of an inviscid fluid or damper wall area that sloshing fluid impacts against Ai coefficients of a polynomial fit, eq.(2.2) A\ linear coefficient in the lift-oscillator model; eq. (4.52) A3 coefficient of the cubic term in the lift-oscillator model, eq. (4.52) AR particle aspect ratio, l /d ; for partitioned rectangular damper V/W a amplitude of propagating free surface wave p p B , By, B body forces per unit mass in the x, y and z directions, respectively x b z coefficient used in the lift-oscillator model to couple the structural displacement with the external excitation; eq. (4.52) C constant used in the definition of linearized velocity potential <j> CA constant associated with the excitation force; CCLT Closed Circuit Laminar Tunnel Ci aerodynamic lift coefficient CL lift coefficient corrected for blockage c wave speed or damping coefficient C c pH L /%-K 2 2 m St 2 m 3 c critical damping c s inherent system damping (without nutation damper) d tube diameter of single toroidal damper; R — R{ di tube diameter of inner toroidal damper d tube diameter of outer toroidal damper d diameter of floating particle D mean diameter of toroidal damper or diameter of cylinderical damper Di mean diameter of inner toroidal damper D mean diameter of outer toroidal damper Djd slenderness ratio of toroidal damper E energy density; energy per unit volume of the damper fluid 0 0 p 0 xvii E rate of energy dissipation per unit volume; dE/dt FFT Fast Fourier Transform F external excitation force on structure F* dimensionless external excitation force on structure; F sloshing force transmitted to walls of a nutation damper F* 3 dimensionless sloshing force transmitted to walls of damper F 0 inertia force generated by system in Figure 2-7 e e 3 C^u^Ci F(x, y, t) functional dependence of <f> f forced excitation frequency, Hz // liquid sloshing frequency f structural natural frequency /„ vortex shedding frequency of structure G(z) dependence of the velocity potential <j> in the z direction g gravitational constant, 9.81 m/s H width of the wind tunnel model h liquid height in nutation damper I inertia of the rectangular damper without liquid II inertia of the nutation damper filled with liquid i, j indices used for nodal diameters and circles, respectively, for sloshing e n 2 in a toroidal damper; also i represents discretization index for integration domain J Jacobian of a transformation Ji Bessel function of the first kind k spring stiffness for a single degree of freedom system LVDT Linear Voltage Differential Transducer L total length of rectangular damper; 21 L length of wind tunnel model I' length of a section in compartmented rectangular damper m xviii I half the total length of rectangular damper l length of floating particle m sequence of a cycle during logarithmic decrement M total mass of oscillating system during experiment p M added mass Mi mass of liquid M equivalent mass of force balance during oscillation tests a 0 M{n) 1 - (xtanh/ci/3) /a 2 Mi [1 - (x;tanh/ci/3) ]To- rn,, mass of oscillating structure in wind tunnel N(ri) 2 o-x/3 2 Ni axiP n total number of elements used during integration p{n) u /2 Pi [( + u )/2] /2 p fluid static pressure p 2 2 2 Ui i+1 ambient pressure at the free surface s Q(U) Qi [(rii+i ~ 77i_i)/(2A)] r radial component in cylindrical coordinates 3? real part of the argument R radius of circular nutation damper Ri inner radius of toroidal damper Ril inner radius of the outer toroid in a double toroidal damper Ri2 inner radius of the inner toroid in a double toroidal damper R outer radius of toroidal damper 5 free surface contamination constant; 1 corresponds to fully 0 2 contaminated surface xix St Strouhal number; f H/U T mean kinetic energy per unit volume of damper fluid; \p/\(V<7)) t time u nondimensional wind speed, V/w H or fluid velocity; \/u + v + w u velocity vector; ui -f- vj -f- iok v 2 2 dz 2 2 n ^u 2 v, u, 2 2 velocity components in the x, y, and z directions, respectively w velocity components at the free surface in the x, y directions, respectively U , V U*, V*, s s + v +w w* nondimensional velocity components in x, y, z directions, respectively; u/c, v/c, w/c, respectively boundary-layer velocity in ai-direction u' velocity u at the i — th element in the integration domain V dimensional wind tunnel velocity Vi liquid volume in a damper w width of rectangular damper w' perturbation in the vertical velocity X horizontal excitation on nutation damper x, x*, y, coordinates in the cartesian system z z* y*, nondimensional coordinates in the cartesian system; x/l, y/l, z / h , respectively x + [a cosh K.(h + z)j sinh KK\ sin KX horizontal curvilinear coordinate; x' amplitude of oscillation at the m t h cycle displacement of the structure in numerical analysis amplitude of structural plunging oscillation during wind tunnel tests Y Bessel function of the second kind z' • vertical curvilinear coordinate; z + [a sinh K{JI + z)/sinh XX nh] cos KX Greek a total damping parameter due to sloshing liquid; (a& + a ) + a + ctf Qj damping parameter due to boundary-layer at the damper bottom af wave damping parameter due to boundary-layer at the surface ct p damping parameter due to the presence of floating particles on the free surface ct v damping parameter due to internal dissipation a damping parameter due to boundary-layers on the damper sides 8 7 h/l z + h/6 Ax width of finite difference mesh 6 boundary-layer thickness; e amplitude of the forced vibration oscillations e 0 initial peak amplitude of the free vibration oscillation ( 3 inherent damping of structure (wind tunnel model) s s e 77 v . free surface liquid elevation from quiescent state or damping ratio; c/c c 77* 77//1 rji free surface height at the i r} a reduced aerodynamic damping ratio; 7] i reduced liquid damping due to sloshing; eqs. (2.2), (2.3), (4.45b) 9 circumferential component in cylindrical coordinates K wave number Kij eigenvalue of the Bessel function for toroidal or circular damper; Tt T element of the damper domain th corresponds toz tfl AirijM/pH L circumferential and j 2 itl K* nondimensional wave number; KI KI resonant wave number; 7i7r/2 A wavelength; 2IT/K p dynamic viscosity xxi 2 transverse modes v kinematic viscosity; p,/p p liquid density p density of air p seeding density of particles at free surface cr wave speed; (tanh K,h)/K,h <Ti resonant wave speed; T* nondimensional time T nondimensionalized period; n/(step — size) T\ parameter used to nondimensionalize time; lj\Jgha\ $, 4> nonlinear and linear velocity potentials, respectively, satisfying a 3 a for K= K\\WKji Laplace's equation V $ = 0, V cri = 0 2 2 tp phase between external excitation, e , and sloshing force F X X e = 3 tanh K,(h + n)/ tanh KK stream function for two-dimensional flow ijj.. vorticity contribution to stream function at the free surface vorticity u) circular frequency for a wave, rad/s; c/c u) forced excitation frequency, rad/s e u> nondimensionalized forced excitation frequency; ui liquid sloshing frequency, rad/s e u /u)\ e co natural frequency of structure, rad/s ivij resonance frequency for i — th circumferential (inline) and j — th n transverse (perpendicular to inline) modes u nv ratio of vortex shedding to structural natural frequency; <jj vortex shedding frequency, rad/s; Ljy frequency associated with structural displacement, rad/s v xxii 2itf v f /fn v subscripts j particles with open ends blocked or boundary-layer c critical e forced excitation i,j , m integers for enumeration L lift force I liquid sloshing n natural structural p particle r radial component r l reduced liquid v vortex shedding 1 fundamental resonant condition superscripts * nondimensionalized quantity xxiii ACKNOWLEDGEMENT 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. G. Schajer, Dr. B. Ahlborn, Dr. S. Calisal and Dr. I. Gartshore. I would also like to acknowledge my parents, Mr. and Mrs. T. Seto, who have always been supportive of my studies. My husband, M. Lefranqois, and his family are deserving of recognition. Their support, patience and understanding made this entire endeavor all the more enriching. This research project was supported by the Natural Sciences and Engineering Research Council of Canada, Grant No. A-2181. xxiv J This thesis is dedicated to the memory of my grandfather, xxv Q. Seto (1898-1976). 1. I N T R O D U C T I O N 1.1 Preliminary Remarks A wide variety of civil engineering structures are susceptible to flow induced vibrations. From large aspect ratio lighthouses, industrial chimneys, cooling towers, and antenna masts to bluff bodies like buildings, bridges and iced transmission lines as well as aerodynamically shaped aircraft wings and control surfaces, are known to oscillate under the action of wind. These oscillations are of engineering concern because of their potentially destructive effects. Industrial chimneys have shown visible cracks due to excessive wind loading, lighthouse and building windows explode outwards, and iced transmission lines are known to have ripped from their towers. The most dramatic example is the collapse of the original Tacoma Narrows Bridge, in Washington State, U.S.A. The bridge was destroyed in 1940, after only a month of operation, through a torsional instability induced by a 67.2 km / h (42 mph) wind blowing for an hour. A combination of critical wind velocity and low structural damping produced catastrophically large oscillations. At times, human comfort, rather than structural integrity, determines the limits on the allowable amplitudes and oscillation frequencies. Tall buildings, like the World Trade Center in New York, can sway up to 30 cm in high winds at the top. These vibrations are usually at frequencies of less than 1 Hz and often not severe enough to compromise the structural integrity of the builidng. However, the occupants have experienced symptoms akin to sea-sickness, vertigo and disorientation. These problems have also been reported by occupants of air traffic control towers [1]. Vibrational accelerations must be less than 0.15g for human comfort. In the past, when safety margins were not clear, engineers overcompensated their calculations by factors of two, five or an order of magnitude to err on the side of safety. Advances in metallurgical sciences and computer aided designs have tended to reduce the safety margin and permitted construction of structures with lower stiffness. Builders prefer lighter materials for reduced cost (not always true) and 1 lower weight induced stresses. In addition, the tendancy today is to build higher and longer structures. The heights of the tallest buildings and the spans of the longest bridges in the world change routinely. The vibrational energies of such structures are difficult to dissipate due to the inherently lower damping. Lower safety margins, lighter materials, with the tendency toward taller buildings and longer bridges all conspire to create structures which are quite prone to vibrations. For a long time, the adverse influence of wind induced vibrations was not appreciated or considered minor. Several major failures and the current building trends have raised sufficient concern and focussed attention on the engineering relevance of flow induced instabilities. Even today, building codes in most countries are still at the evolutionary stage so far as the wind effects are concerned. 1.2 A Brief Review of the Relevant Literature A wind loaded structure can exhibit three basic types of motion: (i) vortex induced oscillations, where the formation of vortices behind a bluff structure causes periodic excitation at a frequency proportional to wind speed, and in the direction normal to that of the wind; (ii) random gust induced oscillations where the structure oscillates at one of its natural frequencies; (iii) self-excited oscillations where the force depends on the motion itself. Flutter and galloping type of instabilities belong to this class. Of interest here are the flow induced transverse vibrations of structures, i.e. vortex induced resonance and galloping. Vortex resonance occurs when the shedding frequency coincides with the natural frequency of the structure causing it to oscillate at large amplitudes. This type of vibrations is an important consideration in the design of a wide variety of structures including lighthouses, smokestacks, cooling towers, periscopes and antenna masts. The transverse oscillations of the structure have the effect of improving the spanwise correlation of the shed vortices. 2 This results in a better organized wake and an increase in the associated unsteady excitation force. The amplitude of motion is governed by two factors: (a) damping of the system (both structural and external); (b) phase difference between the motion and the vortex shedding cycle. Vortex resonance is a self-limiting mechanism. As the amplitude of cylinder vibration is increased beyond the critical value, the symmetric pattern of alternate vortices begins to break up [2]. As a result, oscillation amplitudes greater than one diameter are not observed frequently. This is not to say that these oscillations could not be potentially destructive. The original Tacoma Narrows Bridge was destroyed by vortex induced resonance in the torsional degree of freedom. As mentioned earlier,flutterbelongs to the class of self-excited oscillations. It results when at least two modes of oscillations, say plunging (a transverse oscillation of the structure in the cross-stream) and twisting, are excited simultaneously. Normally it occurs at a frequency between the natural frequencies of the pure plunging and pure twisting modes. This has been extensively investigated as the mechanism for the oscillation of ice-coated transmission lines [3] and suspension bridges [4]. The aerodynamic forces involved in flutter are usually higher than those for other forms of oscillations. This results in a very short build-up time and sudden onset of instability. Galloping is the term used for the self-excited motion in one degree of freedom. The motion leads to changes in the effective angle of attack and the associated force in such a way that the energy is extracted from the fluid stream. The vibration amplitude grows until the rate at which the energy is extracted balances the rate at which it is dissipated. If the dissipation rate is low, the amplitudes can become quite large. Depending on the damping, both vortex resonance and galloping vibrations may occur close to the structural natural frequency. Normally, vortex resonance occurs at lower reduced velocities while galloping manifests itself at higher values [5]. This can change with damping. 3 From practical engineering considerations, the main objective is to suppress or at least delay the onset of the above mentioned instabilities. Broadly speaking, the approaches adopted may be classified into two categories (Figure 1-1): (i) devices such as strakes, shrouds, slats, etc. which interfere with the flow and thus affect the aerodynamics in such a way as to minimize excitation; (ii) damping devices which provide a mechanism for dissipation of energy. Zdravkovich [6] has reviewed the literature pertaining to such add-on devices and their varying effectiveness both in wind and marine applications. They are all fairly successful in reducing the flow induced vibrations of an isolated structure caused by turbulence or vortex shedding. The concept of strakes originated with Scruton in the 1950's who also gave the optimal strake configuration in terms of pitch, protrusion and location [7]. Strakes were found to affect the boundary-layer around the chimney, making it more turbulent thus eliminating the regularity at which vortices are shed. The strakes disrupt the spanwise correlation and hence the excitation force on the structure. Other efficient devices comprise of shrouds and slats that affect theflowaround the structures, particularly in the entrainment region. Their optimal configurations are described by Wong and Cox [8]. Several other devices such as fairings, splitter plates and flags have been developed in order to stabilize the wake behind the structures and hence to reduce the effects of vortex shedding. However, they exhibit less flexibility with respect to the direction of the wind. The suppression of building [9] and bridge [10] oscillations have also been attempted through changes in the aerodynamic contours. Note that these devices often increase drag by a significant amount. Furthermore, there are conditions where the suppression of vortex resonance using aerodynamic means is not possible as in the case of structures with very low damping, or for rows of stacks as discussed by Ruscheweyh [11] and Zdravkovich [6]. The second approach is to increase the damping of the structures. This is a viable option with many structures. While changes in stiffness, mass and inherent 4 Helical Strakes Shrouds Slats Sand Or (a) Damper ~ m 1 i n Tuned Mass Damper Stockbridge Damper Hydraulic Dashpot (b) Figure 1-1 Various devices used to control vortex resonance and galloping types of instabilities. They provide (a) modification in excitation force; (b) the energy dissipation through damping. damping affect dynamical response of the structure [12], they are not amenable to modification with ease after construction is completed. This makes add-on dampers an attractive option. . The conventional hydraulic dashpot has been used.quite often on guyed structures [13]. The use of viscoelastic systems is another alternative discussed by. Gasparini, Curry and Debchaudhury [14]. Impact damping is a passive vibration control technique applicable to attenuate the vibrations of lightly damped, small weight structures. It comprises of. a solid slug placed in a slightly larger cavity attached to the main system. The damper is inactive until the vibration of the main system exceeds a pre-determined clearance between the two systems. Then, a collison occurs with the slug reversing the direction while the mass of the main system decelerates if the two are moving in opposite sense immediately before the contact. If this motion opposition is sustained before each repeated collision, then the impact process acts like an intermittent brake or damper on the main system [15]. The concept of hanging chain impact dampers as described by Reed [16] has proved to . be successful for lightmasts and antennas. The impact mass damper has also been considered for building application [17]. A tuned mass damper system permits the main structural element to stay somewhat unaffected by the exciting forces. It consists of an auxiliary mass attached to the main structure by a coupling which provides stiffness and damping. The mass, stiffness and damping are optimized to achieve minimum response of the main structure to a known excitation. As an example, the tuned mass damper of the 280m tall Citicorp Center building in New,. York City consists of a 410 ton concrete block, 10m on a side and free to slide on oil-film bearings, positioned By pneumatic springs and actively controlled hydraulic actuators [18]. The mass is 2% of the generalized building mass with a damping factor of £ = 0.04. Its efficiency in suppressing wind induced oscillations is also discussed by Wardlaw and Cooper [19], Hirsch [20], Tanaka,..[21] and more recently by Kawaguchi [22]. 6 Ueda [23] compares the relative effectiveness of three different types of tuned mass dampers used on tower-like structures. The concept of tuned mass dampers can be further generalized to include active or semi-active systems with feedback mechanisms to control inertia forces [24]. The formation of ice on transmission line cables changes their cross-sectional profile to that approaching a D-section which is susceptible to galloping under wind excitation [13]. The Stockbridge damper is a tuned mass damper which has been successfully employed to control this form of galloping instability [25]. The damper consists of two weights linked by a cable, the assembly being attached to the antinode point on the transmission line. Vibration control is achieved through internal damping in the cable strands as well as out-of-phase motion of the weights with respect to the transmission line. Extensive analytical and experimental studies have been reported on the effectiveness of this damper [26-30]. Similar to a tuned mass damper is the use of liquid motion within a container as a damper. Liquid sloshing has had limited success with improving the ship stability [31]. Bauer [32] analytically studied oscillating liquid in a rectangular container as a means of damping structural vibrations. Liquid motion has been used successfully to control the nutational motion of satellites. Typically, a near earth satellite has a librational period of around 90 minutes. At the other extreme, a communication satellite, orbiting at the height of 36, 000 km, completes one cycle of oscillation in 24 hours. These represent extremely long period motions. A small change of a few arcseconds in nutation at the geosynchronous orbit would cause the communication beam to sweep a large distance on the earth's surface. This creates alignment problems for satellites aiming their signals at ground based antennas. Torus shaped, ring-type nutation dampers (Figure 1-2) are frequently used to control such long period librational motions. Inspired by this, Modi et al. at the University of British Columbia were the first ones to apply nutation dampers to control vibrations of ground based 7 Figure 1-2 Parameters associated with a toroidal damper. 8 systems [33]. Most earth bound mechanical systems have natural frequencies in the range of hundreds of Hertz: the chattering motion of a machine tool; the shimmy oscillations of automobile wheels; the flutter of aircraft wings, etc. However, the frequencies encountered in wind induced oscillations of bluff bodies, earthquake response of buildings, and wave excited vibration of offshore structures are relatively low. For example, the fundamental natural frequencies of tall buildings, bridges, smokestacks, cooling towers, antenna masts, etc. are usually less than one Hz. Modi et al. initiated the experimental investigations to suppress wind induced oscillations with a liquid damper in the mid-seventies with the first paper appearing in 1980 [33]. Subsequently Modi and Welt [34-42] as well as Modi and Seto [43-45] have explored the subject, both analytically as well as experimentally, to gain a better appreciation of this simple, elegant mechanism for energy dissipation and its effectiveness in arresting wind induced instabilities. For the purpose of suppressing flow induced vibrations, the nutation damper has several advantages over other dynamic vibration absorbers or tuned mass damper blocks: up to five-fold increase in damping efficiency, insensitivity to wind direction, economical, simple, and easy to maintain. Welt explored the parametric optimization of damper geometry, liquid height, excitation amplitude, etc. for maximum damping [34,35]. Briefly, he found the damping to increase with a decrease in the Reynolds number, frequency, and diameter ratio, d/D; and remain essentially insensitive to the cross-sectional geometry of the ring. Internal devices such as screens, baffles, inner tubes, etc., which inhibited the motion of the free surface, adversely affected the energy dissipation process (Figure 1-3). The breaking of free surface waves was found to be one of the principal means of dissipating energy. Viscous forces were observed to be more efficient than inertial ones for damping. Preliminary experiments with floating wood particles appeared promising [35]. Extensive wind tunnel tests in laminar and turbulent flows with two and three dimensional models showed nutation dampers to be quite 9 Figure 1-3 Internal modifications of the nutation damper studied by Welt [34] to optimize damping. 10 effective in arresting both vortex resonance and galloping type of instabilities [3739]. Welt and Modi also developed a potential flow model with viscous corrections to describe liquid response to harmonic excitation. It gave an expression for energy dissipation per unit mass of the liquid per cycle, as affected by the system parameters [40]. Validity of the analysis was assessed through a planned experimental program [41]. More recently, applications of the nutation damper to real-life prototype structures have been reported. The nutation dampers have found applications at several places around the world for the purpose of controlling flow induced vibratons. A few examples include: Narita and Haneda Airport Control Towers in Japan [46]; Nagasaki Airport Tower [47]; Gold Tower at Seto-Ohashi Bridge, the longest suspension bridge in the world at the time; Shin Yokohama Prince Hotel, the tallest residential building in Japan [48]; Yokohama Marine Tower [47], the tallest lighthouse in the world; a communications tower on Mt. Wellington in Tasmania, Australia [49]; and Centerpoint Tower in Sidney, Australia. Its use is considered in marine environments on risers as well [50]. There is also potential for its application in controlling the galloping instability of ice coated transmission lines [51]. The nutation damper is referred to variously by other researchers as a Tuned Sloshing Damper (TSD), Tuned Liquid Damper (TLD), and Tuned Sloshing Liquid Damper (TSLD). The nutation damper achieves its objective of controlling vibrations through two main mechanisms. The phase of the sloshing fluid relative to the structure motion allows it to work like a dynamic countermass. However, when the damper is tuned, the dynamical energy of the structure is efficiently transferred to the liquid as in the case of a system at resonance. This energy in the liquid is dissipated through viscous and turbulent interactions, and most importantly by wave breaking. The need for an analytical model to describe the mechanism of transfer of energy to the damper and its dissipation through sloshing is also recognized. Sloshing 11 process is represented mathematically as an initial-boundary value problem including a moving boundary called the 'free surface'. This is a difficult problem to solve because the position of the free surface varies with time in a manner that is not known a priori, and the boundary conditions at the free surface are represented by nonlinear equations. Hence, most studies are based on the linear theory where the motion of the liquid as well as the container are assumed small. The linearized theory of small free surface oscillations is well established. However, in reality, nonlinearities in the amplitude-frequency response of liquids in containers of various geometries have long been noted [40, 52]. Moreover, the liquid sloshing, resulting from external forces, is shown experimentally to be critical when the excitation frequency is near a natural frequency of liquid oscillation [41]. For such a case of resonance, the linearized theory fails to predict the liquid motion. The present focus is on resonant sloshing in order to take advantage of the optimum damper performance. Hence, attention is focussed upon investigations of finite amplitude liquid motion which include dispersion effects. Obviously, dissipation will also have to be modelled to fully describe the nutation damper. The model incorporates the damper fluid dynamics with those of the structure's. It is useful to study the time resolved response of the nutation damper to excitation. In applications of the damper it is necessary to know how long it takes for steady state conditions to be established. Liquid sloshing problems are of two varieties. The first involves solution of the eigenvalue problem to obtain natural sloshing frequencies, for a given tank geometry, through linearization of the free surface boundary conditions. Moiseyev [53] and others obtained natural frequencies for several tank geometries in this manner. In the second category, the objective is to study the tank response to forced harmonic displacement. Experimental studies include those of Abramson [52] and others. Moiseyev [53] presents a generalized theory using the perturbation method in which one of the free surface boundary conditions is linearized. The 12 problem appears to have relevance to a wide variety of disciplines as apparent from contributing sources. The early engineering interests were in the seismic excitation of dams [54] and liquid storage containers [55], which still remain a problem of interest [56]. Sloshing liquid leading to the harbour resonance phenomena of seiching or surging is of concern to civil engineers. If the wave frequency within a harbour approaches the natural frequency of the mooring system, a large amplitude wave motion can be established in the harbour causing damage to ships, docks and mooring [57-59]. As mentioned earlier, sloshing liquid has also been considered for ship stability [31]. Studies of the natural sloshing frequencies of incompressible fluids have found important applications in the design of space flight vehicles [60-64]. The contents of a rocket fuel tank can sometimes amount to 90 percent of the gross takeoff weight of the vehicle hence fuel sloshing is an important factor in its dynamics. Of course, the fuel sloshing problem is of interest to aircraft engineers as well [65]. The efforts in the aerospace vehicles technology consisted of theoretical calculations for the natural modes and frequencies at small amplitude sloshing of liquids in partially filled fuel tanks of various shapes. The procedure involved determination of the analytical solution of a potential function for a linearized free surface condition under harmonic excitation. The variational principle was used to compute the fundamental frequency of oscillations of liquid in a tank [64]. With spacecraft, there is the added complexity of the microgravity environment [66] and the problem is receiving considerable attention at present [67]. This is partly due to the use of the Shuttle as a space transportation system. There is a trend to increase the liquid fuel phase of a mission [68] thus requiring even larger fuel tanks and hence aggravating the fuel sloshing problem. Problems associated with ground based tankers are also receiving attention due to sloshing loads having adverse effects on vehicles during maneuvors [69,70]. The objective has been to understand the liquid sloshing phenomenon well enough 13 to minimize it because of its potential destabilizing effects. One proposed solution is to install baffles in the tank [69-71]. This tends to reduce the free surface motion as confirmed by Welt [34]. Other preventive methods include modifications of the tank geometry, choice of liquid height, etc. so that resonant sloshing conditions are not easily established. The other notable area where the liquid sloshing plays an important role is the nutation dampers mentioned earlier. However, the objective here is fundamentally different. Now the resonant condition is sought to promote sloshing to increase energy dissipation and thus control the structural instabilities. In the earlier dampers, sloshing was suppressed by having little or no free surface. The present nutation dampers have free surfaces [72] and operate in tandem in a satellite [73]. Attempts have been made to quantitatively study the energy dissipation characteristics [74,75] in such dampers. However, the fluid sloshing in these cases does not occur at very high amplitudes and the flow is modelled adequately as a slug [72-74] in conjunction with a linear analysis. The procedure is rather simple although it seems to provide reasonable answers. The sloshing liquid is replaced with an equivalent mechanical model of lumped fluid masses, springs and dashpots [76]. The mechanical characteristics of the equivalent system are established by following the dynamic similitude of the sloshing fluid through the potential flow theory [77]. The energy transferred to the fluid is dissipated by the viscous action of the fluid. The effect of nonlinearities on the resonant sloshing of water in horizontally oscillated containers started with Moiseyev [53], who studied situations where the water is fairly deep. Chester [78], Ockendon and Ockendon [79], Cox and Mortell [80], and Miles [81] have suggested that the periodic response near resonance is controlled by a second-order, non-autonomous, ordinary differential equation. Ockendon et al. [82] arrive at a similar ordinary differential equation, which represents waves on shallow water near resonance, but it does not allow for dispersion. 14 Verhagen et al. [83] conducted theoretical as well as experimental studies of the steady-state, finite amplitude forced oscillations of a fluid, with shallow height, in a rectangular container. The effects of frequency dispersion and dissipation were neglected in the analysis resulting in only limited agreement with experimental results. Chester [78], in his derivation of a steady state solution for the waves induced in a closed basin due to horizontal excitation, included frequency dispersion and dissipation. He observed improved agreement with the results of his experiment, which were performed near resonance. Miles [84-86] has studied the slightly nonlinear and weakly damped response of the free surface of shallow liquid in rectangular and circular containers subject to resonant horizontal oscillations. The Lagrangian and Hamiltonian for these flows were constructed in terms of the generalized coordinates of the free surface displacement. The kinematical boundary-value problem was solved through a variational method. Miles' results qualitatively support Chester's formulation. Of course, the finite amplitude waves in a container can be described by their full Navier-Stokes equations. Among the procedures used to solve the governing equations are the Finite Difference Methods (FDM), Finite Element Methods (FEM) and Boundary Element Methods (BEM). The first two approaches are computationally intensive. They also present considerable difficulty in extending the fluid sloshing problem to incorporate a structure to assess fluid-structure interactions. However, for the general problem of an arbitrary shaped container with a given liquid height, all the three methods have proved to be fairly successful [87, 88-91]. They are particularly suited to the cases of storage tanks and dams where the liquid height is often variable and usually deep. The boundary element based formulations for sloshing are more amenable to incorporation of the fluid-structure interaction dynamics. Wakahara et al. [48] have successfully used the boundary element method to study thefluid-structureinteraction dynamics involving a rectangular container and a multidegree of freedom structure. 15 Pressure and velocity are used as the dependent variables in the MarkerAnd-Cell (MAC) method developed by Harlow et al. in the 1960's at Los Alamos Scientific Laboratory [92]. This method was the first to successfully treat problems involving complicated free surface motions.- The MAC method uses an explicit Eulerianfinite-differencescheme to solve the time dependent Navier-Stokes equation for a viscous incompressible fluid. Special features of this approach include the use of staggered grids, conservative form of the equations of motion, and the use of 'massless' marker particles to denote the location of the liquid. Over the years several refinements have been introduced. In the mid 1970's, Hirt et al. [93] developed a simplified algorithm (SOLA - numerical SOLution Algorithm for transient fluid flows) based on the MAC method. It allowed the free surface to be a single valued function of the liquid height. The method has proved to be quite successful in defining free surface flows where the surface slope is less than the cell aspect ratio. The method can be used for the numerical simulation when the sloshing is small. However, in real life, the wave surface can have slopes approaching infinity and there may be wavebreaking, which is not accurately modelled by the method. The approach which combines the SOLA algorithm with a versatile fluid surface tracking algorithm was pioneered by Hirt and Nichols [94]. It employs the Volume Of Fluid (VOF) technique. The method can tackle steep and highly contoured free surfaces. The VOF function F is assigned, arbitrarily, a value of unity at a point occupied by the the fluid and zero elsewhere. The average value of this function in a grid cell then represents the fractional volume of the cell occupied by the fluid. Thus, unity value indicates that the cell is full of fluid and zero suggests an empty cell. Cells between zero and one must then Include the interface. Thus only one variable per cell is stored as opposed to several fluid flow parameters in other procedures. The F distribution used in the VOF method has the desirable property of tracking an interface. Surface locations, slopes and curvatures are easily computed for given boundary conditions and the advancing in time of F is achieved 16 by advection through the Eulerian grid. The SOLA-VOF method is currently quite popular in studying systems under seismic excitation. Lepelletier and Raichlen [95] have studied the fluid motion in rectangular tanks using a nonlinear, dissipative, dispersive model with application aimed at lakes and harbours. They obtain the Boussinesq equations through a variational analysis and solve them with the standard FEM technique. The results show good agreement with the experimental data. Hayama et al. [96] obtained an analytical solution to the motion of an inviscid liquid in a rectangular tank subjected to forced horizontal oscillations. They used a perturbation method which retained terms of up to the third degree. The correlation of results was found to be good for deep water cases, however the discrepancy increased at shallow depths. Minowa [56] as well obtained the free surface response of a cylindrical liquid storage tank to random seismic excitation using a perturbation method and found reasonable agreement with experiments. Shimizu et al. [97] and Sun et al. [98] have solved coupled, nonlinear, nonautonomous shallow water equations for an inviscid liquid in a rectangular tank under harmonic excitation. They report good agreement with experimental results at or near resonance. Sun et al. also coupled an oscillating structure with one degree of freedom to their damper. The wavebreaking action in the damper was accounted through empirical constants. The sloshing liquid motion within the damper was posed as a shallow water wave problem with the Navier-Stokes and continuity equations averaged over the depth. The resulting system of equations were solved for the displacement of the water surface and horizontal components of the depth-averaged velocities. From the experimental work to date, it is clear that the nutation damper works well when the liquid is shallow. This also corresponds to the lower frequencies of interest. A model based on shallow water flows allows one to draw on established techniques developed for that class of problems [99, 100]. The assumption of shallow 17 depth also reduces the need to use computationally intensive finite element and finite difference methods. A nonlinear shallow water analysis can capture the physics of the mechanisms involved and optimize the computational effort. Hartlen-Currie [101] and Blevins-Iwan [102] have presented analytical models that simulate the vortex resonance response of two-dimensional structures. The Hartlen-Currie Lift-Oscillator Model employs the flow field properties determined from existing experimental data. The lift coefficient satisfies a van der Pol type equation. The oscillator is self-excited, its natural frequency satisfies the Strouhal relation, and its response is influenced through a simple linear coupling term in the force expression. The model has shown some success in predicting the motion of elastically mounted circular cylinders. The Blevins-Iwan model is developed by introducing a 'hidden flow variable' to describe the effects of vortex shedding. This allows the model variables to be interpreted directly in terms of physical parameters. Model parameters are determined by fitting experimental results for stationary t and forced cylinders. The model is then capable of predicting the response of an elastically mounted cylinder. Refinements to both of these basic models have been made and both are still widely used today. When two structures are close enough to interfere fluid dynamically, the observed effect is a phenomenon referred to as proximity induced galloping. Large amplitude oscillations are possible for the downstream object [103, 104]. The twin strut problem in early aircraft also belongs to the same class. Consider, for example, two equal, similar cylinders, one in the wake of the other. The upstream object sheds vortices which affect the fluid dynamics of the downstream structure. If the vortices are shed at a structural frequency of the downstream object, the resonance condition is established. Note, the downstream structure is excited by the periodic forcing due to its own vortices as well as that due to the upstream structure. The oscillation amplitudes observed are, of course, much larger than those for the usual vortex resonance, hence the incorrect reference to 'galloping'. 18 Studies with circular cylinders of different diameters have also been reported. They exploit flow interference effects by using a smaller downstream cylinder to control the wake of the larger upstream cylinder [105]. Several other researchers [106-110] have also studied the problem of vortex resonance and galloping caused by the presence of nearby structures. Walshe and Cowdry [106] found that for. two identical circular cylinders, perforated shrouds are ineffective when attached to both the cylinders and not signifigantly different when fitted to one of them. Vickery and Watkins [107] found helical strakes also to be ineffective when attached to four in-line circular cylinders. The study by Wong [108] showed the axial slat shroud to be ineffective when the spacing between the two cylinders in tandem was more than six diameters. Zdravkovich [110] concluded that, in some cases, it is better not to use the vibration suppression device as it may have detrimental effects. It appears that flow interference can severely undermine the usefulness of vortex suppression devices normally effective on a single bluff body. - Most of the reported flow interference studies have been performed on circular cylinders. There is also interest in the corresponding information for square cylinders because, of their relevance to bridge and building aerodynamics [111-117]. The above mentioned add-on devices attempt to alter the aerodyanmics associated with a bluff geometry in a favourable way. On the other hand, a damper (coulomb, viscous, structural, etc.) provides a mechanism for energy dissipation [17-23]. The nutation damper has never been applied to interfering structures as a means to control flow interference effects. 1.3 Scope of the Present Investigation The study here builds on the work of previous researchers. It provides a comprehensive understanding of the dissipation mechanisms and contributes to the next generation of nutation dampers which are certain to find increasing applications. The optimal efficiency of nutation dampers for controlling flow induced vibrations is sought through a combination of experimental and theoretical procedures. 19 A synthesis of parametric experiments to identify optimal conditions, development of a model for the energy dissipation in the damper and a fluid-structure interaction process, wind tunnel tests and flow visualization are used to obtain indepth appreciation of the process. The parametric experiments explore the effects of geometry, free surface particle seeding, compartmentalizing, oscillation amplitude, liquid height, etc. on energy dissipation in the damper. Performance of the circular cylindrical, square toroidal and rectangular geometry dampers are compared keeping mass and sloshing frequencies constant. The free surface of the damper is seeded with particles to a varying degree. Both the aspect ratio of the particles as well as their buoyancy were changed systematically. Emphasis was on achieving the condition for maximum damping by operating at the natural frequency of the first antisymmetric sloshing mode during free and forced oscillation tests. The compartmented container studies are aimed at application of the damper to bridge decks. In the past, baffles were placed in tanks to minimize sloshing loads. Here, partitions are used to promote resonant sloshing to increase damping. The objective of the numerical model is to predict wind induced excitation of continuous structures, such as smokestacks, tall buildings, cooling towers, etc. with nutation dampers installed. To that end it is necessary to represent energy dissipation of the nutation damper taking into account nonlinear sloshing behavior manifested by finite amplitude surface waves, dispersion and dissipation in the resonant regime. The shallow water approximation is used to develop a model of the energy dissipation in the nutation damper. The sloshing in a container is posed as a shallow water wave problem where the Navier-Stokes and continuity equations are integrated over the depth. The resulting system is described by the free surface geometry and the horizontal components of the depth-averaged velocity. The numerical results are also animated to provide better visual appreciation of the wave 20 dynamics. The model is further developed to study the conjugate problem of fluidstructure interaction dynamics in plunging. A finite difference scheme is used to solve the nonlinear, nonautoriomous, coupled equations. Effectiveness of the nutation dampers in controlling vortex resonance and galloping types of wind induced instabilities is assessed through an extensive wind tunnel test-program. The structural models were free to undergo two-dimensional plunging oscillations. Effectiveness of the nutation damper in controlling vortex resonance and galloping during wake interference was also explored. The entire test-program was conducted in a closed circuit laminar flow wind tunnel. Finally, a flow visualization study was undertaken to identify the sloshing modes and thus to provide some physical insight into the damping mechanisms involved. It complemented the numerical analysis as well as the wind tunnel tests and assisted in their planning. Figure 1-4 schematically shows the plan of the study. 21 An Investigation on the Suppression of Flow Induced Vibrations of Bluff Bodies Wind Tunnel Tests on Structural Response Parametric Experimental Studies with Damper to to geometry floating particles partitions Two-dimensional isolated structure Visualization Wake Effects Flow within Damper Structural Response Numerical Analysis Animation of Results Damper Dynamics Figure 1-4 Scope of the investigation. Structural Response with Nutation Damper 2. P A R A M E T R I C S T U D Y O F N U T A T I O N D A M P E R S 2.1 Preliminary Remarks The nutation damper is a dynamic vibration absorber more accurately described as a tuned sloshing liquid damper. Tuning consists of choosing the right geometry and liquid height so the sloshing resonance frequency coincides with that of the primary structure. Once the damper is tuned, the energy transfer from the primary to the damper mass takes place efficiently. The problem reduces to dissipating the energy transferred to the damper. In an idealized tuned mass damper system (Figure 2-la), this energy is dissipated through viscosity. In the nutation damper (Figure 2-lb), the energy dissipation mechanisms include the impact force, wave breaking, and viscosity. The impact force refers to the impulsive liquid impact on the damper walls. Energy is dissipated through the inelastic collision. The duration of the impact is typically in the range of milliseconds. Wavebreaking refers to the nonlinear phenomena in a dispersive media where constituent waves with different speeds catch up and cause them to break or spill over. The process results in energy dissipation. Viscosity is the result of shearing action between adjacent liquid layers. It is pronounced when the velocity gradients are large. This is the case in the fluid near a boundary, giving rise to the boundary-layer, where the effects of viscosity are localized. A horizontally excited liquid filled damper produces waves on its free surface. The character of these waves depends on the forcing amplitude, profile, frequency and phase. The free surface waves of interest are both progressive and standing waves. In the case of progressive waves (Figure 2-2a), the fluid particle trajectories are ellipses which decay exponentially with depth. The horizontal distance travelled by the fluid particles is the same. The vertical distance varies linearly from zero at the bottom to almost the quiescent liquid height at the surface. Hence, most of the liquid motion or 'sloshing' occurs at the surface. The trajectories of the 23 (a) (b) Nutation Damper \ Interface between the Damper and Primary Body Figure 2-1 Schematic representation of a tuned mass damper: (a) idealized configuration; (b) the nutation damper modelled as a tuned mass damper. F is the excitation force on the primary body. e 24 (a) Lower Envelope ! Linear Decrease in Particle Motion' with Depth f Figure 2-2 The fluid particle trajectories in two types of waves observed in a sloshing fluid: (a) progressive wave; (b) standing wave. 25 fluid particles in standing waves are vertical lines with no travel in the horizontal direction (Figure 2-2b). A standing wave does not dissipate energy as well as a progressive wave. Both types of sloshing waves transmit a base shear force to the damper and its supporting structure. This is the damping force. Obviously not all fluid activities manifest in an increase in the base shear force. This chapter starts with a description of the test facilities and methodologies used in the parametric study of the nutation dampers. An investigation of three basic damper geometries and a comparison of their relative merits follows. The effects of damper partitioning and floating particles is also included. The chapter concludes with remarks on the dissipative effectiveness of the damper configurations studied. 2.2 Test Facilities, Instrumentation and Methodology The objective of the parametric study is to identify combinations of the system parameters which may lead to optimum dissipation of energy. Of course, effectiveness of the identified damper configurations will have to be substantiated through wind tunnel tests, with dampers installed on models of elastic structures susceptible to vortex resonance and galloping type of instabilities. This aspect is studied later in Chapter 3. In the parametric study, damper dissipation characteristics are assessed using two types of tests: free vibration and forced vibration. With free vibration tests, the system is set oscillating with an initial deflection and the response is measured over time. It assumes dominance of the fundamental mode. Fortunately, this is the case for the systems studied. The reduced damping, rj i, r> is determined through calculation of the logarithmic decrement. Forced vibration tests use a harmonic exciter with variable amplitude and frequency. The system is excited at the desired amplitude and frequency. The main advantage is the ability to measure the damper performance at the fundamental as well as higher harmonics more readily. The 26 free vibration test can provide similar information but would require a systematic variation of stiffness which would involve some effort, at least in the present set-up. 2.2.1 Free vibration tests Free vibration tests determine the reduced liquid damping ratio n j of a r damper mounted on a system of total mass M and stiffness k. Two such facilities are used. The first is for studying the application of nutation dampers to bridge decks. In such an application the deck interior, filled with water, becomes the damper. This facilitiy is a spring loaded box free to oscillate in the vertical plane about a bearing at one end and supported by a vertical spring at the other (Figure 2-3a). When deflected from rest the plunging motion simulates a bridge section near an end. This facility is rather versatile. It can be modified as desired by replacing the bearing with a universal joint, mounted underneath the damper's center, to simulate all the three rotational degrees of freedom (Figures 2-3b, 2-3c). When deflected from rest, the motion simulates the vibration modes in pitch roll and yaw of a bridge deck. A different free vibration facility is used to test damper configurations undergoing nutational motions. With this facility, the damper is bolted to a horizontal platform at the end of a vertical bar (Figure 2-4). The bar rotates about a bearing and is spring loaded to oscillate in a vertical plane. An initial displacement is imparted to the system and the resulting damped oscillations are measured. In both the free vibration facilities, the motion is sensed through a strain gauge based transducer integrated into the spring mounting as shown in Figure 2-4. The differentially mounted strain gauges on the transducer form half of the Wheatstone bridge in a Bridge Amplifier Meter (BAM, Ellis Associates). The analog displacement output from the BAM goes to an A / D converter where the signal is recorded for subsequent analysis of the logarithmic decrement. 27 W (c) K i K / - \/\N\r L^yJ V, / J I? Figure 2-3 The bridge deck free vibration experimental setup: (a) plunging oscillation mode; (b) torsional oscillation mode (roll); (c) torsional oscillation in transverse direction (pitch). 28 NUTATION DAMPER c 2 RECORDER OR OSCILLOSCOPE BAM SPRING A STRAIN GAUGE . MOVEABLE COLLAR ADJUSTABLE TO CONTROL SUPPORT NATURAL A N D HEIGHT FREQUENCY BEARING //////////////'/ Figure 2-4 A free vibration test facility for studying efficiency of nutation dampers. 29 The displacements in free vibration are essentially decaying sinusoids. The logarithmic decrement, a measure of damper dissipation, is determined from _ ln(ai/a ) J; ~ -J, m (2-1) Vr,l - zirm I normalized for the relative inertia of the fluid. Here x for the mth cycle and m is the system amplitude is the fluid to total inertia ratio for the pivoted system (equivalent to M\jM for translation). Equation (2.1) is valid for discrete values of amplitude corresponding to integral values of m. In the limit as m tends to 0, . . n i(x)= T hm— m->0 \a.x\/x Ii - , m 27T771 I — 1 dx/dmli ~ 2TT 7' x where x is the amplitude function x{m) = x . A polynomial fit for the amplitude m decay envelope in the form x (m) = A + Aim + A m 2 0 2 + is now applied to facilitate the analysis on the digitized displacement as shown in Figure 2-5. This finally yields the logarithmic decrement as 1 [Ai + 2mA + ... + p m A A + Aim + ... + A m 2TT ( y _ 1 ) 2 p 0 p 2.2.2 Forced excitation tests Forced vibration tests determine responses of the damper over a range of frequencies. To this end, another facility, originally built by Modi et al., was modified (Figure 2-6). Essentially, the facility is a Scotch-Yoke mechanism connected to a 30 start zero mean of oscillatory part (a) determine offset (b) digitized free vibration data record (c) time window determine maximum in a given time window At At select At neighbors before and after max mean fit polynomial through the 2At+1 values offset from mean choose extremum from polynomial fit and record examine next time window N fit polynomial through all extrema values calculate rj Figure 2-5 Free vibration data analysis: (a) flow chart illustrating procedure for determining r, j ; (b) digitized free vibration decay data; (c) typical time window used in the analysis. T Spectrum Analyser Oscilloscope ZJT LZ Filter Damper BAM BAM i r _l L Strain Gauge CO to A- IV / / 1/ —/ / - / i s •Strain Gauge Moving Frame Scotch-Yoke Mechanism I t<ij Bearings .3" DC. Drive Figure 2-6 A schematic diagram of the forced vibration test set-up to obtain reduced liquid damping, r) i, and added mass, M , ratios. Ti a horizontal platform. A force balance is rigidly fixed to the platform and a damper is mounted on the force balance. The Scotch-Yoke steadily forces the platform to oscillate at a desired amplitude and frequency. The time history of the displacement of the platform is recorded so the phase between excitation and sloshing forces can be determined. A new motor was installed specifically for high torque and low frequency performance (1/4 hp, maximum 2 cycles per second, Bodine) in order to assess the damper behaviour in the range of practical importance. A force balance (Figure 2-7) was designed and constructed to test a wide variety of nutation dampers. It consisted of two galvanized steel plates supporting the horizontal stainless steel platform which, in turn carries the damper. Strain gauges, mounted differentially on both sides of each vertical plate supporting the platform, sense the sloshing force. Care was taken to calibrate the balance every time it was used in the event the plates were not exactly vertical. The sloshing force, F , transmitted from the liquid to the damper walls in the direction of excitation 3 cause the supporting platform to deflect proportional to its magnitude (provided the system is elastic and operates far from its resonant state; the lowest resonance frequency is 12 Hz). An additional force F generated by the system's own inertia Q is proportional to the amplitude and frequency square of the excitation, F 0 = M W e 6 COSWet, 0 e where M is the equivalent mass of the support. F contributes to the overall strain Q D recorded by the sensor. Appendix 1.1 shows the static calibration plot, with the system at rest undergoing bending under a known stress. In addition, a dynamic calibration procedure, involving measurement of the output voltage after loading the support with a dead weight, under various conditions of amplitude and frequency, was adopted to estimate M as well as the slope of the response curve, (Appendix 0 1.1). 33 nutation damper -« £ • e cos c o t e Figure 2-7 The force balance used to measure the reduced liquid damping of nutation damper models under steady state excitation. 34 The oscillating platform displacement is sensed with a Linear Voltage Differential Transducer (LVDT). The transducer consists of two concentric cylinders, the inner one of steel and outer one made of ferromagnetic materials. Relative motion between the cylinders results in generation of the proportional voltage. Appendix 1.2 presents a typical calibration plot for the LVDT for measurement of the time resolved displacement of the oscillating platform and oscillating wind tunnel model. Experimentally, the condition of sloshing resonance is established when the force F lags the displacement e by 90°. A correlator (Hewlett Packard, 3721A) s e was used to measure the relative phase between the signals in real time. The two signals, one from the force balance measuring the sloshing force F 3 and the other from the platform displacement measuring the excitation e , were e recorded with no analog filtering. The quantities determined from these signals are: F, e, u a e e (excitation frequency), and (p, the relative phase between the sloshing force and the platform displacement. These parameters identify, the nondimensional sloshing force or added mass, M a -- v = T% a l \ F s — , and reduced damping r\ \ : M jM\ o—1 F, cos ip; ri i = —— r . ^smtp. (2.3) Figure 2-8 shows typical traces of the excitation, e , and sloshing force, F . e s 2.2.3 Damper models The Plexiglas nutation damper models used in parametric experiments are described in Table 2-1 and schematically shown in Figure 2-9. The measure of slenderness for a toroidal damper is the ratio of its mean diameter (D) to the tube diameter (d) as defined in Figure 1-2. Therefore, slender toroids have higher D/d. Occasionally, the inner to outer radii ratio is used as a measure of slenderness as well. Toroidal dampers with Ri/R 0 > 0.5 are considered slender. 35 CO time Figure 2-8 Example of data traces obtained during the steady state excitation study: (a) displacement of the platform, e ; (b) sloshing force, F . e a Table 2-1 Nutation dampers models used in the parametric study TOROIDAL DAMPER MODELS Model Outer Dia. D (cm) Inner Dia. Di (cm) 30.0 30.0 26.4 20.0 19.4 10.0 26.4 20.0 22.2 12.0 12.4 7.2 0 1 2 3 4 5 6* Slenderness Ratio, D/d 28.2 22.2 24.3 16.0 15.9 8.6 / / / / / / 1.8 5.0 2.1 4.0 3.5 1.4 = = = = = = 15.8 4.44 11.6 4.00 4.54 6.14 CIRCULAR DAMPER MODELS Model Diameter, D (cm) 7 8 9* 30.0 20.0 10.0 RECTANGULAR DAMPER MODELS Model 10 11 12 13 14* Length, L (cm) Width, W (cm) Aspect Ratio, L/W 91.4 37.0 25.5 27.0 10.0 10.2 15.2 15.3 27.0 10.0 0.6, 1.2, 3 and 6 1.2, 2.4 1.7 1.0 1.0 * refers to the dampers used during wind tunnel tests The bridge deck model shown in Figure 2-3 is Model 10 in Table 2-1. This model is a long rectangular damper of dimensions 0.9m x 0.102m x 0.152m. The damper is provided with partitions so that it can be subdivided into smaller rectangular compartments. The partitions are removeable and can be placed at different intervals along the damper length. This damper is used for the free vibration bridge deck study in plunging and torsional modes. Several other rectangular geometries 38 ) were used during the forced excitation study. For example, Model 12 is just a Plexiglas box. Model 11 is a rectangular box with a single removeable partition. It has the same dimensions as the optimally compartmented Model 10 in the frequency range of interest. However, now it is used to assess the forced vibration performance whereas Model 10 is for free vibration tests. A 30 cm diameter circular tank was used to construct, in modular fashion, a family of nutation dampers. Plexiglas rings of different diameters were used to create either smaller circular cylinder dampers (Model 7, 8) or toroidal dampers (Models 1-4). This method of modular construction was deliberate in order to obtain a family of similar dampers. The Plexiglas rings were held in place with floral clay. Model 5 is the largest D/d damper used by Welt [37]. Smaller nutation damper models were tested as well. The wind induced instability studies in the wind tunnel involve mounting a nutation damper on an elastically supported structural model. As structural models are not full scale, the dampers need to be proportioned appropriately. The smaller nutation dampers for wind tunnel tests are identified with an asterisk in Table 2-1 (Models 6, 9 and 14). Several additional dampers were used during the wind tunnel tests as discussed later in Chapter 3. 2.2.4 Data acquisition and analysis methodology An electronic strip chart recorder was used to collect the data. The data can be free vibration amplitudes, forcing amplitudes, sloshing forces, structural model displacements, etc. This strip chart recorder is assembled from a 12 bit, 8 doubleended channels, analog to digital converter (Data Translation 2800, Mass., U.S.A.) interfaced to an IBM Personal Computer. The frequencies of interest are < 1 Hz. An effort was made to capture a large number of cycles of oscillation to obtain a clean spectrum. Generally, ninety cycles of oscillations or about 1.5 minutes of sampled data was collected for each run. The data sampling frequency is an order of magnitude higher than that necessary to minimize error. The results were 39 acquired at a rate of 100 samples per second per channel, processed and stored on the computer hard drive as an ASCII file. Programs written in DOS and MATLAB performed the demultiplexing, processing and filtering of the digitized signals. The processing of free vibration displacement data included identification of the peaks of the damped sinusoid and fitting them with a third order polynomial to calculate the reduced damping (Appendix II). With forced vibration tests, the data processing included determination of the phase between the forced oscillation and the sloshing force, as well as the amplitude of the forced oscillation in order to calculate reduced damping and added mass ratios. Spectral analysis of the digitized signals determined excitation frequencies, sloshing force amplitudes, etc. (Appendix II, Program getfreq.m). A block diagram of the data acquisition system is shown in Figure 2-10. Generally, the parametric tests were performed systematically changing damper geometry as well as variables associated with the liquid (height, Reynolds number) and excitation (frequency, amplitude). 2.2.5 Flow visualization In order to get better physical 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 and floating particle parameters was carried out. The forced vibration facility was used. During a test, a damper was mounted on the Scotch-Yoke mechanism (Figure 2-6) and the excitation frequency varied. The fundamental as well as subharmonics and higher harmonics were observed and recorded using still camera and S-VHS video. In particular, the video showed quite vividly evolution of the progressive waves, their collision amongst themselves and with the wall, disintegration and regeneration. . The presence of floating particles and associated parameters clearly revealed their influence on the above process and thus helped understand the conditions leading to higher dissipation. The effectiveness of the dampers during structural dynamics tests in the wind tunnel, with vortex resonance and galloping, was also recorded. 40 switch to trigger data collection 1/4 HP, low speed Bodine motor for Scotch-Yoke LVDT excitation displacement correlator, output JxF (t-x)dx s bridge amplifier meter slosh force transducer analog to digital converter spectrum analyzer Figure 2-10 Data acquisition system for forced vibration test facility. IBM 386 computer 2.3 Damper Geometry and Dissipation 2.3.1 Preliminary remarks A practicing engineer faced with the design of a structure safe against wind induced instabilities primarily turns to meteorological data, building code (if available), and procedures used by other fellow professionals with experience. Once the design criterion in terms of the maximum permissible motion under the design wind conditions is identified, the problem crystallizes to the rate at which the energy imparted by the wind to the structure must be dissipated to limit its oscillation within the specified bound. Of course, the structures do have some internal damping, however, in general, it is not considered adequate. This brings in the nutation damper for additional dissipation of energy, and its design. Obviously, one would like the damper to be efficient, i.e. achieve the maximum dissipation of energy with a given mass of liquid. To that end, three basic geometries are studied: circular cylindrical damper; toroid with circular or square cross-section; and rectangular box-type damper as shown in Figure 2-11. The dimensions of the specific damper used during the tests were given earlier in Table 2-1. The investigation with each class of dampers has three distinct aspects to obtain better understanding of their performance. To begin with, experimental results are presented which show parametric influence on the system dissipation function, referred to as the reduced damping (r) ,l)- The influence of floating particles, added r to the free surface, is also explored. Furthermore, the effect of partitioning a liquid volume into smaller compartments is investigated with a possible application to bridge decks. Here, a rectangular damper is progressively divided into smaller sections to arrive at an optimum configuration. A similar study was also conducted for the toroidal damper with square cross-section. This is complemented by a flow visualization study of the liquid free surface which provides, rather qualitatively, some understanding of the wave geometry and the associated energy dissipation. Finally, the approximate, classical, well-established small wave theory [77] is used 42 Figure 2-11 Photograph showing some of the nutation damper models used during the parametric study: (i) rectangular; (ii) toroidal; (iii) partitioned rectangular; (iv) circular cylindrical. to provide practising engineers with a simple procedure that may help in selection of an appropriate damper, at least during the preliminary stage of design. 2.3.2 The circular cylindrical damper Using the test facility described earlier and shown in Figure 2-6, tests were conducted with a cylindrical geometry damper, systematically varying the liquid height and excitation frequency. The results are presented in Figure 2-12. It is apparent that the dissipation of energy improves at the lower end of the frequency range studied (0.4 Hz < fe < 0.55 Hz) as the liquid height diminishes. Note, the peaks are associated with resonance at the liquid fundamental frequency and higher harmonics. With an increase in the liquid height, the onset of the surface waves seems to be delayed and the dissipation level is reduced. The flow visualization study confirmed this general behaviour. Figure 2-13 shows a sample of the free surface response at an excitation frequency ratio fe/fi — 0.8. It shows formation of a wave train in the direction of excitation. The excitation frequency was gradually increased to observe the surface wave dynamics, which can be better appreciated on the video taken at the time. More important phases in the surface wave evolution around the fundamental sloshing resonance are schematically sketched (based on the video) in Figure 2-14. At low frequencies, the standing wave mode is observed with the free surface pivoting about the line of nodes perpendicular to the direction of oscillation (90° and 270°) as shown in Figure 2-14(a). As the excitation frequency is increased, the free surface becomes flexible (i.e. nonplanar) and a propagating wave train appears as indicated in Figure 2-14(b). This changes to the colliding wave-type sloshing motion. Here the two circumferential waves follow the curved walls of the damper (in phase) and collide at the instant of the oscillation extrema (Figure 2-14c). At the sloshing resonance, the spectacular breaking of waves lead to a significant dissipation of energy as observed earlier in Figure 2-12. With a further increase in the frequency, the sloshing mode changes to stronger propagating waves 44 10 1 -i——i r 1 ~i 1 r ~ i i ~ T optimum height Q A o—- 1 1 r f, (Hz) 0.603 0.514 0.420 h/R 0.094 0.069 0.047 e / R = 0.066 e J 0.4 I I -I L 0.5 I I 0.6 - J L 0.7 I - _ I L. 0.8 Excitation Frequency, f , (Hz) e Figure 2-12 Reduced liquid damping, n i, for a circular nutation damper (Model # 7) at different liquid heights. Note, a distinct improvement in dissipation as h/R decreases in the excitation range of f <0.55 Hz. r e Figure 2-13 Circular cylindrical damper showing formation of a wave train in the direciton of excitation: We/wj = 0.80. 90° (c) c o l l i d i n g w a v e s (d) s t r o n g e r (resonance) propagating waves with swirl Figure 2-14 Top and side views of the free surface modes in a circular damper as the forcing frequency, u> , increases from below to beyond the sloshing resonance. The damper is shown accelerating to the right: (a) standing wave; (b) propagating wave; (c) toroidaltype sloshing (resonance); (d) stronger propagating wave; (e) propagating, waves with swirl; (f) standing wave. e 47 with their ends attached to modes mentioned before (Figure 2-14d). Next, the liquid acquires a swirl component which contributes little to energy dissipation (Figure 2-14e). Finally, with further deviation from the resonance, the free surface returns to the standing wave with rocking motion about the line of nodes (Figure 2-14f). The effect of changes in h/R was also observed. Although, in general, the trends in the surface dynamics remained essentially the same, there were situations where certain phases became accentuated or weakened, and some cases got completely eliminated. This substantiated the variation in the damper performance with h/R as observed in Figure 2-12. For example, with h/R — 0.047, the colliding wave condition (Figure 2-14c) was so violent that liquid elements left the main body, remaining suspended for a while, before returning with a splash. On the other hand, with larger h/R, the wave trains that preceeded and followed the resonant sloshing were essentially absent. A practicing engineer can readily use the well-known small wave theory to design such circular dampers for real-life application. Although approximate, the theory can serve as a useful tool during the early stage of design. According to the theory [77], the wave dispersion relation is a , ^ —tanh—, with the liquid volume in terms of height given by Vi = irR h. 2 Thus W y = ^ Here: ujij = resonance frequency for i — t a n h ^ . tfl Kij = wave number for i th circumferential and j th circumferential and j th 48 transverse modes ; transverse modes; it! = container radius; V\ = liquid volume; g = gravitational acceleration. The eigenmode of interest being the fundamental sloshing resonance, ACIO = 1.8412 [77] and hence (2.4) Thus for a given liquid frequency, which is governed by the liquid height and hence the volume, radius of the container can be obtained through iteration. 2.3.3 T h e toroidal damper The toroidal damper consists of sloshing liquid between two concentric circular cylinders. The earlier study on the nutation damper by Welt [37] was limited to the toroidal damper with D/d = 1.89, a relatively small value. This is surprising because, in general, larger D/d leads to an improved performance. Based on his extensive study, Welt also concluded that variation of the Reynolds number is not a dominant parameter affecting the damper performance. With this as background, it was decided to assess the effect of excitation amplitude on damping using the Scotch-Yoke facility. A relatively large value of D/d = 6 was taken with the liquid height and the excitation frequency held fixed (h/d — 0.5, u> — 1.15). The excitation amplitude was gradually increased and e the reduced damping parameter measured. The results are shown in Figure 2-15. Several important observations can be made: (a) A toroidal geometry can serve as an effective dissipation device with a relatively large value of reduced damping parameter. (b) The energy dissipation is dependent on the amplitude of excitation, particularly in the range of small e /d e (0 < e /d < 0.4). However, for large values e of excitation (e /d > 0.4), the reduced damping parameter, e sentially constant. 49 r) i, Tt remains es- 12 T r 1 r T r -i 1 1 T—H—~i r r 11 co = 1.15 h/d = 0.5 D / d = 6.0 e 10 experimental data — polynomial fit a O 8 j 0.0 i L. 0.2 0.4 e./d 0.6 1.0 0.8 e Figure 2-15 Variation of the reduced damping parameter with the excitation amplitude for a large value of D/d = 6. The liquid height and excitation frequency were held fixed at h/d'= 0.5 and u) = 1.15, respectively. e (c) The damper geometry parameter D/d appears to have considerable effect on the damper performance. The results obtained by Welt [37] for D/d = 1.89 showed a different trend. In his case, not only the peak dissipation coefficient was found to be lower but also it varied with the excitation amplitude significantly. This clearly suggests careful choice of D/d in the damper design to obtain improved performance. The next logical step was to assess the effect of excitation frequency and liquid height on the damper performance. This is shown in Figure 2-16. At the outset, for a given h/d, the presence of the resonance peak, where the liquid frequency coincides with the excitation frequency, is as expected. It leads to the peak dissipation. Among the liquid heights tested, a smaller value led to higher reduced damping at a given excitation frequency. This, indeed, is important information from damper design considerations. Concurrent with the reduced damping tests, the free surface modes of oscillations were also observed to help gain, at least qualitatively, some appreciation of the energy dissipation process and its correlation with the surface wave dynamics. The typical surface modes with an increase in the excitation frequency are sketched in Figure 2-17. At a very low frequency, the surface pivots like a plane about the line of nodes, extending from 90° to 270° points, and the fluid particle motion is described by a standing wave (Figure 2-17a). As the excitation frequency increases, the surface plane becomes 'flexible' and two progressive wave trains appear simultaneously, propagating in opposite directions circumferentially, colliding at 0 and 180° positions (Figure 2-17b). As the frequency approaches the resonance, the wavelets in the train merge to give two larger amplitude waves which collide in a spectacular fashion (Figure 2-17c). The merging of the waves continues leading to a single propagating wave, on each side, at resonance with maximum dissipation of energy at collision (Figure 2-17d). Beyond the sloshing resonance, the two colliding waves merge into one and swirl as indicated in Figure 2-17(e). The direction 51 12 -i—r—i—r—•—i—r—r — • — A 10 - ' — O —" ~i—i—i—i—i—r—i—i—i—i II—i—i i i i—r T—I—i—r h/d 1/3 1/2 4/5 f, (Hz) 0.548 0.595 0.679 e / d = 0.20 e D / d = 6.0 8 Cn to 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 frequency, f (Hz) e Figure 2-16 Effect of excitation frequency and liquid height on the reduced damping parameter. The excitation amplitude and the damper geometry parameter were held fixed. line of nodes (d) single propagating wave (maximum dissipation) (e) swirling single wave (0 standing wave y . £ c o s co t e e Figure 2-17 Top and side views of the free surface oscillation modes in a toroidal damper as the forcing frequency, w , 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. e e 53 of swirl is determined by the initial flow perturbations. The swirl mode involves negligible dissipation of energy. It eventually returns to the standing wave at a still higher frequency (Figure 2-17f). Any further increase in excitation frequency leads to higher harmonics [37] which contribute little to the energy dissipation. Figure 2-18 presents typical flow visualization photographs. As pointed out before, one can use inviscid small wave analysis to assist in the design of the toroidal damper. It would involve solution of the Laplace equation for the potential function with appropriate boundary conditions at the wall and free surface (Appendix II.3, whtor.m). In brief, the eigenvalue equation for the free surface motion is given by Yi( )j[{ Ri/R ) Klj Klj - j[( )Yi( Ri/R ) 0 Klj Klj 0 = 0, (2.5) which leads to the resonant, sloshing frequency for a toroidal damper as Here: J\,Y\ — Bessel functions of the first and second kind of order one respectively; K\J = eigenvalue (wave number) for the jth transverse mode with the 1st (i.e. fundamental) circumferential mode; Ri = inner radius; R = outer radius; 0 h = liquid height; g = acceleration due to gravity. The volume of liquid in a toroid can be written as Vj = 7r(E - R )h. 2 2 (2.7) In the toroidal damper design uij, V\ and R are specified while #c»y, Ri and h are 0 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). 54 Figure 2-18 Typical flow visualization photographs for a toroidal damper showing: (a) two waves, on either side, approaching collision to the right, fe = 0.60 Hz; (b) single wave at resonance, fe = 0.65 Hz. The program entitled solvtor.m, presented in Appendix II, determines the range of inner and outer toroid diameters that would satisfy a given sloshing frequency and liquid volume. It solves eqs. (2.5) - (2.7) simultaneously in an iterative fashion. The range of diameters available is limited. For example, with the water volume and frequency specified as V\ = 730 ml and fi = 0.58 Hz, it gives the R 0 range of 14.4 - 16.2 cm with the R{ varying from 9.1 - 13.6 cm as shown in Table 2-2. Above this diameter range, the results are imaginary, and below it there is no convergence. The tolerance for convergence was set at 0.01 cm. Note, the diameter ratio varying from 0.63 - 0.84 represents a set of relatively slender geometry dampers. This is deliberate as the attention was focussed on this class of solutions. Table 2-2 Toroidal damper design for given liquid frequency and volume. Toroidal Dampers, ft = 0.58 Hz, VJ = 730 ml Ri (cm) R (cm) 9.106 9.565 9.883 10.289 10.656 10.830 11.113 11.377 11.744 11.787 12.037 12.270 12.559 12.824 12.813 13.034 13.171 13.524 13.608 14.400 14.500 14.600 14.700 14.800 14.900 15.000 15.100 15.200 15.300 15.400 15.500 15.600 15.700 15.800 15.900 16.000 16.100 16.200 0 56 Ri/R 0 0.632 0.660 0.677 0.699 0.720 0.727 0.741 0.753 0.773 0.770 0.782 0.792 0.805 0.817 0.811 0.820 0.823 0.840 0.840 2.3.4 The rectangular damper As against the circular damper, whose performance is unaffected by the direction of the wind induced excitation due to the symmetry, the rectangular geometry may prove to be sensitive to the direction of oscillation. To put it differently, the character of the surface wave, and hence the energy dissipation, may strongly depend on the excitation direction requiring consideration of this aspect during the design. In general, this can be accounted for quite readily. In most cases, meteorological data, extending over several decades, for the region where the structure under consideration is located are consulted during the design for magnitude of the wind speed and its direction. In the case of structures like bridge decks, the permissible directions (plunging, swaying) as well as modes of oscillations (linear, torsional) are limited. Furthermore, rectangular dampers are readily amenable to partitioning leading to a possibility of improved performance (for the same amount of liquid) as discussed later in Section 2. Above all, their simplicity is indeed quite attractive. Figure 2-19 shows variation of the reduced damping parameter as affected by the excitation frequency and the liquid height. The excitation direction is indicated on the diagram. As can be expected, with an increase in height the resonance peaks move to the right due to an increase in the liquid frequency. However, the change in the peak r} i is not substantial. This is a desirable feature as the same damper can Ti be tuned to a desired frequency without a significant reduction in performance. Photographs of typical free surface vibration modes are presented in Figures 2-20 and 2-21. The rectangular damper has an aspect ratio L/W — 1.64. Figure 2-20 shows formation of a wave train for u) /u)i = 0.95. Photographs in Figure 2-21 e attempt to capture free surface dynamics at the excitation frequency around the resonance value (i.e. w = e The harmonic excitation is along the length (L) of the damper. The direction of propagation of the wave is indicated by an arrow. 57 7 forced excitation frequency (Hz) Figure 2-19 Effect of excitation frequency and liquid height on the dissipation for a rectangular damper. Figure 2-20 Wave train in a rectangular damper of aspect ratio L/W — 1.64 at U) /ui = 0 . 9 5 . e 60 The propagating wave to the right at w < coi is presented in Figures 2-21(a), e (b). The resonance condition is depicted in the rest of the photographs where the surface wave travelling to the left gradually approaches the wall (Figures 21c, d, e) and eventually breaks resulting in large dissipation of energy (Figure 2-21f). Figure 2-22 presents sketches of the free surface dynamics based on visual observations as the excitation frequency is gradually increased from below to above the resonance value. The modal character changes from the plane standing wave undergoing rocking motion (Figure 2-22a) followed by a propagating wave train (Figure 2-22b), and the larger amplitude resonant condition leading to breaking (Figure 2-22c). In the post resonance phase, a single propagating wave appears eventually returning to the rocking mode at a higher frequency (Figures 2-22d, e). The design methodology for rectangular dampers using the simple small amplitude wave theory is rather straightforward. From the wave dispersion relation [77], •2 "ij = V^[(^2 + ^2) -2 i. -2 2 "2 t a n h T T ^ + ^2) - - ' J , 2 with the liquid volume given by Vj = LWh. At resonance r1 7r/i] ^10 = V^S ["^tanh —j 2 . (2.8) From these relations, the damper geometry can be established quite readily. It should be emphasized that the inviscid, small amplitude solution cannot be used to predict free vibration modes accurately, particularly near the resonance, as the assumptions inherent in the theory are no longer valid here [118]. On the 61 •< - • d a m p e r length (a) s t a n d i n g w a v e j * \ l i n e of n o d e s i i (b) p r o p a g a t i n g w a v e t r a i n (c) r e s o n a n t s l o s h i n g (d) propagating single pulse (e) s t a n d i n g w a v e e e c o s co e t Figure 2-22 Schematic diagrams based on visual observations showing the free surface geometry as affected by the variation of the excitaiton frequency around the resonance: (a) standing wave; (b) propagating wave train; (c) resonant sloshing; (d) propagating single pulse; (e) standing wave. 62 other hand, it predicts the resonance frequency with considerable precision for all the three geometries considered for a parametric study. 2.3.5 Effect of damper geometry The results clearly show that the damper geometry does have significant effect on the energy dissipation (Figures 2-12, 2-16, 2-19). Among the three geometries studied, the circular cylindrical damper appeared to be the most promising, at least under the conditions used during the parametric study. Of course, this must reflect on the free surface dynamics. In the case of a toroidal damper, at resonance, the energy dissipation mechanism involved collision between two waves moving circumferentially. On the other hand, for the rectangular damper, the wavebreaking involved collision between the wave and the damper wall. The wave dynamics in the case of a circular cylindrical damper is a bit more involved. It has features of both the toroidal geometry as well as the rectangular damper. At resonance, there are circumferential waves of the toroidal damper type together with propagating waves observed in the rectangular damper (Figures 2-17d, 2-21e,f). These collisions between the waves as well as between the waves and the walls appear to result in higher energy dissipation. 2.4 Effect of Partitioning a Damper The main objective in an efficient damper design is to achieve the maximum level of dissipation for a given mass of liquid. To that end, the concept of partitioning of a damper to improve dissipation appeared attractive, particularly with respect to the rectangular and toroidal geometries. Each compartment now has its own free surface dynamics, which if suitably controlled through local tuning can lead to an increased level of damping. Of course, at the outset, it must be recognized that the structural frequency is a known parameter. 63 2.4.1 Partitioning the rectangular damper The rectangular damper (Model # 10, Table 2-1) was selected for this set of tests. In absence of any partitioning, it has an aspect ratio L/W of 6. Through introduction of partitions, this was systematically changed to cover the range 6,3, 1.2 and 0.6 spanning an entire order of magnitude. For a given structural frequency (i.e. wind induced excitation frequency at vortex resonance), one can proceed to find the liquid height that, for a given number of compartments, would result in the liquid sloshing resonance. The objective is to find the optimum number of compartments that would result in the highest dissipation for a given volume of liquid. Figure 2-23 shows the effect of partitioning a rectangular damper for a given system frequency and volume of liquid. The results clearly show optimum number of compartments (five in the present case) for a given length, L, of the damper. The solid dots correspond to the resonance condition when the liquid natural frequency coincides with the system natural frequency (5.78 rad/s, 0.92 Hz) for the damper configuration under consideration. Increase in the damper parameter from around 1.2 to the peak value of 6 for the five compartment case is indeed remarkable. As pointed out before, such partitioned rectangular dampers are ideally suited for bridge deck applications. In the present study, roll motions of the damper may correspond to that of the bridge deck close to the anchored end (i.e. near the bridge tower). Obviously, for a given compartment length, the resonant liquid height would give the highest dissipation. However, it is important to recognize that the peak value of dissipation at resonance is not the same for different compartment lengths. In fact, among the four compartment lengths studied, the one with an aspect ratio of 1.2 performed the best not only at resonance but under all the nonresonant conditions tested. 64 0.00 0.05 0.10 0.15 h / w 0.20 0.25 0.30 Figure 2-23 The effect of compartmenting the rectangular nutation damper (Model # 10) , for a given system frequency (5.78 rad/s) and oscillation amplitude ( e / w = 0.17), on the damping ratio. AR is the aspect ratio of an individual compartment. 0 The liquid sloshing modes at different partition separations have a degree of similarity with those observed during the horizontal excitation as presented in Figure 2-22. The optimal compartment length is half the length of the free surface wave. With a large number of partitions (say 10), the compartment length is relatively small, the liquid cannot slosh, the standing wave mode appears and the energy dissipation is poor. The concept is often used to reduce the sloshing load on tankers and water towers [70,71]. In the case of too few partitions, the phase difference between the force and the response is no longer optimum (90°) to enhance dissipation. Thus, from energy dissipation considerations, establishing the condition of resonance for a damper with several small compartments represents more efficient use of liquid than exciting the resonance of liquid in a longer length of rectangular damper. In part, this may be attributed to the fact that smaller compartment lengths have lower resonant liquid heights. Test results at two different system frequencies, as shown in Figure 2-24, support this conclusion. The figure compares variation of the dissipation function r) j with the liquid height for the rectangular T damper with an optimum number of five comaprtments (AR=1.2) at two system natural frequencies of u n tively). = 5.78 and 4.50 rad/s (/„ = 0.92 and 0.73 Hz, respec- Note, the peak dissipation occurs essentially at the same liquid height, however, the value is higher at the lower system frequency. Furthermore, it appears that the partitioned damper has a fairly broadband response, which is a desirable feature. It would be appropriate to point out one additional advantage of partitioning a rectangular damper. According to the simple, approximate theory of waves in shallow water [77], for a rectangular damper under consideration, from eq. (2.8) 66 0.00 0.05 0.10 0.15 . . 0.20 h/w 0.25 0.30 Figure 2-24 Performance of the optimally compartmented rectangular damper (Model # 10, AR = 1.2) at two natural system frequencies, fn = 0.73 and 0.92Hz, as affected by the liquid height. Thus, for a given structural frequency, longer lengths (L) of the damper would require higher h for resonance, i.e. larger amount of liquid. During oscillations, this may lead to collection of a large amount of liquid at one or the other end of the damper causing unbalance of the system because it is statically unstable. Obviously, partitioning minimizes such possibility. A series of systematic tests were carried out to assess the effect of excitation amplitude e . With the system frequency fixed at u = 5.78 rad /s as before and 0 n the liquid height held constant at h/W = 0.08, the excitation amplitude was varied progressively to measure its influence on the reduced damping. The results are presented in Figure 2-25. Except for the optimum five compartment case, the effect of excitation amplitude is essentially negligible. The high dissipation associated with the five compartment case suggests large scale sloshing motion of the liquid when the nonlinear effects are likely to dominate. This would make the dynamic response dependent on the initial conditions. One would expect little effect of partitioning when the rectangular damper is free to undergo pitching oscillations, i.e. torsional motion about the longitudinal axis as shown in Figure 2-3(c). Note, the free surface motion, which is governed by the liquid frequency, is now normal to the partitioning direction. The results of this study are summarized in Figure 2-26. With the system natural frequency, w , held n fixed at 5.78 rad/s, the reduced damping results for three values of liquid height (h/W) and for a given excitation amplitude (e /W) show virtually no effect from 0 partitioning a rectangular damper. Note, the reduced damping level is rather low, hence the dependence on the excitation amplitudes is also negligible. This does not mean that a partitioned rectangular damper cannot be used to control torsional oscillations of a bridge deck. It only suggests, as one would expect, that now the damper axis should be transverse to the longitudinal axis of the bridge. 68 8 T—i—i—i—I—i—i—i—i—i—i—r | I S K p-rH I I I | I I ""I T T compartments W no. 17 w o I-—H T 6 — — 2 3 — ° — 5 1.2 "— 10 0.6 o) = 5.78 rad / s h / w = 0.08 A 0 r,l n 0 0.00 iD-r°i—•- 0.05 -I 0.10 I rI • -L - t - - i 0.15 £„/w -QM---I--I--4--I---T 0.20 --I---,----- 0.25 0.30 Figure 2-25 Effect of initial excitation amplitude, e and compartment aspect ratio on the reduced damping of a rectangular damper (Model # 10). The experimental set-up is shown in the inset. The system frequency u) was held at 5.78 rad/s, with the liquid height parameter h/W — 0.08. 0 n • 1 2 i i 1 4 3 1 — 5 6 0.20 ( b ) h / w = 0.05 0.15 ii 0.10 0 0.05 i 0.00 0.20 . 1 • 1 1 i i • > 1 2 . • • —— i — i 1 . . i . . 4 3 2 • . > 3 . . • . . . , 5 •— 4 compartment a s p e c t ratio, I / w Figure 2-26 Performance of the rectangular damper (Model # 10) in pitch motion as affected by the compartment aspect ratio and excitaition amplitude: (a) h/W = 0.033; (b) h/W = 0.05; (c) h/W = 0.10. 70 6 2.4.2 Partitioning of the toroidal damper Earlier in Section 2.3.3, the performance of the toroidal damper was evaluated. A natural question arises: Is it possible to improve energy dissipation of a toroidal damper by partitioning it as in the case of a rectangular damper? To answer this question an experiment was carefully designed based on some useful information already available: (i) The concentric character of the toroid must be maintained to take advantage of circular symmetry in practical applications where the vortex induced excitation can act in any arbitrary direction in the plane of the damper. (ii) As the excitation frequency is fixed by the wind speed, the resonant sloshing frequency should be the same in each toroid. (iii) The earlier study by Welt [37] as well as the present investigation suggest that slender toroids (large D/d, Figure 2-2a) generally lead to higher damping. (iv) The inviscid small wave analysis [99], though approximate, is sufficiently accurate and can assist in the design of partitioned nutation dampers. Even for the case of two concentric toroidal dampers (i.e. one partition), a large number of variations exist. To facilitate comparison with earlier results, it was decided to focus attention on a specific situation as follows: Outer diameter of the toroid, D = 2R = 30cm; 0 0 Liquid resonant frequency in each toroid, fi =0.58 Hz; Total liquid volume Vj=730 ml. The question then is: How to divide the given volume of liquid between the two toroids for a given /;? The solution of eqs. (2.5) - (2.7) was obtained using a simple program described in Appendix II whtor.m and is presented in Figure 2-27. It shows possible combinations of diameters as functions of volume fraction in the two toroids. It was decided to conduct experiments with 56% of the liquid volume in the outer toroid giving the configuration as shown in Figure 2-28. The single toroid and circular cylinder dampers are also included for comparison. 71 1.0 p 1 1 1 1 0.55 1 1 1 1 1 1 1 1 1 1 i 1 0.60 0.65 fraction of liquid in outer toroid 1 1 1 0.70 1 1 r 0.75 Figure 2-27 Geometry of the double toroidal damper as given by the inviscid small wave theory: / ; = 0.58 Hz, V}= 730 ml of water and D = 30 cm. 0 liquid sloshing frequency = 0.58 H z total liquid volume = 730 mL (a) double toroid to 0.743 (b) single toroid (c) circular cylinder 1.00 -0.882 1.00 dimensions normalized with respect to D 0 = 30 cm Figure 2-28 The dimensions of the double toroid, single toroid (Model # 3) and circular damper (Model #7) studied at ft = 0.58 Hz, Vj = 730 mL. The relative performance of the three dampers is compared in Figure 2-29. The excitation frequency was gradually changed and the reduced damping parameter measured. It is apparent that the toroidal damper provides the highest dissipation even though it is not operating at resonance. The circular damper also has a higher damping value at resonance compared to that for the double toroid, whose desired feature appears to be a broadband response around the resonance value. Obviously more tests are necessary to arrive at any definite conclusion. It should be pointed out that an efficient damper does not have to be a partitioned damper. Once an optimum damper configuration (circular, toroid, rectangular) for specified conditions is established, one can use the required number of identical dampers to attain the energy dissipation rate required, which is the current practice [48]. 2.5 Influence of Floating Particles With some information concerning the damper performance in hand, the next logical step was to explore the possibility of improving the dissipation rate further. To that end, introduction of floating particles appeared attractive. The premise is that they may promote dissipation of energy through an increase in exposed area, higher particle drag and inelastic collisions among themselves as well as with the wall. To assess the effect of floating particles, a systematic experimental program was formulated. It was immediately recognized that the number of variables involved can be rather large. To get some appreciation of the dissipation process, it was decided to focus attention on what appeared to be more important parameters. Table 2-3 shows the parameters involved in the study aimed at influence of floating particles on the damper efficiency. Of course, there are other variables associated with the particle geometry, wavelength of the free surface wave relative to the particle dimension, particle float- 74 i 1 1 1 1 1 1 1 1 1 1 1 1 1 r Figure 2-29 Effect of damper geometry and excitation frequency on the dissipation of: (a) single toroid (Model # 3); (b) double toroid; (c) circular cylinder (Model # 7). A fixed volume of 730 mL of water was used. ing orientation, the phase of the relative motion between particles and liquid and others. The investigation was limited to a set of particles described in Table 2-4. Table 2-3 Important parameters in the study of floating particles to enhance dissipation. Parameter Nondimensional quantity aspect ratio diameter lp j dp dp/d, dp/R, dp/w seeding density particle projected area / free surface area particle density particle mass PP/P mp/Mi liquid frequency liquid height damper geometry excitation amplitude u>i/u) e h/d, h/R, h/w toroidal, cylindrical, rectangular e /d, 0 ,e /R, e /w 0 0 Table 2-4 Floating particles used during the test-program. Material Diameter (in) Aspect Ratio surgical tubing poly-flo tubing 0.066 0.25 1.0 0.5, 1.0, 2.0, 4.0 1.1 0.8 Perler beads 3/16 2/16 1.0 1.0 0.7-0.8 0.5 straw Specific Gravity Table 2-5 Liquid height and corresponding resonant sloshing frequencies used in the test-program. h/d Frequency (Hz) 0.47 0.37 0.25 0.6348 0.5615 0.4883 76 The hollow cylindrical particles used in the study have diameter d and length p l p with particle aspect ratio l /d p as shown in the inset of Figure 2-32. Figures 2- p 30 to 2-35 show the effect of floating particles on dissipation for the three damper geometries: toroidal; cylincrical; and rectangular. Note, the toroidal damper clearly shows that, for a given aspect ratio of particles, there is an optimum value of seeding density leading to the peak reduced damping (Figure 2-30). Furthermore, in general, for smaller aspect ratio particles the peak is higher and well defined as shown in Figure 2-31. For the optimum aspect ratio of 1, the peak reduced damping of 1.27 represents an increase of around 40% with respect to the plain water case. As particle aspect ratio was increased, the particles tended to organize into concentric rings impeding the free surface motion and the progressive wave was suppressed. This, of course, is reflected in the reduced dissipation. Such behavior was particularly noticeable at l /d p p = 4.0 with a seeding density of 50%. It was also observed with low aspect ratio particles at high seeding densities but now the clustering was not as strong as with the higher aspect ratio case. The effect of particle aspect ratio tends to be relatively small as the liquid height, h/d, increases for a toroidal damper. Among the liquid heights tested, the significant effect of particle geometry (aspect ratio) was observed at h/d = 0.3. There appears to be optimum values for both the aspect ratio (l /d p p = 1.0) and seeding density (p = 30%) for the particles with open ends. In general, the p optimum p shifts to a higher value as h/d increases, however, it may now be p associated with a different optimum value of the particle aspect ratio. From the practical application consideration, a lower h/d is desirable. Fortunately, for a given particle density, an increase in the damping ratio is higher at lower h/d tested. The trends remained essentially the same at higher excitation amplitudes with an optimum increase in the damping ratio, through presence of particles, by about 40%. 77 I I I I . I I , - - o I Particle aspect ratio —Cr— 4.0 - - O 2.0 — h / d = 0.3 1.0 S = 1.0 e D / d = 4.54 e„/d = 0.17 oo I 0 I I 25 50 75 100 Figure 2-30 Representative plots showing the effect of particle geometry and concentration on the reduced damping of a toroidal damper (Model # 6). The liquid height was held fixed at h/d = 0.3. -I 1— I I 1.05 1.00 0.95 0.90 co 0.85 ' D / d = 4.54 e / d = 0.25 e 0.70 0 d P 2.0b 1.0b 1.0 2.0 4.0 0.80 0.75 - / P 20 h / d = 0.3 co =1.0 e 40 60 80 particle seeding density (% free surface) 100 120 Figure 2-31 Representative plots showing the effect of oscillation amplitude and particle geometry on the reduced damping for a toroidal damper (Model # 6, h/d —0.3 D/d = 4.5; u) = 1.0, e /d — 0.085). Suffix 'b' in the aspect ratio refers to particles with ends sealed or blocked. e e Experiments with particles of different densities showed the higher specific gravity particles to be more effective. It appears that the optimal particle specific gravity is the one that is close to that of the fluid. The desirable floating position appears to be when the particle is half submerged. This was particularly apparent in the tests that involved particles with trapped air inside. Figure 2-31 also shows the effect of covering the particle openings with a thin membrane. This caused the particles to float at a much higher position above the water line due to trapped air resulting in poor damping at all particle aspect ratios studied. As before, now the particles organized into concentric rings and prevented the surface from breaking as observed for particles with higher aspect ratio. Figure 2-32 presents response results for a typical circular damper with the optimum particle aspect ratio of 1.0 and no membranes at the openings. The results show a broad peak suggesting a relative insensitivity to the particle seeding density. The improvement in damping with the particles in this case is approximately 30%. As seen earlier in Figure 2-23, the rectangular damper has an optimum number of compartments leading to a peak value of damping for a given liquid volume. Figure 2-33 shows the effect of seeding density and excitation amplitude on reduced liquid damping, for the optimally partitioned rectangular damper. The particle aspect ratio was fixed at 1.0 since it gave better performance with other damper geometries. Results suggest that the presence of floating particles has little effect on damping. It may even deteriorate the performance in the excitation amplitude range considered. Only at a seeding density of about 70% there was a modest increase in the dissipation. Now the particles clumped together in clusters near the center of the compartment and the system behaved like an impact damper. In fact, the sound due to collisions was distinctly audible. Figure 2-34 shows the effect of increasing seeding density at two liquid heights, h/W — 0.10 and 0.15, in a rectangular damper under forced excitation tests. In general, the optimal seeding density appears to increase with an increase in the 80 1 1 1.0 0 20 40 1 1 _1 1 60 I 80 100 Figure 2-32 Representative plot showing the effect of particle concentration on reduced damping for a circular damper (Model # 7). Note, the liquid height is held fixed at h/R = 0\047. I I I [ 00 CO Rectangular damper in resonant condition (o=co. = 5.40 rad/s h / w = 0.067 5 compartments 0 i Particles l /d =1.0 specific gravity = 0.8 p J i 20 40 I I I L 60 p I L 80 100 Figure 2-33 The effect of oscillation amplitude and seeding density on the damping efficiency of a rectangul ar damper (Model # 10) optimally compartmented (AR = 1.2). T 1 I L. 1 P 00 co h/w •—-- 0.10 — o — - - 0.15 e /'w = 0.09 d 1.0 e V = P t 0 • ' ' I 20 • I I I I I I I_ J ' 40 60 seeding density (% of free surface area) 1 I 80 100 Figure 2-34 The effect of seeding density, for liquid heights h/W = 0.10 and 0.15, on the performance of a rectangular damper (Model # 13, 1 compartment). liquid height particularly in the region of peak performance. Unlike the toroidal damper case, now the particle density has significant effect at a larger liquid height. Generally, the influence of p appears to be less for the rectangular geometries. p It may be of interest to point out that results in Figure 2-34 are for a rectangular damper with less than optimal partitioning (1 compartment). Figure 2-35 compares the case of h/W = 0.15 with plain water against 100% particle seeding over a range of excitation frequencies. A fully seeded surface slightly increases the peak damping value and delays the appearance of the resonance peak. This has the implication that lower liquid heights can be used with the consequence of better reduced liquid damping. The fully seeded surface also suppressed the appearance of the standing wave between the fundamental and first harmonic merging the two modes into one big resonance peak as observed during the flow visualization. This is especially useful at a lower liquid height when the resonance peak is relatively narrow. The effect of particle diameter and specific gravity manifests in the degree with which a particle 'sticks' to the liquid, i.e. behaves as an integral part of it. If the particles move independent of the liquid or out of phase, more energy is dissipated as in the case of particles with larger diameter and higher specific gravity. With the particle motion not in phase with the liquid, the number of collisions increases. The inertia force exerted by the particles increase as (volume) 3 whereas the drag is proportional to (velocity) . As the particle density diminishes, 2 the inertia force decreases more rapidly compared to the drag. Hence very small and light particles behave like tracers and follow the fluid motion closely. This is not conducive to energy dissipation. When the particle seeding density is small the fluid field (velocity distribution) is not significantly affected by the presence of the particles. Hence the increase in damping is either negligible or modest. Experiments suggest improved dissipation with relative motion between the particles and the liquid. Such a relative motion is promoted by heavier and larger diameter particles, leading to an optimum condiiton for energy dissipation. To summarize, the optimal 84 plain water " — o - — 100% of surface particle seeded rD— J h / W : = 0.15 * e / w = 0.09 e oo Cn 1.0 1.5 forced excitation frequency (Hz) 2.0 Figure 2-35 Comparative performance of a rectangular damper (Model # 13, h/W = 0.15) for the cases of plain water and fully.seeded surface conditions over a range of frequencies. seeding tends to promote out of phase motion of the particles. At higher seeding densities, the free surface motion is inhibited, consequently the damping decreases. 2.6 Summary of Results (i) The objective of the experimental study was to arrive at parameters that would maximize dissipation of energy in nutation dampers. The results suggest that this can be accomplished through control of the damper geometry, partitioning of the liquid volume, and addition of floating particles. A methodology was developed to arrive at a set of dampers with the same liquid volume and sloshing resonance in order to investigate geometrical effects. The damper geometry determines characteristics of the resonant free surface sloshing modes. Certain modes, like those in the circular damper, are more conducive to wavebreaking, and hence improve damping. For the three damper geometries studied, lower liquid heights (i.e. lower sloshing frequencies) and lower oscillation amplitudes lead to an improvement in energy dissipation. (ii) If the depth, h, is significantly less than the wavelength A = 2TTC/U)I, then the flow may be considered as shallow and the waves are described as long. The classical definition for the shallow water waves is h/X < 1/20 [99]. In general, the experimental results suggest that the nutation dampers are most efficient for low liquid heights in the range where the shallow water approximation is valid. This information proved to be quite useful as it would considerably simplify analysis of such a complex fluid dynamical problem. This aspect is treated at length in Chapter 4. (iii) Partitioning of the rectangular damper aims at making the most efficient use of a damping liquid volume. It also suggests the applicability of the nutation damper to control bridge oscillations. In general, it is more efficient to excite the resonance of a series of smaller dampers than that of one large damper. Partitioned dampers also tend to be more stable due to the reduced volume of sloshing liquid in the individual compartment. However, partitioning of the rectangular damper in the direction perpendicular to the sloshing has little effect on damping. The liquid height and compartment length are more important in determining the damping response within a given range of system 86 frequencies. Furthermore, partitioning not only leads to higher dissipation but also gives near peak response over a wider bandwith. A combination of toroidal and circular dampers tuned to the same sloshing resonance, would combine the narrow frequency response of the toroidal damper with the broadband response of the circular cylindrical geometry, (iv) Seeding of the free surface with particles permit more energetic collisions and lead to an increase in the bandwidth through merging of the fundamental and first sloshing harmonic. Generally lower aspect ratio, smaller diameter, and high specific gravity particles resulted in improved dissipation. Half- submerged particles perform better than those with a higher floating position above the water line. The optimal seeding density can range from 30% 60% depending on the geometry of the damper, liquid height and oscillation amplitude. As the presence of floating particles delays the onset of the sloshing resonance, lower liquid heights (seeded with particles) can be used to achieve better reduced damping ratio. The increase in damping with particle seeding can be around 30 - 40% of the plain water case. Before closing the Chapter, a comment concerning application of the results to design of a prototype damper would be appropriate. Design of the structure, where the damper is to be installed, is of course available; so is the wind profile from meteorological data. Furthermore, building code provides information as to the peak amplitude (stress) that can be acceptable. Thus the amount of energy to be dissipated per cycle of oscillation to limit the amplitude can be calculated. For example, during vortex excitation the resonance condition would be critical. One can now select the liquid height and geometry (say diameter of the circular damper) so that the peak sloshing condition is established at the structural resonance. The reduced damping data obtained in this Chapter can now be used to calculate the energy dissipated per cycle. This would then establish the total number of dampers needed to meet the building code requirement. Welt has explained in detail the design procedure [37, 38]. 87 3. WIND INDUCED INSTABILITY STUDIES 3.1 Preliminary Remarks With the information concerning parametric performance of the nutation dampers in hand, the next logical step would be to assess effectiveness of the dampers in suppressing wind induced instabilities through dynamical tests with models of bluff bodies. The two-dimensional structural models were permitted to undergo vortex resonance, galloping and wake induced oscillations. The three basic damper configurations used during the parametric study (circular, toroidal, and rectangular), without as well as with floating particles, were used to this end. This chapter starts with a description of the test facilities, instrumentation, and methodology used during the wind tunnel studies. The dynamic tests ex- amine the damper effectiveness with reference to structures that undergo: vortex resonance; galloping; both vortex resonance and galloping; and wake induced resonance. The chapter ends with some remarks on the relative performance concerning damper effectiveness in arresting wind induced instabilities. 3.2 Wind Tunnel Test Facilities, Instrumentation and Methodology The experimental setup for dynamic wind tunnel tests is shown in Figure 31. The structural model, mounted with a nutation damper, is spring loaded onto a heavy, rigid, free-standing frame constructed to fit around the outside of the wind tunnel (in the plane perpendicular to wind direction). Light aluminum guide tubes slide through low friction air bearings on the frame. These tubes are rigidly connected to the structural model by slender mounts protruding through slots in the tunnel floor and ceiling as shown in Figure 3-1. Four helical coil springs, stretched from the frame to the guide tubes, provide elastic support for the system permitting the structural model to oscillate in the cross-stream direction as required. 88 NUTATION DAMPER DISPLACEMENT TRANSDUCER AIR BEARING SHAFT LU Q O AIR BEARING BLOCK oo CO V\A- ANALOG TO DIGITAL CONVERTER AND COMPUTER Figure 3-1 Wind tunnel test setup to assess effectiveness of m i t ; ^ ™ / ^ ™ . . • • and P a l l o n i n t r T V , ™ . A\* . J " or nutation dampers m suppressing vortex resonance and galloping. Three different damper geometries were used during the tests. c u c t L l v e n e s s The dynamic tests are performed in the closed circuit laminar flow wind tunnel shown in Figure 3-2. 2.44m. The tunnel has a working cross-section of 0.69 x 0.91 x 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 7:1 ratio is measured with a Betz micromanometer with an accuracy of 0.2 mm of water. The rectangular testsection, 0.69x 0.91m, is provided with 45° corner fillets which vary from 13.25 x 13.25 cm 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. The choice of structural model is governed by the nature of the instabilities under study; circular cylinder models for vortex resonance and square cylinder models for galloping. Square cylinders are also used, occasionally, to study vortex resonance. The aerodynamic structural models and nutation dampers used are outlined in Tables 3-1 and 3-2, respectively. Structural model displacement Y(t)/H serves as a measure of the system response. The displacement of the structural model was measured using a linear voltage differential transducer (Schaevits Model 3000 H.R.) with supporting circuitry. The resulting output voltage is linear with the deviation within 1.5% over a range of 63 mm, and it is independent of velocity over the frequency range used in the tests (less than 6 Hz). Appendix 1.2 shows the calibration plot for the transducer. Figure 3-3 is a typical sample of the data trace. Note, the circular nutation damper with a liquid / total mass ratio of 2% reduces the model oscillation by over an order of magnitude. During wind tunnel tests, a real time analyzer is used (SD335 Spectrascope II, Spectral Dynamics Corp., California) to obtain a spectrum of the structural motion, which can be used as a diagnostic tool. For example the vortex resonance condition is identified through the peak response observed at f . The transition to galloping n 90 Figure 3-2 Schematic diagram of the closed circuit laminar wind tunnel-facility used in wind induced instability studies. . to Figure 3-3 Typical traces taken during wind tunnel tests showing variation of the structural model displacement with time: (a) without damper; (b) with a circular damper of mass ratio 2%. is reflected in strong amplitude modulation of the displacement signal (Figure 3-4, inset). Table 3-1 Structural models used during wind.tunnel tests TWO-DIMENSIONAL WIND'TUNNEL AERODYNAMIC MODELS* MODEL NO, crosssection •. ; 1 • • • ' 3 o 4 . o material balsa and aluminum balsa and aluminum PVC balsa, paper and aluminium mass (g) 786 749 745 600 *Dimensions H = 102 mm, L m = 673 mm; Endplates: 204 x 305 mm Table 3-2 Dampers used in wind tunnehtests DAMPER MODELS MODEL NO. 6 9 14 GEOMETRY DIMENSIONS (cm) " toroid circular rectangular D =10.0, d= 8.6 D = 10.0 / =11.0, w =8.8 The modulation frequency increases until the transition from vortex resonance to galloping is completed, and the structural response at large amplitudes sets in. As can be expected, in the case of the structure located in the wake of another structure, two such peaks were observed in the displacement response spectrum. One represented vortex shedding from the rigidly mounted upstream cylinder and the other corresponded to the natural frequency of the downstream structure. The coincidence of these two peaks represent the resonance condition. Close proximity to the resonance condition was indicated as beats where two peaks are centered about the natural structural frequency. 93 0.060 0.050 0.040 0.030 Re = U H / v = 14,000 0.020 U — 0.010 -L H ~r • 4 Y(t) -L 0.000 1 2 coY/co n 3 Figure 3-4 Spectrum of elastically mounted square cylinder displacement, Y(t) / H, under wind loading. The square cylinder is about to make the transition to galloping. The inset shows the time history of displacement for the square cylinder. Note the presence of strong modulations during transition from vortex resonance to galloping. The reduced aerodynamic damping, 77^ , is a measure of the energy dissipated a by a vibrating structure. It is defined as _ 47r(c/c )M c p li'L a m Here c represents the sum of inherent internal structural damping, viscous dissipation at the joints, etc. including the eddy current damping, which is also viscous in character. Of course, in addition, now we have the energy dissipation through a nutation damper, F *lr,l = ^ 7 a 2 s i <P- n ' (2-3) Prom the above relations, it follows that the aerodynamic damping is related to the reduced liquid damping through, ^ 1 LH ~ \ 47T M/p M/PaLmH 2 a S 2 m \ 1 VM,/M" W Thus energy dissipation of the aerodynamic structure can be increased by introducing an eddy current damper. This can be used to advantage in avoiding overlap of the vortex resonance and galloping regions. The eddy current dissipation was provided by surrounding the oscillating aluminum guide tubes with steady magnetic fields. The system damping ratio, ( , with no magnetic field applied was 3 0.0002 and could be increased to over 0.006 using the electromagnetic dampers. The level of damping was measured by observing, with no wind, the decay of free vibrations following a displacement. The effective mass of the moving system was determined by observing the natural frequency of the model with an additional mass added to it. A plot of additional mass versus 1/w , where u) is the corresponding 2 n structural frequency, was used to determine the effective mass of a model [119]. 95 3.3 Structural Response with Nutation Dampers 3.3.1 Isolated structure The effectiveness of a circular cylindrical damper in controlling both the vortex resonance and galloping instability is illustrated in Figure 3-5. Note, in absence of the nutation damper and with low inherent damping of rj a Ti = 2.90, the galloping sets in at a low wind speed of around U = 0.8. With the addition of eddy current damping ( 7 7 r>a = 5.21), the vortex resonance peak appears and the onset of galloping is delayed to U « 3.6. However, in presence of the nutation damper with a liquid mass ratio of only 1 % (liquid mass ratio = liquid mass / total oscillating mass), the vortex resonance and galloping are successfully controlled. Note, even at a very high wind speed of U « 6, it was not possible to excite the galloping instability. Figure 3-6 shows the effect of increasing the circular damper's (Model # 9) liquid mass on the vortex resonance and galloping response of the square cylinder (Model # 1). Clearly, the structural oscillations decrease with an increase in the liquid mass. For the cases studied, the resonance peak was not very broad and it was easy to increase the wind speed past the peak. As expected, the effect of more liquid was primarily on the vortex resonance response, the galloping being essentially controlled at 1% of the mass ratio as seen earlier in Figure 3-5. Figure 3-7 illustrates the effect of damper geometry on the control of the square cylinder undergoing galloping instability. To facillitate comparison, care was taken to maintain the same liquid mass ratio and, where necessary, the springs were adjusted to maintain the same structural natural frequency. Each damper geometry was tuned to resonance within a velocity bandwidth of 11 < U < 13 and a liquid/total system mass ratio varied from 1.0 - 4.0%. A typical set of results for the 3% case is shown in Figure 3-7. Note, each of the three dampers studied showed excellent performance. The circular geometry proved to be the most successful with a reduction in amplitude of over 90 %. 96 I 1 ' I ' I 1 2 1 1 1 I 1 3 1 ' 1 1. I 1 4 ' 1 ' I 1 5 1 1 I I I I 6 ure 3-5 The effectiveness of a circular nutation damper in arresting both vortex resonance and galloping instability of a square cylinder. -•- represents the system response with little damping hence the square model gallops from the start. - A- presents the displacement data with system dissipation increased through introduction of eddy current dampers in order to separate vortex resonance from galloping. - O r shows the nutation damper suppressing both the vortex resonance and galloping over the entire range of the wind speed. T 0.6 1 1 i—| 0.8 r — i r—r-—| 1.0 1 1 1 r—| 1.2 r—i 1 y r—| 1.4 r—i 1 1 1 1.6 1 1 r r 1.8 Figure 3-6 Wind tunnel test results showing the effect of liquid mass ratio on the vibratory response of square prism (Model # 1) with a circular cylindrical damper (Model #9). • 1.5 T—r|—i—i—|—i—|—i—i—i—i—[.—i—i—i—i—|—i—i—r—i—|—i—i—i—r—^—i—i—i—i—|—r—i—i—i—|—i—i—i—i—|—i—r limit Damper —o— — -o— -•— f Y(t) Y(t)/H 1.0 geometries no damper square toroidal circular Model: 1 liquid / total mass = 3% CD CO 0.5 0.0 8 10 11 U 12 13 14 15 Figure 3-7 Comparative performance of three different geometry dampers (Model # 6, 9, 14) showing their effect on the galloping response of a square cylinder (Model # 1). 3.3.2 Effect of floating particles With the understanding of the structural behavior when coupled to a pure water damper, the attention was now turned to evaluate the system performance in the presence of a damper with floating particles. Figure 3-8 shows a typical set of results for a toroidal nutation damper mounted on a circular cylinder model undergoing vortex resonance. In all the cases, the natural frequency of the system was held constant at the liquid resonance. Dramatic decreases in plunging amplitudes were observed even with a small amount of water. The vibratory response reduced further by 30% with an optimal value of particle density. Similarly, for the models susceptible to galloping, the onset of galloping was delayed and the associated amplitudes were significantly lower (Figure 3-9) at a damper mass ratio of about 2%. Circular and rectangular geometry dampers, with optimum particle density, showed essentially the same trend. 3.3.3 Structures affected by wake Experiments were also carried out to assess effectiveness of the nutation dampers under more realistic situations where a structure is located in the wake of another structure. This set of tests used two identical circular or square cylinders. In a given test, for example with a pair of circular cylinders, the two models were held parallel to each other, one behind the other, in the direction of the wind (Figure 3-10). The upstream cylinder was held fixed and generated the wake, while the downstream one was elastically mounted on the frame of Figure 3-1 and hence free to oscillate under wind loading. A nutation damper was mounted on the downstream cylinder. The wind speed as well as distance between the two models were varied systematically to arrive at a critical condition leading to peak response of the elastic model. As pointed out earlier by Zdravkovich [110], vibration suppression procedures that affect the aerodynamics of the structure can have undesirable effects on the system dynamics. Thus, it appears that flow interference can severely undermine the usefulness of vortex suppression devices normally effective with an 100 I I I 1 , 1 I I I I I I I I I I I I I I I I I I 0.75 - Y(t)/H 0.50 - 0.25 - 0.00 2.0 2.5 ,j 3.0 3.5 4.0 Figure 3-8 Wind tunnel test results for a two-dimensional circular section model (Model # 3) i a toroidal damper (Model #6) with particles. I I Figure 3-9 Effectiveness of the toroidal damper (Model # 6 ) in arresting galloping oscillations of a twodimensional square cylinder (Model # 1). The effect offloatingparticles is also shown. '//////// Figure 3-10 Plan view of the interfering structural models arrangement: (a) inline square cylinders; (b) inline circular cylinders. 103 isolated bluff body. On the other hand, the nutation damper being a dissipative device should be unaffected by the environmental conditions. This observation was substantiated by tests with inline circular and square structural models. The results show a dramatic decrease in oscillation amplitudes of the downstream structure. To begin with, the critical vortex resonance wind speed for a square (or a circular) cylinder was established (in absence of the nutation damper). With the critical wind speed held fixed, the spacing between the in-line cylinders was progressively reduced until the peak response of the downstream cylinder was observed. The critical distance corresponded to 6.5H for the case of square cylinders and 3.75H for the circular cylinders. Figure 3-11 looks at the effect of wind speed on the inline square cylinders at a fixed separation of 6.5H. Figure 3-ll(a) presents the spectrum of the downstream cylinder's motion at a wind speed below the resonance. Notice that in addition to its own resonance condition, the downstream cylinder also detects the vortex shedding of the upstream cylinder at wy/wn ~ 0.7. Figure 3-ll(b) shows the time history of the downstream cylinder's response at a wind speed just below the resonance. The beat phenomenon showing amplitude modulations at the differential frequencies is apparent. The carrier frequency is the average of the sum of the two frequencies and the modulation frequency is the average of the difference. The two frequencies involved are the natural structural frequency, and the vortex shedding frequency, u> , of the rigidly mounted upstream cylinder. Figure 3-11(c) shows the response s spectrum at the resonant wind speed, i.e. the vortex shedding from the upstream cylinder occurs at the natural frequency of the downstream cylinder. Note the amplitude of oscillation increases by an order of magnitude and the two peaks have merged. The inset in Figure 3-11(c) shows the response spectra as affected by the liquid in the circular cylinder type nutation damper. The decrease in amplitude by more than 90% for a liquid mass ratio of 3.75 % is indeed impressive. Figure 311(d) shows the response spectra at a post-resonant wind speed where the separation 104 0.030 i - i r- T 1 r ™i 1 r 1 (a) ^ Y(t) / (H) \ Y(t) ^ u V / 77777/ 6.5H 0.020 vortex shedding structure 0.010 0.000 0.0 ^H-^ 0.5 1.0 1.5 2.0 Figure 3-11 Effect of wind speed on the response of a square cylinder (Model # 2) critically located in the wake of the other square cylinder (Model # 1): (a) undamped response with the wind speed lower than the resonance. - 0 20 40 * Figure 3-11 Effect of wind speed on the response of a square cylinder (Model # 2) critically located in the wake of the other square cylinder (Model # 1): (b) undamped beat-type response for the wind speed close to resonance. 6 t 0 8 0 1 0 0 0.30 0.25 - Circular Damper Mass Ratio, % (C) - - - - - 0.0 — — 1.25 2.50 3.75 0 1 5 Y(t)/(H) 0.20 0.10 I D 0.15 0.05 u 0.10 0.05 • Y(t) • -6.5H 0.00 —H '4. 0.00 0.0 0.5 1.0 co /cb Y n 1.5 2.0 Figure 3-11 Effect of wind speed on the response of a square cylinder (Model # 2) critically located in the wake of the other square cylinder (Model # 1): (c) damped response at resonance for the circular geometry damper (Model #9). 0.030 (d) U Y(t)/(H) 0.020 Y(t) 97777/ 6.5H structure vortex shedding 0.010 0.000 co /co Y n Figure 3-11 Effect of wind speed on the response of a square cylinder (Model # 2) critically located in the wake of the other square cylinder (Model #1): (d) undamped response with the wind speed beyond the resonance value. between the natural frequency of the downstream cylinder and the vortex shedding frequency of the upstream cylinder is again significant. Figure 3-12 compares the relative performance of the three dampers for the inline square cylinder response at resonance. Note, both the wind speed as well as the spacing between the cylinders correspond to the critical resonance conditions. The amount of information presented here is rather extensive. Each damper's performance is evaluated under four different conditions of the liquid mass, ranging from 1.0% to 4% of the system mass. It is apparent that the dampers are quite effective, even with 1.0% of the liquid (Figure 3-12a), reducing the amplitude of oscillation by 45% (toroidal), 58% (square), and 65% (circular geometry damper). With an increase in the mass ratio, the performance improves further and the vibration is essentially suppressed (reduction in amplitude by 90%) for the circular geometry damper with a mass ratio of 4% (Figure 3-12b). Figure 3-13 illustrates the effectiveness of the dampers in reducing dynamical response during the vortex resonance due to interference between inline circular cylinders. Again, the three damper geometries were assessed at the critical conditions of wind speed and spacing. Each damper geometry was tested with the liquid mass progressively increasing in increments of 1.0% of the total system mass as before. Thus, for a square damper with a liquid mass of 1.0% (Figure 3-13a) the peak amplitude Y(t)/H reduces from around 0.67 to 0.52, while for the circular cylinder damper it diminishes to 0.25. Note, for the circular cylinder damper, with a liquid mass ratio of 4.0% (Figure 3-13b), there is a further drop in the amplitude to 0.08 ! 3.4 Concluding Remarks The superior performance of the circular cylindrical damper may be attributed to its broadband damping character as observed in Figure 2-12. Sun et al. [120] have proposed the use of multiple tuned dampers, each tuned to a slightly different resonant frequency. The circular cylindrical damper appears to accomplish this, 109 I I I I I I I I I I I I I I I I I I I I. I I I I liquid mass 1% of total mass I I •— I I I I I I I ""I I I no damper square damper toroidal damper circular damper o • • Re = V H / v = 14,000 • Y(t) 2* 6.5H i—i 0.5 0.6 0.7 0.8 0.9 i i i 1.0 i — ii i 1.1 i i i i I .1.2 i i ••• i i i i i i 1.3 Figure 3-12 Effect of the damper geometry on the resonant response of the inline square cylinder (Model # 2): (a) mass ratio of 1 %. 0.5 0.6 0.7 0.8 0.9 y 1.0 1.1 1.2 1.3 Figure 3-12 Effect of damper geometry on the resonant response of the inline square cylinder (Model # 1,2): (b.) mass ratio of 4%. Note the circular damper appears to be quite promising with a reduction in the amplitude by 98%. 0Q • 3 • • i ' • I 4 ' 1 ' ' ) 5 ' I I I I I L-l I l—J I I I I I I I I 6 7 8 9 in-line circular cylinder separation / H I L_J I I I 10 I I I I I 11 I I I I 12 Figure 3-13 A comparison of damper performance at resonance for inline circular cylinders (Model # under critical conditions of wind speed and separation: (a) liquid mass ratio is 1%. 0.7 T—i—III "T—I i—i—i—r—I—i—i—i—r "T 1—T T 1 1 II I 1 1 1 1 1 1 1 1—1 1 T no damper square damper toroidal damper circular damper liquid mass 4% of total mass 0.6 1 0.5 Y(t) / H 0.4 0.3 0.2 • 0.1 0.0 6 i 3 4 5 i i' i i i i' ~ i i I^I J L 6 7 8 9 in-line circular cylinder separation / H 10 11 12 Figure 3-13 A comparison of.damper performance at resonance for inline circular cylinders under (Model # 3, 4) critical conditions of wind speed and separation: (b) liquid mass ratio is 4%, The circular damper (Model # 9) continues to be the most efficient. through its broad damping peak, without such elaborate tuning exercise. Thus, here we have a damper that is effective over a broad range of forcing frequencies often encountered by prototype structures exposed to natural wind environment. To summarize, the structural vibration in vortex resonance being a broadband phenomenon makes a damper with the broadband performance more effective. On the other hand, control of the galloping response is governed primarily by the level of the damping and not so much by the damper's frequency response. It may be pointed out that the dynamical response of the model when located in the wake was significantly higher than when operating in isolation. Thus the former represents a more demanding situation. It is indeed, remarkable for the dampers to perform so well under such adverse situation. Before closing, a comment concerning the blockage condition during the wind tunnel tests would be appropriate. Prototype structures such as bridges, buildings, smokestacks, etc. normally operate under unconfined or small blockage condition. On the other hand, wind tunnel walls during model tests restrict the flow. This increases the local wind speed. Boundary-layer growth on the walls and the surface of the model, its separation, wake and vibration of the model further complicate the problem. A reliable scheme for blockage correction for bluff bodies has not been reported in literature yet except for some experimental results for specific geometries [121]. Under such situation, the results are purposely not corrected for blockage. However, it is of interest to recognize that the higher local velocity would increase the excitation force and vortex shedding frequency. Thus the results presented tend to be conservative, i.e. the damper performance is likely to be better in real-life situation. 114 4. N U M E R I C A L A P P R O A C H T O S Y S T E M DYNAMICS WITH NUTATION DAMPING 4.1 Preliminary Remarks The experimental investigations aimed at parametric analysis of the dampers and their effectiveness in suppressing wind induced instabilities have firmly established their potential. The experimental program was primarily aimed at formulation of the test methodology so as to evaluate effectiveness of the concept. It was not meant to compile a large body of performance information by variation of system parameters in an organized fashion so that it may serve as a handbook for design engineers. From a practising engineer's point of view, undertaking of the test-program would be demanding in terms of time, effort and cost. With the advent of computers, a simple, numerical approach, if available, can prove to be quite attractive. However, energy dissipation through sloshing liquid is a complex process involving a large number of parameters and their intricate interactions. Hence to develop a relatively simple model that can capture the physics of the problem is a challenging task. Fortunately, the information obtained during the parametric tests, wind tunnel experiments and flow visualization study can prove to be useful in formulating a model that may have some promise. Of course, one can test its validity quite readily as test results are already in hand. 4.2 Model The experimental results clearly suggest the following: (a) The nutation damper appears to be quite effective at low liquid heights. This indicates that the relatively simple shallow water wave theory [99] suitably modified to account for viscous dissipation and dispersion, may present one avenue to follow. 115 (b) Viscosity and wave breaking seem to be the major factors contributing to dissipation. With this as background, it was decided to approach the problem of sloshing in a rectangular damper as a shallow water wave phenomenon (Figure 4-1). Characteristic features of the analysis are summarized below: (i) The liquid is considered homogeneous, irrotational and incompressible. (ii) The walls of the damper are treated as rigid. (iii) The influence of surface tension is taken to be negligible. For water with its surface in contact with air at ambient temperature, the surface tension effect becomes significant for a wavelength < 1.7 cm. For nutation dampers used in the tests, and their prototypes for real-life applications, this condition is not usually satisfied so surface tension is negligible. (iv) The analysis accounts for nonlinearities in the governing equations as well as boundary conditions. Thus it is applicable at resonance when a large amplitude sloshing motion is encountered. (v) The analysis considers dispersion as well as dissipation in the damper. Effect of floating particles is also accounted for. (vi) It should be emphasized that the structural response acts as an excitation for the damper thus affecting the sloshing motion of the liquid and its dissipation. On the other hand, the nutation damping affects the structural response. The analysis accounts for this conjugate character. (vii) Such a comprehensive treatment of the subject accounting for nonlinearities, resonance conditions, and conjugate interactions between the structure and the damper is indeed rare. To begin with equations governing sloshing motion of a liquid in a rectangular cross-section container, subjected to harmonic excitation, are derived and their potential flow solution is obtained consistent with boundary conditions. The solution is subsequently modified to account for viscous dissipation and dispersion of 116 z the waves. The solution obtained through a finite difference scheme provides information about the free surface dynamics. Finally, dynamics of a vertical cantilever beam with the damper at its tip is studied, accounting for the conjugate character of the problem, and the numerically predicted results are compared with the experimental data obtained earlier. 4.3 Potential Flow Analysis 4.3.1 Equations of motion For nonviscous, incompressibleflowsthe governing continuity and momentum equations can be written as: V • U = 0, i.e. V <£ = 0; and 2 du ~dt dv dt (4.1) (4.2a) ldp pdy + By] dw (4.26) (4.2c) ~dt Here: U = velocity vector, ui + vj + iok; <f> — velocity potential; p = pressure; B ,B ,B x y z — body forces per unit mass in x, y, z directions respectively. Taking the velocity potential in the form $ = F(x,y,t)G(z) 118 (4.3) and substituting into equation (4.1) gives dF d^F d?G_ _ dx dy dz 2 c G 2 F 2 Q 2 ' i.e 1 (d F F [a* 1 dG d F\ 2 2 + 2 = 2 = c o n s t a n t = ~ K Thus the continuity relation requires the solution of two uncoupled equations: dG 2 dz 2 dF - K G = 0; 2 (4.4a) dF 2 2 consistent with boundary conditions. 4.3.2 Initial and boundary conditions The shape of the liquid surface is represented by 77 = w(x,y,t). Thus 77 = 0 represents the initial quiescent condition of the free surface in the horizontal x, y plane. Now at time t = 0, the damper is subjected to a sinusoidal excitation in the x direction, X = e sino; i, e e where: X = displacement of the damper in the z-direction; e = amplitude of the damper motion; e w = frequency of the damper motion. e Thus the initial conditions are: n{x, y, 0) = 0; 119 (4.5) U(aj.y.O) = 0. There are two distinctly different boundary conditions at the free surface (Appendix III): (i) Kinematic Condition dr] d<j> drj dt dx dx i.e. d<f> dr) d<j> dy dy drj w dz drj ' drj , » = ^7 dt + »~E~ dx + *T~dy U (- ) V 4 6a (ii) Dynamic condition Considering gravity to be the only body force in the z direction (B = — g), z ^ + M = \(V<!>) . 2 (4.66) As the liquid is not permitted to flow through the wall, at the wetted surface d<j>/dn = 0, where n represents the direction normal to the wall. In particular, there is no flow through the damper bottom and hence at z = -h, w = 0. (4.7) Furthermore, at the side-walls, u(l,t) =u(-l,t) = 0. (4.8) The nature of the waves excited at the surface of the sloshing liquid depends on the physical properties of the liquid, its height, geometry of the container (rectangular in the present case), and the external excitation. Standing waves, progressive waves, solitary waves, hydraulic jumps, or a combination of these are typical. It is of 120 interest to point out that a wide variety of free surface motion is governed by the equations and boundary conditions discussed above. 4.3.3 Solution of the potential flow equations It is apparent that the problem is reduced to the solution of equations (4.2) and (4.4) with the appropriate initial and boundary conditions. To begin with, consider the uncoupled equation in the variable G obtained before, dG 2 ^ - « » O = 0. (4.4„) A function in the form G(z) = cosh K,(h + z) satisfies equation (4.4a) as well as the boundary condition (4.7). With this as the solution, the velocity potential (eq. 4.3) becomes 4> = F(x,y,t) cosh K,(h + z). (4.9) Here constant K corresponds to the wave number. Thus: d <t> dF d6 dF dy dy . u = —— = —— cosh Kin + z): u = —^- = — cosh K(h + z); , (4.10) dd> w — — = KF sinh /c(/i + z); dz i.e. at the surface (z = 77): dF u = —— cosh/c(/i + 77); 3 dx dF . v — — cosh K(h + 77); 3 dy 121 (4.11a) where the subscript, V , refers to the value of the variable at the surface. It is desirable to obtain the corresponding expression of w in terms of u and v . To 3 3 3 that end, one may turn to the continuity eq. (4.1), dw du dv ~dz~ dx dy' dF dF 2 2 + dx 2 cosh K,(h + z), dy 2 1 dF dF 2 i.e. w K 2 + dy dx 2 2 1 du dv K dx dy Thus at the free surface sinh n(h + z), tanh K,(h + z). du 3 dv dx dy 3 It would be convenient to develop quantities w(dw/dx) t a n h « ( ^ + 77). (4.116) and w(dw / dy) here as they are used later in the analysis. From eq. (4.10), w = KF sinhre(/i+ z), i.e. dw . . —— = KU tanh K(h + z). dx Using the expression for w from eq. (4.11b) and applying the condition of irrotationality gives: w dw dx d (U 2 dx\2 l + — (uv) tanh K,(h + z). ) 2 7J d 2 J 2 dy' (4.12a) Similarly, dw w dy d (u 2 dy\2 Vv \ 2 O . 2 J) + —dx (uv) 2/ dx 122 .1 i , 2 / . \ tanh K{h + z) (4.126) 4.4 Integration of the Governing Equations along z-axis 4.4.1 Continuity equation Consider first integration of the continuity equation (4.1) du dv dw dx dy dz (4.1) Substituting for u, v from eq. (4.10), integrating from the bottom of the container to the liquid free surface and switching the order of integration and differentiation gives 9 f v , dF f dF dy J_ h dy dy AC . d v —— cosh K[h + z)dz + —— / — / dx J_ h dx . n , —— cosh K(h + z) dz + / dw J_ h , —- dz = 0, dz i.e. d -u tanh K ( / I + 77) + dx K 3 -v tanh K.(h + 77) + w s s = 0. Applying the kinematic boundary condition from eq. (4.6a) leads to d dx tanh«;(/i + 77) tanh K(K + 77) d dy K dri AC dr\ £ -£ -£= - + +u +v , dri a < 4 - 1 3 ) 4.4.2 Momentum equations Consider the momentum equations in (4.2). Applying the condition of irrotationality, they can be rewritten as: d ,U ^ du I dp 2 dv d ,u 2 idp s 123 „ » „ l t _ . dw d U ldp 2 (4.14c) Integrating eq. (4.14c) with respect to z up to the free surface 77 and considering gravity to be the only body force gives "dm, U 2 - U . 2 , 1 . where p is the ambient pressure at the surface. Thus 3 , , u?-u\ , P-P. paw dz. (4.15) Differentiating the above equation first with d/dx and next with d/dy gives: d (P~P> fdr] dz\ Id dx dx) 2 dx \ (^)=»( dx dw to -u )+— r 2 J 3 dxj (4.16a) dz: dt z d_fpdy\ Integrating eqs. (4.14a) and (4.14b) with respect to z, as in the case of eq. (4.14c), and substituting from eq. (4.16) gives: r fdr] d fU(V\ du 2 1 dz\ 1 d t -\ f d 2 s dv r fdn d fU \ 2 3 u 1 5 / dz\ 2 t t 2 u \ dw v to (T) = " I (s -to)+ 2to( - - ) 3 + d telM * P dw d + , dt which further simplify to: du 3 ~dt + dn g— -| dx 1 d . . . d 2 dx 124 ndw • (4.17a) x dv s 1/2 where U« = u + 2 2 and wdw/dy in eq. (4.17) from eq. (4.12) gives: 1 - tanh K(h + z)] 2 at dx dv lH dy~ +g r 1 + tanh 2 g + 2 K(H + z) (4.18a) 2 3 ox J dt z rl —tanh AC(/J + z) 2 ^ 2/ dx \ c? f"^ duo tanh K,(h + z) + — / — dz = X; d — (u v„) dy dn a d fu \ dx\2 9 - (4.176) represents the resultant surface velocity. Substituting v +w 2 for wdw/dx = 0: + ~dt l + tanh /c(/i + .z)] d r 2 fv 2 a ) ay'Ki: + a dx , 2 „ a p x dw (u i; )tanh K(h + n) + — / s s , (4.186) —-dz = 0. dy J dt z Assuming second and higher degree derivatives of w to be negligible, the integral term in eq. (4.18) can be evaluated quite readily as follows: d dx with dwjdt J P dw z dr) dt dx^ dt) (4.19a) Z=T\ as (Appendix III. 1), dw 1 r (d r) 2 d r]\] 2 , , (4.196) Substituting from eq. (4.19b) into eq. (4.18) finally gives the desired integrated form of the momentum equations as: du h<7 dt dx y 1 — tanh n(h + n) 2 dv s h dx \ 2/ 125 + 1 + tanh K,(h + rj) dx\2 ) (4.20a) - - ( u , ) tanh a + *) + {-) s + x 1 -tanh K(fe + r7)] d (v ^ 2 J^lyJ 2 9t T dy -~(u v ) s 3 2 tanh + „) = X ; r 1 + tanh/s(ft + 17)] a / T J \ +i 2 \dvV2) tanh «(fc + z)+ ( £ ) ^ 2 2 + tanh ( 4 - 2 0 6 ) + „) = 0. The set of equations (4.13) and (4.20) govern the free surface motion in a rectangular damper. The next logical step would be to develop effective models for dispersion and dissipation of the free surface waves. 4.5 Dissipation Model Waves at the free surface of a liquid experience attenuation due to three different energy dissipation mechanisms: (i) damping at the bottom and side walls where the viscosity effects are dominant; (ii) dissipation within the body of the fluid where the viscous effects are not as dominant and the flow may be considered as irrotational; and (iii) dissipation at the free surface due to wave interactions. This section touches upon contributions from each of the above mentioned sources. The approach is based on classical fluid mechanics concepts which are well documented [99, 100, 118]. The main contribution lies in application of known basic principles to the complex situation of energy dissipation through sloshing liquid. Ultimately, the objective is to modify the original nonlinear momentum equations (4.20), derived earlier through incorporation of the energy dissipation terms. To that end, a general expression for energy dissipation in a sloshing liquid is obtained first. It is then applied to assess contributions from the three sources of dissipation mentioned above. For brevity, only more important steps in the analyses are touched upon here. 126 4.5.1 Energy density function This section derives velocity potential, <f>, for a small amplitude wave in order to determine the total energy density in a shallow nutation damper. A small wave solution for the velocity potential, <j>, is needed for a two-dimensional sinusoidal wave propagating along the x axis (Figure 4-1) with the phase velocity, c. Taking the small amplitude wave solution for <j) as 4> = $(z) < u t K X e \ and recognizing from eq. (4.9) that <f>(x, y, z, t) — F(x, y, t) cosh AC(/I + z), it follows that <f> can be written as (j> = C cosh «(/! + *) e^O"*-**), where C is a constant. Applying the kinematic and dynamic free surface boundary conditions in eqs. (4.6a, 4.6b) gives (Appendix III.3) C= ™ /csinh K/V where a = to cosh K,h/g. Thus </>= aw coshAC(/I + z); AC . ; 7 sinh KK , . cos(o;t - KX).- (4.21) Now, the mean kinetic energy per unit volume, T, for a damper filled to height h can be written as 1 ~Fn =Hi™ 127 ) dz. 2 With the shallow water approximation and small amplitude waves I ~2 J T i f® P ( ^) k V 2 { = ^Pw^^^tanh/c/i) -1 where the bar indicates average over the period. Recognizing that in a conservative dynamic system of small wave motion, the mean kinetic energy equals the mean potential energy E = -p(wo) (/ctanh/c/i) . 2 2 (4.22) _1 Hence dE 1 da, — = -pu 2a—(/ctanh/c/i) at 2 dt 2 o , , . i . . . (4.23) As E oc a for small waves, 2 dE da dt dt , — oc 2a— where: a = a e 0 _Q!i (4.24) K ' ; ao is the initial amplitude; and a is the damping parameter. The corresponding general expression for energy dissipation due to fluid viscosity can be written in tensor notation as [118] dE _ _1 ~dt f° (dm 2 . J_ \dxj duj\* dxiJ h 4.5.2 Dissipation at the damper bottom and side walls The energy dissipation from the boundary-layer due to the shear stress exerted on the bottom face of the damper is due to the flow propagating in the x direction and varying in the z direction. In this case, eq. (4.25) reduces to ~dT ~ ~2 J_ \dz' PV h + ~dx~) ' Z \ 128 As the boundary-layer thickness, 6, is much smaller than A 27r /« > £ «dz giving dE ~dt r i (4.26) Introducing the boundary condition w' = 0 at z = — / i 8$ ii = — dx -h sinh K,h sin(rea; — <vt). Now, as the flow is irrotational outside the boundary-layer, u' must tend to 0 at its edge. This requirement can be satisfied quite readily by modifying the expression for u' as u where 6 = y/2vju) ' _ ^ ° _ -e ^ *+(* J) sinf/ca; i ( sinh AC/I = z+/l e 8 n / - ut — 7) is the boundary-layer thickness and 7 = (2 + /i)/(5. Therefore, the integrand in (4.26) can be written as /5u'\ \dz / 1 2 Vsinh/c/7./ . 1 COs(AC2! — U)t + ~(z + h)) + Sm(K2! — £J7J + -(z 0 0 6 2 + h)) giving 1 dE f° (du>\ 2 1 dz — — —I 46 1 = -pv 4 wa 2 5 2 1, -(e sinh nh 0 2h 6 !)• ( ^ — ^ __a 2 I e * \sinhre/i/ ( z + f c ) 0 z=—h (4.26) Equating eq. (4.26) to eq. (4.23) gives, da K ILJU 1 . _2h. = - o - J — . . - . ( 1 - e «)• 2 V 2 sinh2K/i 129 (4.28) Taking 6 « h, da K — = —a dt 2 v 1 2 sinh2/c/i Hence the damping parameter aj due to the bottom surface can be written (4.29) Note, because of the shallow water consideration the present analysis is valid for the boundary-layer thickness much smaller compared to the wavelength A = 2IT/K. Physically this implies that the change of wave amplitude during one cycle of excitation is small. The same approach is applied to the boundary-layers formed on the side walls but scaled by a factor of h/w to obtain damping per unit width. Thus the total damping, a„ due to two side walls can be written as (4.30) 4.5.3 Dissipation within the body of the fluid In the liquid outside the boundary-layers the flow is considered irrotational, i.e. U = Vc£. Applying the continuity relation, eq. (4.25) becomes (4.31) <j) being known (eq. 4.21), the above integral can be evaluated giving dE ~dt pvna u) -2 tanh K,h 2 130 2 As before, comparing the above relation with eq. (4.23) gives i.e. the damping parameter a corresponding to the main volume of the fluid is v a = 2i/« . (4.32) 2 v 4.5.4 Dissipation due to free surface dynamics Evaluation of energy dissipation due to the complex interactions between the surface waves and damper walls is indeed a challenging task. A rather simple approach described here attempts to capture the essential physical features of the process. The objective is to arrive at an expression for the dissipation parameter which can be applied quite readily with a measure of confidence. It is based on the solution of the diffusion equation with appropriate free surface boundary conditions. The dynamic character of the free surface with waves suggests the choice of a body fixed moving coordinate system having the origin attached to the surface with the x' coordinate along the wave and z' orthogonal to it as shown below. Using Undisturbed Free Surface Travelling Wave equation (4.21) giving the potential for a sinusoidal wave propagating along the x— axis with a velocity c, the body fixed moving coordinates at the free surface are 131 defined as: . a cosh K(H + z) . — — sin sinh KH , a sinh K(H + z) — ; COS KX. sinh Kh x = x -\ Z — ZH Note, to the first degree, z' KX; = 0 corresponds to the free surface z = acosKx, while at the damper bottom z' fa z — —h. The Jacobian of the transformation, to the first order, becomes J d(x',z') = — o(x,z) 1+ r - = 2a/ccosh K(h + . , smh/c/i z) 1 , COS KX . Recognizing that the stream function for the irrotational field (major portion of the fluid away from the walls and the free surface) is — cz , where c is the wave speed, 1 the stream function ^ at the surface can be written as * = -cz' + ij}, where ip is the stream function corresponding to the vorticity contribution from the surface layer where the viscous terms in the momentum equation are comparable with the other terms [122]. The thickness of the layer, 6, may be taken to be extremely small compared to the wave amplitude, i.e. one may consider it as a thin sheet generating vorticity. Now the vorticity (Q) equation in the body coordinate system has the form [122] - C M = ^ > ( which is the classical diffusion relation. As az !] = V x (du u = ox dw\ ( a I - a J =*»+ * - 132 4 - 3 3 ) Hence, in the moving coordinate system, SI Since d/6\z' » V > >). = - J(* v + z x x d/dx , 1 n « —jy > >, z z which serves as a boundary condition. Setting w = C K at the surface (z' = 0), vorticity at the surface can be approximated as [122] Q « —Jipz'z' — —2aKU) cos KX 1 at z = 0. The solution to eq. (4.33) satisfying this boundary condition and vanishing as z' jb becomes large, can be written as, /6 Q = -2aKu> e- 'l z z 1 (4.34) —), COS(KX' - o where 6 = ^/2v/u). The average value of the tangential component of the velocity in the surface layer can be obtained quite readily by integrating Q with respect to z' giving ' u = Sana e~ I z s z' COS(KX' — - ) 6 z' + sin(/cx' —-) 6 Note, u' —> 0 as z'/6 becomes large. Velocity fluctuations induced normal to the surface are relatively small and hence can be neglected. Now, contribution to the vorticity comes mainly from the du /dz 1 term, as dw /dx' is relatively very small. 1 1 Using eq. (4.25), the rate of energy dissipation can be written as dE ~dt f° -a I J-h 2 , / f° fdu'\ , , /2J7 Q dz = —fj, I ( ——- ) dz = —p\ —a J- \dz'I V u 2 2 h 133 2 2 K U) , giving, with eq. (4.23), da 2u , — = —au\ — K tanh/c/i, V(j dt and hence the damping parameter due to a clean free surface as I2v a.f — vy—K , . tanhre/i. 3 This contribution is normally small. A question arises concerning energy dissipation in presence of floating particles. The approach to this class of problems would essentially remain the same with an appropriate change in the free surface boundary condition. For example, representing the closely packed floating particles as a film of liquid unable to resist the tangential stress caused by the wave motion, ty — ty 'dx'/dz + ^z'dz'/dz x z — —c. Therefore, _ , dz 1 *z' = -c / for — dz ip > « x ip >. z Retaining only the first degree term in the series expansion and removing the constant reference frame velocity, this reduces to [122] cos KX' V / = aKC-tanh Kh z The solution to eq. (4.33) with this boundary condition gives the tangential velocity in the boundary-layer as v! = u)ae~ / z z' coth Khcos(/cas' — - ) . o One can now obtain the dissipation parameter ct to account for the presence of p particles using the same procedure as before. dE /° = —fj, I dt J_ h 2 J , f° (du'\ 2 U dz = —a I , a 2 I —— I dz = J-h^-oz'J 134 J 26 2 .T2 l -/^w coth Kh, giving, with eq. (4.23), da dt VK = —a- 26tanhAc/i' hence ' = [uv 1 2 V T t ^ h ^ K a ( 4 " 3 5 ) Thus the total damping parameter based on energy dissipation from the three sources (neglecting the term due to the clean free surface) takes the form a = (a + a ) + a + a / , b \2\ s v 2 sinh2Ac/i + * w2 \ 2 sinh2/wJ + UK + 2 V 2 tanhre/i' = |Yl + 2—) + cosh/c/i]-AI . / — + 2iJi//c . IA m / J 2 V 2 sinh2K/i U 2 r It may be pointed out that cu appearing in the above equation can be determined from the dispersion relation to be obtained in Section 4.6. Several researchers have attempted to account for the free surface "cleanliness" by incorporating an empirical parameter refer to as the "contamination factor" (5). One can readily accomodate such an approach by introducing the parameter as indicated below, K / fa~ KV 1 h W r + 2i//c. a = 1 + 2w h Scosh Kh -2W W 0 - 1 2 sinh2/Wi 9 . (4.36) v ; 4.5.5 Added mass and reduced liquid damping The force with which liquid impacts the damper walls is referred to as the sloshing force, F . It can be determined by integration of the pressure distribution s on the vertical walls, 135 where A is the wall area and p is the pressure, which can be obtained from the momentum equation d u +v + w dU ~dt 2 + 2 2 dz~( I dp 2 ' (4.14c) p dz The expression for dw/dt as derived in Appendix III. 1 is dw ~dt \dx ^ tanh K,(h + z). (4.19b) dy . 2 K 2 The pressure is a result of two contributions: vertical displacement and acceleration brought about by the presence of surface waves. Note, the vertical acceleration is 180 0 out of phase with the free surface displacement. These contributions modify the purely hydrostatic case. The resultant pressure can be obtained by integrating eq. (4.14c) with respect to z from the nutation damper's bottom to the free surface. Ignoring the effect of local sinusoidal waves at the boundaries, f_ lTz P= dZ H I V = h 2 ^ ) } i.e. ^sh (h+z) (u +v +w )}\ _ , 2 K + 2 2 V z= h dy 2 Recognizing that Mi _ Mi "Vf ~ (2l)bh' 1 Mi dn Albh dx dm 2 2 2 where: Mi — mass of the liquid; Vi — volume of the liquid; X = tanh /c(l + n/h)/tanh Kh; a — tanh Kh/Kh. 136 + dy . 2 \ri + h) 2-x<rh]}, (4.37) This is the sloshing force acting on one of the two damper end walls. The other two walls do not experience any sloshing force due to the two-dimensional character of the wave propagation considered in this analysis. The net force on the damper is the difference between that exerted on the wall at x = —I and x = In further discussions F is taken to be the resultant force acting on the damper. Defining the 3 added mass ratio M /Mi a M\€ u) l e e and reduced damping r) i as: r Alohu)fe < oy 1 \. K Vox 0 J J x=-l z (4.38a) 2Mfeu>j e %lbhu)fe { K Vox 2. 0 2, oy 1 i iJ z x=-l (4.386) 4.5.6 Modified equations of motion The next logical step would be to incorporate the damping parameter in momentum eq. (4.20a) obtained earlier. Furthermore, it would be desirable to nondimensionalize the set of governing eq. (4.13) and (4.20) to make the analysis applicable to a large class of rectangular dampers operating under a wide variety of conditions. The length and time scales were selected as indicated in Table 4-1 to emphasize shallow water consideration of the analysis. Table 4-1 Nondimensionalization of the shallow water equations of motion. quantity horizontal length vertical length variable scale 1/ 2 of damper length quiescent liquid height time time for nondispersive wave to travel distance I velocity non-dispersive wave speed 137 I h c= \/gh/ol Putting aJ = y; I X> = X/l; y = y; l Y' = YIH 2 = - ; a t = t/r ] n = -; n 1 .' = .1; ^ Kh ff= ; x = u* =-; c u = t a n h ^ ( l + ,-) ^ tanh re/i ; - c ft t the nondimensionalized equations of motion can be written as: dr] rdu^ V 5JC dt du dt s dvs\ 5y / dv dx (o_ _ \o"i A (du^ / \ dx dv s dn — - -\ dt r_d_ _ <J\ \.dx a_ 3 _d_ dy 8 rl — (xtanh Kh)' 1 + (xtanh/c/i)^ dx V 2 / + I 2o~i 2(Tl (%tanh Khy dy dvs\ dy) dr]\d r] 5a; dx 2 2 (U V ) S S + 2 2 2 s d rdh dy idx (%tanh KK)^ dx OCTlU 3 2ai 2(71 dy + 1 + (xtanh Khy 1 - (xtanh«;/i) ] d (v? \ + d V dy . 2 1 L + 2 0"! dh dy 1 2 -aiX; d fv dy\2 f) V (4.39c) = 0. Here o\ represents the wave speed corresponding to the wave number K\. It corresponds to the resonant liquid sloshing condition. Note, for x 7^ 1) the equations retain nonlinear contributions consistent with the shallow water analysis. OLT\ is the dissipation parameter while 3 associated with the second order partial derivatives of 7] is the measure of dissipation. The details are presented in Appendix III.2. 138 4.6 The Dispersion Relation A linear analysis of the final equations (4.39) would yield the dispersion relation and hence resonant wavelengths [99,100]. Linearizing this set of equations gives: ^ dt ^ dx + ^ L dv + + ( — - l ) ( ^ L \<7i J \ dx du + (440) ^ L ) Q . dy J ' = dn 3 dv dr) + IdyT = ° ; 3 dt ( 4 4 2 ) where: tanh Kh a = Kh ' and <Ti = resonant wave speed. Take the potential function as (j> = /(a;, y)e cosh /c(/i + z), lu with the time dependence decoupled. Integrating eq. (4.41) and substituting for u 3 in terms of <f> determines the wave height 77 as 77 = — / ^ J dt dx — — ( uoe ^J dx %wt cosh K(h + z) dx + cosh = -iue f iwi + z) + K(y), K(y). Differentiating with respect to time results in cosh K(h + z). = u) e f 2 dt lu>t Substituting this into eq. (4.40) leads to df df 2 0-1 dx 2 z dy z <J 139 , , The potential function must also satisfy the Laplace equation, S 0 + ^ + = O. (4.44) Comparing eqs. (4.43) and (4.44) gives the desired relation for the surface waves as a u = A I—K. V °"i (4.45) Taking the amplitude A in the form f = A sin KX and applying the boundary condition u(l, t) = u(—I, t) = 0, i.e. u(x = ±2) = -^-e ox xwi cosh K(K + z) = (AK cos KI) e lwi cosh AC(/I + z), the nontrivial solution condition requires l T Kl = TI — , 2 i.e. K* = n - , (4.46) where n is a positive integer. The fundamental resonant sloshing, i.e. n = 1, is expected to result in the maximum dissipation. The next task is to model dissipation in the boundary- layers at walls, the irrotational body of the liquid, as well as the free surface and incorporate them into the equations of motion. 140 4.7 Numerical Solution for Sloshing Liquid Equations 4.7.1 Two-dimensional equations at resonance It is important to emphasize that the governing equations of motion are threedimensional in character, however averaged over liquid height in the z-direction. The assumption of two-dimensionality was used only during the evaluation of dissipation. The equations are highly nonlinear and coupled, hence one is forced to resort to a numerical procedure for their integration. The computational time and effort involved for such a complex system would be, indeed, quite demanding. The challenge faced was to capture the essential physics of the phenomena, through judicious simplifications, using modest computational tools (workstation). To that end, it was decided to consider the wave motion as essentially two-dimensional, i.e. along the x- axis. This corresponds with the model used during the dissipation analysis. Furthermore, preliminary tests suggested stronger wavebreaking during such two-dimensional motion. Putting d/dy = 0 and v = 0, eqs. (4.39a) and (4.39b) governing the free s surface dynamics are reduced to: du s dn M 8P at where: M(r/)= [ l - ( d ox , a\ ox tanh «i/3) ]/or; N(r,)=a 6 ; 2 X •• 2 X P(u) = ^f; Q(u) = For the resonance condition: 7r «i = - ; tanh/ci/3 tanh 0-1 = — - — — — ; (ACI/3) ' 141 xi Ki/3(1 + tanh/ci/3 77) (^V therefore: (4.48a) dn •i d , . + u dn , 3 + TJ* = °- < 4 4 8 I > A finite difference scheme was selected for solution of the above set of equations [123, 124]. The tank width 2/ was discretized using the nodes at the free surface as indicated in Figure 4-2. A different grid was used for each of the dependent variables u and 77. The grid for u is staggered, in a non-overlapping fashion, downstream relative to the grid for 77. Such arrangement avoids unrealistic periodic solutions that also satisfy the equations of motion. The x-raomentum equation (4.48a) is integrated over the grid elements from i — 1 to i. Eq. (4.48b) is integrated over the grid elements i to i +1. Therefore M, N, x and u at even nodes are integrated from z— 1 to i. On the other hand, P, Q and 77 are integrated over the odd nodes in order to satisfy the boundary conditions and to prevent index i from becoming less than zero. The integration step Aa; associated with odd and even nodes is staggered by the length Ax/2 as is normal. Thus the free surface is discretized into a uniform difference mesh (Eulerian grid) consisting of cells with width Ax. The resulting finite difference equations have the form: ^- = -T-im-i ~Vi + Mi(Pi-i - Pi) + Ni(Qi-i - Qi)} -trXat I\x at , lUi i = [1, n]; (4.49a) CLTI " 1 1 Ui' -TT = -^(XiUi - Xi+iUi+i) + — -^(vi-Vi+i), at Ax o~\ I\x i = [1,71-1]; (4.496) with the boundary conditions: ^ dt = -^-xiiti; Ax at =-T-XnU ; Ax n for i = 0 and n, respectively. 142 (4.49c) free surface (b) X = -1 x = Ni X N M t U U 1 M N| Xj i u M M„ N N i+1 i+1 " i+1 i+1 Po Q n Q Pi Q 5 p Q n 5C n u ^•1 +1 u / i+1 1 1+1 / i+1 / n ^n-1 Tin Pn-1 Qn-1 Pn Qn free surface Figure 4-2 Modelling nonlinear sloshing in a rectangular nutation damper: (a) variable definition; (b) discretization over the solution domain. • 143 Here Xi, Mi, Ni, Pi, Qi, rji and ii; are taken to vary linearly over Aa;: Xi — t a n h K\B[1 + (?7i_i + 77;)/2] tanh K\B; Mi = [ l - ( x i t a n h K i / 3 ) N{ = a 8 2 ] / c r ; : i = [l,n]; 2 Xi (4.50a) Pi = [(ui + u i)/2] /2, 2 i+ Qi = [(r/i+i - 7 _ )/(2Ax)] /2 : i = [1, n - 1]. 2 7i 1 Note, at i = 0, n P 0 = Pn = 0; Qo = [ ( - 3 7 7 + 4771 0 (4.50b) m)/(2Ax)] /2, 2 Qn = [(Vn-2 ~ 477n-l + 3 7 7 „ ) / ( 2 A z ) ] / 2 . 2 Taking the initial surface condition as 77(0) = 0 , ii(0) = 0 and with a given excitation force, the above equations were integrated simultaneously to yield the time dependent surface displacements 77^ and velocities Ui. With the time dependent variation of the surface established, added mass and reduced liquid damping can be calculated quite readily. It may be pointed out that the system under consideration is not stiff. A hybrid Adams-Bashforth/Runge Kutta-Gill procedure was selected for integration. The Gill method controls the growth of round-off errors and thus minimizes the truncation error. This enabled the capture of steep slopes at the surface near the impact. The program was written in the C language and executed on a Sun Workstation. The iteration process, in general, converged quite rapidly. A typical case required around 0 . 3 second (~ 15 interation cycles) to give the transient solution at a given instant of time. Typically 1.5 hours were required to obtain the steady state solution. Figure 4-3 summarizes important stages in the derivation of the equations of motion describing the free surface dynamics. The associated equation numbers are also indicated. 144 kinematic free surface b.c.'s (4.6a); dynamic free surface b.c.'s (4.6b); wall b.c.'s (4.7, 4.8) determine expressions for u,v,w at the free surface (4.11) find.expressions for U , w3w and w3w (continuity eqn.) dz (4.13) s 3x 1 p = | (z-momentum eqn) dz 3y (4.12) derive expressions for (4-15) f dz # - - ' ^ d z and 3x I 3t 3y * z Take 3. and 3. of eq. (4.15) 3x 3y J* 7 (4.19) T (4.16) substitute eqs. (4.12) and (4.19) into eq. (4.18) ' (4.20). substitute eq. (4.16) into x,y eqns. of motion, set z=n. and integrate ( ) derive small wave expression for total energy density (4.31) 4 1 7 obtain dispersion relation from linearized governing eqs. and determine resonance wave number ^4 27) compare with dissipation due to viscosity . (4.32) determine dissipation due to boundaries (bottom a n d side walls) (4.35,4.36) model dissipation nondimensionalize free surface eqs. of motion and introduce dissipation (4.45,4.46) derive reduced liquid damping and added mass * • discretize free surface equations of motion (4 49) determine dissipation in main volume of fluid (4.39) determine dissipation due to free surface boundarylayer ( 4 4 1 ) determine total dissipation (4.42) Figure 4-3 Flow chart illustrating procedure to derive the nonlinear shallow water sloshing equations for the nutation damper. The number in the parenthesis refer to the equation resulting from executing the step indicated in the box. 145 4.8 Fluid-Structure Interaction Dynamics The next step would be to incorporate the damper model in the structural dynamics study. The conjugate character of the system dynamics should be recognized. Motion of the structure acts as an excitation for the damper thus affecting the sloshing motion; while the free surface dynamics, in turn, governs the damping and hence response of the structure. Consider a cantilevered beam-type structure undergoing vortex induced motion normal to the flow (Figure 4-4) represented by: m 'x + c x 3 3 + kx = F s 3 + F. e Here: m , c , k = mass, damping and stiffness of the structure, respectively 3 3 3 x t = displacement of the structure normal to the flow; 3 F = aerodynamic excitation force transverse to the flow; e F = sloshing force. 3 Representing the aerodynamic force on the structure as: = C { -pV ){HL ) l L where: Ci = lift coefficient; V = wind speed; H = width of the structure; L m = length of the structure. 146 2 m d a m p e r H __7C 1 L I m AA/vV% s p n r ig ari b e a n r ig s w e g ih t b e a n r ig u tb e Figure 4-4 Schematic diagram showing the test arrangement for the fluid. . structure interaction model. 147 It is convenient to write the equations of motion in a nondimensional form using the definitions as follows: /*T x 4 st x = —i T = \ t H / , =tU} , .f ; Wnv = 2m u) 3 X „ v c Qs = ~ n \ rn s c s n u = ( M n \j HJ ]! pH L 2 m ^4 ~ 2 = ^72 ; 87r m St* z n 3 where: x t = nondimensional structural displacement; 3 ( = structureal damping ratio; 3 u} = undamped natural frequency, \Jh jm 3 n 3 = 27r/ ; n = dimensionless vortex shedding frequency; giving x + 2( x + x = F * + F *. 3t 3 3t 3t 3 (4.51) e Here: F * = nondimensional sloshing force, eq. (4.37) /m Hu} ; 2 3 3 n F * = nondimensional wind induced excitation force, C ^ u ^ C t . e Now the sloshing force F * can be determined quite readily from eq. (4.37) derived 3 earlier. Thus it remains to determine the transient force F * that can effectively e represent time dependent aerodynamic excitation during dynamic response of the structure. This is accomplished through a relatively simple model proposed by Hartlen and Currie [101]. Accounting for the fact that: (i) the vortex shedding frequency is proportional to the wind speed, i.e. the Strouhal number remains essentially constant; (ii) the system has self-exciting and self-limiting (limit cycle) type of response; the lift variation is characterized through an equation C - Au C L x nv L + — (C f L W O T 148 + u> C = bx , 2 nv L 3t (4.52) where A\, A 3 and b are constants with values based on information compiled over years [101]: = 0.02; Ai b= 0.4; A = 4A /(3a4 0.2 ) = 0.67. 2 3 1 ; The analysis is confined to the subcritical range of the Reynolds number with the solution sought in the range of oj nv « 1.0 (i.e. vortex resonance case). The set of governing eqs. (4.51) and (4.52) were solved simultaneously using an ANSI C compiler on a SUN SPARC 2 Workstation. The free surface wave height as well as velocity over the entire domain were computed over 40 oscillation cycles. Appropriate time step-size was taken to assure convergence of the solution to a stable value. In general, a step-size in the range of 1/60 — 1/120 of the oscillation cycle resulted in the acceptable accuracy. Important system parameters were varied as indicated in Table 4-2. Furthermore, for the fluid-structure interaction investigation, the damper to structure mass ratio was varied from 1-4%. Figure 4-5 presents a flow-chart showing important steps involved in the numerical algorithm. Table 4-2 The cases studied numerically for steady forcing from rest. Forcing Frequency 0.90, 0.95, 0.99, 1.0, 1.01, 1.05, 1.1 Liquid Height h/l 0.1, 0.2, 0.3 Forcing Amplitude te/l 0.0025, 0.005, 0.01, 0.077 4.9 Results and Discussion The amount of information obtained through a systematic variation of parameters is rather extensive. For conciseness only some typical results useful in establishing trends are presented here. The effect of variation of the excitation frequency on the surface wave near resonance is shown in Figures 4-6 to 4-10. Both the numerical results as well as their 149 start input h, (Dg, E ,\); e damper dimensions; boundary, conditions set initial conditions and .trial time-step At / determine dissipation from initial values, boundary conditions and input parameters integrate damper free surface velocity and liquid height vectors over the interval integrate damper free surface velocity and liquid height vectors to the interval midpoint as well as to interval end calculate damper base shear force determine excitation on elastically mounted structure eqs. (4.51, 4.52) determine the time step-size for the next integration using the adaptive step-size driver save time integrated surface velocity and free surface liquid height vectors no set trial time interval to At / 2 integrate structural equations of motion save time-step value used save structure and damper fluid response data no Figure 4-5 Solution algorithm for implementing nonlinear shallow water sloshing in a rectangular nutation damper. 150 0.2 1 = 0.0, 1.0 i = 0:5 0.0 -0.2 x=l x=-l 1 t = 0.6 = 0.1 x = 0.2 1 = 0.7 i = 0.3 1 = 0.8 i = 0.4 1 = 0.9 Figure 4-6 Time history of the free surface profile for a rectangular damper subjected to an excitation frequency w — 0.90: (a) numerical results based on shallow water model; (b) animation of the numerical data. Note the-standing wave character of the free" surface. e 151 00 Figure 4-6 Time history of the free surface profile for a rectangular damper subjected to an excitation frequency w = 0.90: (a) numerical results based on shallow water model; (b) animation of the numerical data. Note the standing wave character of the free surface. e 152 x=l i = 0.2 1 = 0.3 1 = 0.4 1 = 0.7 1 1 = 0.8 = 0.9 Figure 4-7 Free surface motion over a period at <2> = 0.95: (a) numerical . results based on shallow water model; (b) animation of the numerical data. The distinctive feature is the presence of a wave train. • e 153 T = 0.0, 1.0 00 , = 0.5 Figure 4-7 Free surface motion over a period at u> = 0.95: (a) numerical results based on shallow water model; (b) animation of the numerical data. The distinctive feature is the presence of a wave train. e 154 (a) Figure 4-8 Free surface dynamics at resonance: (a) numerical results based on shallow water model; (b) animation of the numerical data. The train with two waves is apparent. Note larger wave amplitude at the wall (r = 0.5). 155 Figure 4-8 Free surface dynamics at resonance: (a) numerical results based on shallow water model; (b) animation of the numerical data. The train with two waves is apparent. Note larger wave amplitude at the wall (r — 0.5). 156 (a) 0.500 X=-l 0.000 0.500 0.000 0.500 11 0.000 0.500 0.000 0.500 0.000 Figure 4-9 Time histories of wave height at five spanwise locations of the rectangular damper at resonance. 157 " 4 0 0 5 10 15 x 20 25 Figure 4-10 The linear shallow water solution for a rectangular nutation damper at resonance clearly showing its limitation. Note, it is unable to give the propagating wave solution. The sloshing force also shows discrepancy. 158 30 animated counterparts are shown for better appreciation of the wave dynamics. For Cj = 0.9 the presence of standing waves with rocking motion of the free surface is e apparent (Figure 4-6). The flow in this mode does not dissipate energy well as evidenced by small sloshing forces observed in Chapter 2. With a small increase in the excitation frequency (u> = 0.95) progressive waves appear as anticipated e through the flow visualization information represented earlier (Figure 2-20). A train with as many as three waves were observed (r = 0.4). Note the amplitude at the wall shows some increase compared to that for the standing wave condition. This, in turn, results in a small increase in the damping. This trend is further accentuated at the resonance (Figure 4-8, r = 0.5) with two distinct surface waves. As can be expected, the damping now is near maximum as observed earlier through the experimental data (Figure 2-19). Time histories of the wave height at resonance for five different spanwise stations along the length of the rectangular damper are shown in Figure 4-9. It clearly reveals the cnoidal character typical of shallow water flows. Note, as expected the time histories at stations located symmetrically with respect to the centerline are similar. It is of interest to point out that corresponding results for a rectangular lake, obtained through the variational method by Lepelletier and Raichlen [91] compared quite favourably with the present data. The strength of the present nonlinear shallow water analysis can be better appreciated by recognizing the misleading results given by the linear model. The linearized solution can be obtained quite readily by setting % = 1 and M = N = 0. The results are presented in Figure 4-10. The time history of the sloshing force F , a is also indicated. Comparison with Figure 4-8 clearly shows the discrepancy. The linear analysis arrives at standing waves, i.e. it is unable to predict the presence of progressive waves such as the wave trains. Note, the sloshing force is also in error. Thus the use of linear analysis, particularly at and near resonance, as has been the case until recently, can lead to misleading conclusions. 159 The experimental results in Figure 2-19 showed the peak dissipation at ui e slightly beyond the resonance value of & = 1.0. The flow visualization study also e showed stronger wavebreaking with a single wave at u> fa 1.05 (Figure 2-21). With e this as background, the numerical simulation of the free surface dynamics was also studied at £) — 1.05. With an increase in u> beyond resonance the strength of the e e second wave progressively diminished, and at ui = 1.05 it disappears completely. e This results in significantly larger wave amplitudes at the wall (Figure 4-11, r = 0. 2, 0.6) leading to higher values of damping. Note, an increase in the sloshing force compared to that observed in Figure 4-8. Experimentally, the effect of increasing liquid height is to increase the sloshing resonance frequency and subsequently, delay the appearance of the free surface oscillation modes to higher frequencies. This is supported by the model. As liquid height is increased, the wave forms become smoother and closer to sinusoidal. Physically, the liquid is less shallow and the wave profiles lose their cnoidal character. Wavebreaking is harder to induce unless oscillation amplitudes are large. Consequently, the reduced liquid damping decreases. This also agrees with experiments. Figure 4-12 presents variation of the reduced damping with liquid height as obtained using the numerical procedure. Experimental data are also presented to assess accuracy of the numerical analysis. Considering the complex character of the sloshing dynamics and relatively simple character of the numerical model, the agreement is surprisingly good, particularly for h/w > 0.2. This suggests that wave dynamics and viscosity, which are accounted for in the numerical analysis, are the major parameters contributing to the energy dissipation. Note the numerical results are conservative compared to the experimental data at lower liquid heights suggesting a change in relative contributions from different sources. It appears that at a smaller h/w, the energy dissipation is essentially due to wavebreaking, 1. e. viscous dissipation at the walls and from the irrotational body of the liquid are relatively small. On the other hand, the present analysis does not account for 160 400 s 0 -400 10 15 x 20 25 Figure 4-11 The nonlinear shallow, water numerical solution for a rectangular nutation damper oscillated at a frequency u) = 1.05: (a) numerical results based on shallow water model; (b) animation of the numerical data. Note the train with a single wave and higher amplitude at the wall. The sloshing force is also higher. e 161 30 x = 0.5 x = 0.6 0.2 T = 0.7 = 0.3 x = 0.8 0.1 x = x = x x x = 0.4 = 0.9 Figure 4-11 The nonlinear shallow water numerical solution for a rectangular nutation damper oscillated at a frequency u> = 1.05: (a) numerical results based on shallow water model; (b) animation of the numerical data. Note the train with a single wave and higher amplitude at the wall. The sloshing force is also higher. e 162 1.50 I 1 1 1 r i i i i i i i i i i •— > co = 1.050), —o— co = 0.95 co, — v — numerical result e / w = 0.05 e e 1.25 e 1.00 0.75 0.50 0.0 i 0.1 i i • 0.2 h /w 0.3 i 0.4 0.5 Figure 4-12 Experimental results for a rectangular cross-section box damper showing the effect of liquid height and excitation frequency on the damping ratio. Note V refers to the numerical results corresponding to resonance. Correlation with experimental results is good. the impact dynamics of the wave striking the damper wall. However, during the experiment, the strain gauge transducer is able to record the sloshing force rather accurately. A comment concerning the effect of liquid height and excitation amplitude as given by the numerical analysis would be appropriate. Experimentally the influence of an increase in the liquid height is to raise the sloshing resonance frequency thus delaying appearance of the various free surface oscillation modes to higher frequencies. The numerical analysis substantiated this observation. An increase in the liquid height was found to reduce cnoidal character of the waves, i.e. the waveform becomes smoother, thus rendering the waves closer to sinusoidal. Furthermore, at a larger liquid height, it was more difficult to induce wavebreaking unless the oscillation amplitude was quite large (e /w > 0.5). Thus for a small ame plitude excitation (e /w < 0.1), the damping showed a significant reduction with an e increase in the liquid height. In general, the effect of an increase in the excitation amplitude at resonance was found to improve dissipation in the range considered (e /w = 0.0045 - 0.14). e It would be of interest to show some typical cases of surface waves as obtained numerically and their animation, and compare them with the free surface character observed during the flow visualization study. In a sense, this would help in assessing the accuracy of the numerical model. This is shown in Figure 4-13 for two characteristic surface conditions associated with a progressive increase in frequency: propagating wave train (approximately three waves); and a single wave. Further increase in the frequency lead to the recurrence of the standing wave. Next the attention was directed towards assessment of the numerical model in predicting response of a cantilevered structure, undergoing vortex induced oscillations, in presence of a rectangular nutation damper. The model support system is schematically shown in Figure 4-4. As mentioned earlier in Section 3.2, the model is supported by a system of four air bearing with a controlled stiffness provided by 164 Numerical Animation Flow Visualization Figure 4-13 A comparison between the numerical model, its animation and flow visualization results for a rectangular damper free surface showing: (a) wave train; and (b) single wave. 165 Numerical Animation Flow Visualization Figure 4-13 A comparison between the numerical model, its animation and flow visualization results for a rectangular damper free surface showing: (a) wave train; and (b) single wave. 166 a set of four springs. The amplitude of the model displacement, x t, was computed s over 30 cycles and its rms value determined. Figure 4-14 presents typical time histories of the model response and the liquid wave height at the wall (r) ) for the 0 excitation condition just beyond resonance. The damper parameters are indicated in the figure. It is encouraging to note that the numerical model is able to predict the expected beat response which was also observed during the experimental study described in Chapter 3. Note the cnoidal character of the sloshing liquid. Equally encouraging is the fact that the wave frequency is identical to the structural frequency, as it should be. Furthermore, the wave dynamics accurately reproduces the beat character of the structure with the same period. Thus the numerical model appears to capture the physical character of the response rather well. Numerically calculated response of the structure with a systematic increase • in the wind speed is shown in Figure 4-15(a), for both the conditions of with and without a rectangular damper. The damper to structure mass ratio is 2% with the structural frequency / „ = 1.05 Hz. The numerical model is able to capture the classical resonance phenomena quite well. The reduction in amplitude by over 85% at resonance clearly shows the nutation damper to be effective. Of course, it would be desireable to compare the numerical prediction with the experimental data. However, as pointed out in Chapter 3 (p. 114), that would require correction of the wind tunnel results for blockage. This is a difficult task as no reliable analysis applicable to an oscillating bluff body is available. Under this situation, the results were corrected using a rather approximate approach proposed by Maskell [121], C = C (l + 1.95Aby Le L where Cj is the lift coefficient and A is the blockage ratio (projected area / testJc b section area) of the stationary model. The relation was obtained using a simple momentum balance in the planes upstream and downstream of the body. The oscillating character of the model further complicates the problem as now the effective 167 Figure 4-14 Typical plots illustrating interaction between the damper fluid and structure during the vortex resonance: (a) fluid wave height at the damper wall; (b) displacement of the structure, x t, with the damper. The undamped peak value of x was found to be 1.0. Modulated character of the x t signal agrees well with the experimental observations reported in Chapter 3. 3 3t a 1.00 I i i ' '. | I . . . | i i i i | i i i i | i i i i i | Y(t)/ 0.00 1 2.0 ' ' ' ' ' 2.5 1 ' ' J ' ' 3.0 1 ' ' ' 1 ' ' 3.5 ' ' 1 4.0 Figure 4-15 Comparison between experimental measurements and numerically obtained structural displacement Xgt/H, with massratio M / / m , near vortex resonance (u) = 1.05): (a) Mi/m = 0.022; .(b) Mi/m = 0.04. The agreement is acceptable. s 3 e 3 1.00 i i i i i i i i i i i i i i i i i i undamped response Expt. Num. i (b) - -4- —P— damped response 0.75 Y(t) A H - Expt. - -o- - Num. —V— h / w = 0.12; T) = 2.78 ra f = 1.05Hz; n M,/m = 0.04 s 0.50 0.25 0.00 i i 2.0 2.5 U 3.0 3.5 4.0 Figure 4-15 Comparison between experimental measurements and numerically obtained structural displacement x t/H, with mass ratio Mi/m , near vortex resonance (cD = 1.05): (a) M\jtn, = 0.022; (b) M\/m = 0.04. The agreement is acceptable. a a 3 e a blockage would be higher. In the present study the blockage based on the stationary model was around 11%. Considering complex character of the fluid-structure phenomenon, simplicity of the numerical model, and approximate nature of Maskell's correction procedure, the correlation may be considered rather good. A similar plot for a 4% mass ratio is shown in Figure 4-15(b). The correlation continues to be acceptable for all practical engineering applications. This is a useful development as now we have a numerical tool to design a nutation damper for real-life applications. 171 5. C L O S I N G COMMENTS 5.1 Concluding Remarks Based on nutation damping experiments, numerical analysis and flow visualization, the study lays a firm foundation for the design of this class of dissipative devices. The focus of the study has been on understanding, at the fundamental level, energy dissipation through sloshing liquid. To that end four distinctly different but complementing approaches were used: (i) Scotch-Yoke shaking table based vibration tests to assess the effect of damper parameters on the system's energy dissipation; (ii) flow visualization of the surface wave character as affected by the damper parameters as well as excitation frequency and amplitude; (iii) wind tunnel tests to establish dampers' effectiveness in suppressing vortex resonance and galloping type of instabilities; (iv) development of numerical algorithms for prediction of the sloshing modes, their animation, energy dissipation, and fluid-structure interaction dynamics in the presence of nutation damping. The comprehensive investigation provides methodology for the design of a large class of nutation dampers which should prove useful to practising engineers. Each of the above mentioned phases contain innovation and present contributions not reported before. Some of the more important results based on the study and their signifigance can be summarized as follows: (i) Among the three fundamentally different damper geometries studied, the circular cylindrical damper promises to have the highest energy dissipation efficiency, perhaps due to the presence of the combined propagating and circumferential waves near resonance. The broadband frequency response makes it all the more attractive. 172 (ii) Damping can be improved through a suitable choice of floating particles. In the present study, improvement in the reduced damping was as large as 32 % for the circular damper with a particle density of 50 % and an aspect ratio of 1.0. In general, the floating particle geometry and concentration that does not inhibit the free surface motion improve the damping. (iii) For a given amount of liquid, energy dissipation can be improved through optimal partitioning of a rectangular damper, in the direction of the wave motion. In the present study with a rectangular damper of 91.4 x 10.2 cm, dividing it into five smaller rectangular dampers of 17.9 x 15.5 cm (aspect ratio 1.2) resulted in the increase in damping by anywhere from two to fivefold . As can be expected, optimal partitioning condition is directly related to the desirable free surface motion. In general, partitioning of a toroidal damper results in a broadband but slighly reduced level of damping. (iv) Flow visualization study provided better physical appreciation of the free surface dynamics leading to the efficient dissipation of energy. In general, breaking of waves leads to higher damping. (v) All the three damper geometries proved to be successful in damping both vortex resonance and galloping type of instabilities. They were also quite effective in damping the instabilities when a structure is located in the wake of another structure. In general, the circular damper performed the best. (vi) The numerical simulation, based on the nonlinear dissipative as well as dispersive shallow water model, is able to predict the free surface oscillations and reduced damping quite well considering the complex character of the problem. Furthermore, it proved to be successful in predicting vortex resonance response, a challenging conjugate problem. 173 5.2 Recommendation for Future Work The investigation reported here, though comprehensive in character, presents several avenues for extension which are likely to add to its effectiveness and versatility. A few of them are listed below: • The numerical model developed here is for a damper with rectangular geometry. Although the procedure would remain essentially the same except for minor differences in the boundary conditions, it would be useful to arrive at the corresponding algorithm for a circular geometry damper. • The use of more sophisticated fluid-structure interaction model would make the computational effort more demanding. However, it would also represent a desirable development. To that end extension of the shallow water theory would be the first step. • For a rectangular damper, it would be useful to explore several attractive partition geometries which promise to promote wavebreaking (Figure 5-la). • Preliminary tests suggest that judiciously selected geometry and location of "speed bumps" can induce higher intensity wave collisions at the wall. This aspect needs to be explored more thoroughly to arrive at the optimal configuration for peak damping (Figure 5-lb). • Combinations of damper geometries leading to broad-band response without signifigantly reducing the damping level should prove to be an attractive development from practical consideration. • Initial study indicates that introduction of a pendulum, suitably designed to excite the free surface, can improve damping. A carefully planned study to validate this concept is likely to be rewarding (Figure 5-lc). 174 speed bump (c) pendulum Figure 5-1 Proposed novel damper geometries which may lead to increase dissipation: (a) curved partitions; (b) speed bumps; (c) peridu. lum. 175 REFERENCES [1] Goto, T., "Studies on Wind-Induced Motion of Tall Buildings Based on Occupants' Reaction," Journal of Wind Engineering and Industrial Aerodynamics, Vol. 13, 1983, pp. 241-252. [2] 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. [3] Sturm, R.G., "Vibration of Cables and Dampers," Electrical Engineering, Vol. 55, 1936, pp. 637-688. [4] Scanlan, R.H., "On the State of Stability Considerations for Suspension-Span Bridges under Wind," Practical Experiences with Flow- Induced Vibrations, Springer - Verlag, New York, pp. 598-618. [5] Blevins, R.D., Flow Induced Vibrations, 1990, pp. 54-82. Van Nostrand Reinhold, New York, [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] 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, London, U.K., Report 335, 1957. [8] Wong, H.Y., and Cox, R.N., "The Suppression of Vortex Induced Oscillations on Circular Cylinders by Aerodynamic Devices," Proceedings of the 4ih 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. [9] Dutton, R., and Isyumov, N., "Reduction of Tall Building Motion by Aerodynamic Treatments," Journal of Wind Engineering and Industrial Aerodynamics, Vol. 36, 1990, pp. 739-747. [10] Tanaka, G.H., and Davenport, A.G., "Wind-Induced Response of Golden Gate Bridge," Transaction of the ASCE, Journal of Engineering Mechanics Division, Vol. 106, 1983, pp. 296-312. 176 [11] Ruscheweyh, H., "Straked In-line Steel Stacks with Low Mass Damping Parameter," Proceedings of the 4 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. 195-204. th [12] Vickery, B.J., Isyumov, N., and Davenport, A.G., "The Role of Damping, Mass and Stiffness in the Reduction of Wind Effects on Structures," Journal of Wind Engineering and Industrial Aerodynamics, Vol. 11, 1983, pp. 153-166. [13] Den Hartog, J.P., Mechanical 305-309. Vibrations, McGraw-Hill, New York, 1956, pp. [14] Gasparini, D.A., Curry, L.W., and Debchaudhury, A., "A Passive Viscoeleastic System for Increasing the Damping of Buildings," Proceedings of the ^th Colloquium on Industrial Aerodynamics, Editors: C. Kramer, H.J. Gerhardt, H. Ruscheweyh, and G. Hirsch, Published by Fluid Mechanics Laboratory, Department of Aeronautics, Fachhochschule, Aachen, Germany, 1980, Part 2, pp. 257-266. [15] Bapat, C.N., and Sankar, S., "Single Unit Impact Damper in Free and Forced Vibration," Journal of Wind Engineering and Industrial Aerodynamics, Vol. 99, 1985, pp. 85-94. [16] Reed, W.H., "Hanging-Chain Impact Dampers: A Simple Method for Damping Tall Flexible Structures," Proceedings of the International Seminar - Wind Effects on Structures, University of Toronto Press, Ontario, Canada, 1968, Vol. 2, pp. 287-320. [17] Ogawa, K., Sakai, Y., and Sakai, F., "Control of Wind-Induced Vibrations Using an Impact Mass Damper," Journal of Wind Engineering and Industrial Aerodynamics, Vol. 41-44, 1992, pp. 1881 - 1882. [18] McNamara, R.J., "Tuned Mass Dampers for Buildings," ASCE Structural Division, Vol. 103, 1977, pp. 1785-1798. Journal of the [19] Wardlaw, R.L., and Cooper, K.R., "Dynamic Vibration Absorbers for Suppressing Wind-Induced Oscillations," Proceedings of the 3rd Colloquium on Industrial Aerodynamics, Editors: C. Kramer and H.J. Gerhardt, Fluid Mechanics Laboratory, Department of Aeronautics, Fachhochschale, Aachen, Part 2, 1978, pp. 205-220. [20] Hirsch, G., "Control of Wind-Induced Vibrations of Civil Engineering Structures," Proceedings of the 4th Colloquium on Industrial Aerodynamics, Editors: C. Kramer, H.J. Gerhardt, H. Ruscheweyh, and G. Hirsch, Published by Fluid 177 Mechanics Laboratory, Department of Aeronautics, Fachhochschule, Aachen, Germany, 1978, part 2, pp. 237-256. [21] Tanaka, H., and Mak, C.Y., "Effect of Tuned Mass Dampers on Wind Induced Response of Tall Buildings," Journal of Wind Engineering and Industrial Aerodynamics, Vol. 14, 1983, pp. 357-368. [22] Kawaguchi, A., Teramura, A., and Omote, Y., "Time History Response of a Tall Building with a Tuned Mass Damper under Wind Force," Journal of Wind Engineering and Industrial Aerodynamics, Vol. 41-44, 1992, pp. 1949 - 1960. [23] Ueda, T., Nakagaki, R., and Koshida, K., "Suppression of Wind-Induced Vibration by Dynamic Dampers in Tower-Like Structures," Journal of Wind Engineering and Industrial Aerodynamics, Vol. 41-44, 1992, pp. 1907 - 1918. [24] Healey, M.D., "Semi-Active Control of Wind-Induced Oscillations in Structures," M.A.Sc. Thesis, University of Illinois, Urbana, 1983. [25] Wagner, H., Ramamurti, V., and Sastry, R.V., "Dynamics of Stockbridge Dampers," Journal of Sound and Vibration, Vol. 30, 1973, pp. 207-220. [26] 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. [27] Dhotarad, M.S., Ganesan, R., and Rao, V.B.A., "Transmission Line Vibrations," Journal of Sound and Vibration, Vol. 60, 1978, pp. 217-237. [28] 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. [29] Rowbottom, M.D., "The Optimization of Mechanical Dampers to Control SelfExcited Galloping Oscillations," Journal of Sound and Vibration, Vol. 75, 1981, pp. 559-576. [30] Every, M.J., King, R., and Weaver, D.S., "Vortex Excited Vibrations of Cylinders and Cables and their Suppression," Ocean Engineering, Vol. 9, 1982, pp. 135— 157. [31] Hunt, J.B., "Applications of Rectilinear Passive Dynamic Vibration Absorber," Dynamic Vibration Absorbers, Garden City Press, Great Britain, 1979, Chapter 4, pp. 72-83. 178 [32] Bauer, H.F., "Oscillations of Immiscible Liquid in a Rectangular Container: A New Damper for Excited Structures," Journal of Sound and Vibration, Vol. 110, No. 11, 1984, pp.1645-1649. [33] 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. [34] Welt, F., "A Parametric Study of Nutation Dampers", M.A.Sc. Thesis, University of British Columbia, Vancouver, Canada, 1983. [35] Modi, V.J., and Welt, F., "Nutation Dampers and Suppression of Wind Induced Instabilities," 1984 ASME Winter Annual Meeting, Symposium on Flow Induced Oscillations, New Orleans, Louisiana, Dec. 1984, Paper No. 84/045. [36] 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, BHRA Publisher, London, England, pp. 369376. [37] Welt, F., "A Study of Nutation Dampers with Application to Wind Induced Oscillations," Ph.D. Thesis, University of British Columbia, Vancouver, Canada, 1988. [38] Modi, V.J., Welt, F., and Irani, M.V., "On the Suppression of Vibrations Using Nutation Dampers," Proceedings of the 1988 Pressure Vessels and Piping Conference, Pittsburgh, PA, U.S.A., June 1988, PVP Vol. 133, Editors: F. Hara and S.S. Chen, pp. 17-25. [39] Modi, V.J., and Welt, F., "On the Nutation Damping of Fluid- Structure Instabilities," International Journal of Offshore and Polar Engineering, Vol. 1., No. 3, 1991, pp. 167-175. [40] Welt, F., and Modi, V.J., "Vibration Damping Through Liquid Sloshing, Part 1: A Nonlinear Analysis," Transactions of the ASME, Journal of Vibration and Acoustics, Vol. 114, 1992, pp. 10-16. [41] Welt, F., and Modi, V.J., "Vibration Damping Through Liquid Sloshing, Part 2: Experimental Results," Transactions of the ASME, Journal of Vibration and Acoustics, Vol. 114, 1992, pp. 17-23. 179 [42] Modi, V.J., Welt, F., and Seto, M., "Control of Wind Induced Instabilities Through Application of Nutation Dampers: A Brief Overview," Journal of Engineering Structures, Vol. 17, No. 9, 1995, pp. 626-638. [43] Modi, V.J., and Seto, M., "Energy Dissipation Through Liquid Sloshing and Suppression of Wind Induced Instabilities," Proceedings of the 9th International Conference on Wind Engineering and Industrial Aerodynamics, New Delhi, India, Jannuary 1995, pp. 1619 - 1630; also Journal of Wind Engineering Aerodynamics, and Industrial in press. [44] Seto, M., and Modi, V.J., "Nutation Damping of Wind Induced Instabilities: Experimental and Numerical Analysis", Proceedings of the ASME/JSME Pressure Vessels and Piping Conference, Symposium on Flow-Induced Vibrations, Honolulu, Hawaii, U.S.A., July, 1995, Editor: K. Karim-Panahi, PVP-Vol. 309, pp. 31 - 41. [45] Modi, V.J., and Seto, M., "Nutation Damping of Wind Induced Instabilities," Proceedings on the Sixth Asian Congress on Fluid Mechanics, Singapore, 1995, Editors: Y . T . Chew and C P . Tso, Vol. 1, pp. 113 - 125. [46] Tamura, Y., Kousaka, R., and Modi, V.J., "Practical Application of Nutation Damper for Suppressing Wind-Induced Vibrations of Airport Towers," Journal of Wind Engineering and Industrial Aerodynamics, Vol. 41-44, 1992, pp. 1919- 1930. [47] Fujii, K., Tamura, Y., Sato, Y., and Wakahara, T., "Wind-Induced Vibration of Tower and Practical Applications of Tuned Sloshing Damper," Journal of Wind Engineering and Industrial Aerodynamics, Vol. 33, 1990, pp. 263 - 272. [48] Wakahara, T., Ohyama, T., and Fujii, K., "Suppression of Wind- Induced Vibration of a Tall Building using Tuned Liquid Damper," Journal of Wind Engineering and Industrial Aerodynamics, Vol. 41-44, 1992, pp. 1895 - 1906. [49] "Sloshed, but not Dizzy," FOCUS, published by Commonwealth Scientific and Industrial Research Organization, Melbourne, Australia Autumn 1991, p. 2. [50] Venugopal, M., "A Toroidal Hydrodynamic Absorber for Damping Low Frequency Motions of Fixed and Floating Offshore Platforms," Sectional Paper presented before the New England Section of the Society of Naval Architects and Marine Engineers, 1989, pp. 1 - 13. [51] Welt, F., "Optimization of Mechanical Absorbers Using Liquid Damping for Transmission Line Vibrations," Proceedings of the Asia-Pacific Vibration Confer- 180 ence '93, Kitakyushu, Japan, Japan Society of Mechanical Engineers Publisher, 1993, pp. 306 - 311. [52] Abramson, H.N., "The Dynamic Behaviour of Liquids in Moving Containers," NASA SP-106, 1966. [53] Moiseyev, N.N., "On the Theory of Nonlinear Vibrations of a Liquid of Finite Volume," Journal of Applied Mathematics and Mechanics, Vol. 22, No. 5, 1958, pp. 860-870. [54] Westergaard, H.M., "Water Presssures on Dams during Earthquakes", Transaction of the American Society of Civil Engineeers, Vol. 98, 1933, pp. 418 — 433. [55] Housner, G.W., "Dynamic Pressures on Accelerated Fluid Containers," Bulletin of the Seismological Society of America, Vol. 47, 1957, pp. 15-35. [56] Minowa, C , "Surface Sloshing Behaviours of Liquid Storage Tanks," ASME Transaction, Sloshing and Fluid Structure Vibration, Editors: D.C. Ma, J. Tani, S.S. Chen, and W.K. Lin, PVP-Vol. 157, 1989, pp. 165 - 171. [57] McNown, J.S., "Waves and Seiche in Idealized Ports," Gravity Waves Symposium, National Bureau of Standards, Circular 521, 1952, pp. 153-164. [58] Raichlen, F., "Harbour Resonance," Estuary and Coastal Hydrodynamics, A.T. Ippen, McGraw-Hill, New York, 1966, pp. 281 - 340. Editor: [59] Lee, J.J., "Wave -Induced Oscillations in Harbours of Arbitrary Geometry," Journal of Fluid Mechanics, Vol. 45, Part 2, 1971, pp. 375-394. [60] Budiansky, B., "Sloshing of Liquids in Circular Canals and Spherical Tanks," Journal of the Aerospace Sciences, Vol. 27, No. 3, 1960, pp. 161-173. [61] Bauer, H.F., "Theory of the Fluid Oscillations in a Circular Ring Tank Partially Filled with Liquid," NASA Technical Note D-557, 1960. [62] Hutton, R.E., "An Investigation of Resonant, Nonlinear, Non-Planar Free Surface Oscillations of a Fluid," NASA Technical Note D-1810, Washington, 1963. [63] Bauer, H.F., "Fluid Oscillations in the Containers of a Space Vehicle and Their Influence Upon Stability," NASA T R R-187, 1964. 181 [64] McNeill, W.A., "Fundamental Sloshing Frequency for an Inclined, Fluid-Filled Right Circular Cylinder," Engineering Notes, Vol. 7, No. 8, 1970, pp. 1001-1002. [65] Graham, E.W., and Rodriguez, A.M., "The Characteristics of Fuel Motion which Affect Airplane Dynamics," Transaction of the ASME, Journal of Applied Mechanics, Vol. 19, 1952, pp. 381-388. [66] Dodge, F.T., and Garza, L.R., "Experimental and Theoretical Studies of Liquid Sloshing at Simulated Low Gravity," Transaction of the ASME, Journal of Applied Mechanics, Paper No. 67-APM-14, 1967, pp. 1-8. [67] Vatistas, G.H., Yan, W., and Sankar, T., "Dynamic Behaviour of Liquids in Cylindrical Containers Under Zero Gravity Conditions," Canadian Aeronautics and Space Journal, Vol. 40., No.3, 1994, pp. 131-139. [68] Meserole, J.S., and Fortini, A., "Slosh Dynamics in a Toroidal Tank," Journal of Spacecraft and Rockets, Vol. 24, 1987, pp. 523-531. [69] Bauer, H.F., "Dynamic Behaviour of an Elastic Separating Wall in Vehicle Container, Part 1.," International Journal of Vehicle Design, Vol. 2, 1981, pp. 44-77. [70] Popov, G., Sankar, S., and Sankar, T.S., "Dynamics of Liquid Sloshing in Baffled and Compartmented Road Containers," Journal of Fluids and Structures, Vol. 7, 1993, pp. 803-821. [71] Berlamont, J., and Vanderstappen, N., "The Effect of Baffles in a Water Tower Tank," Proceedings of the 5th International Conference on Wind Engineeering, Colorado, U.S.A., 1979, Editor: J.E. Cermak, Vol. 2, pp. 1195-1201. [72] Alfriend, K . T . , "Partially Filled Viscous Ring Nutation Damper," Journal of Spacecraft and Rockets, 1974, Vol. 11, pp. 456-462. [73] Alfriend, K.T., and Hubert, C.H., "Stability of Dual-Spin Satellite with Two Dampers," Journal of Spacecraft and Rockets, Vol. 11, No. 7, 1974, pp. 469-474. [74] Agrawal, B.N., and James, P., "Energy Dissipation Due to Liquid Slosh in Spinning Spacecraft," Proceedings of the 3rd VPI and SU/AIAA Symposium of Dynamics and Control of Large Flexible Spacecraft, Blacksburg, Virginia, U.S.A.,1981, Editor: L. Meirovitch, pp. 439 - 452. [75] Bryson, A.E. and Banerjee, A.K., "Stabilization of a Spinning Spacecraft with Liquid Slosh," AIAA Guidance, Navigation and Control Conference, Williamsburg, Virginia, 1986. 182 [76] Kareem, A., "Reduction of Wind Induced Motion Utilizing a Tuned Sloshing Damper," Journal of Wind Engineering and Industrial Aerodynamics, Vol. 36, 1990, pp. 725-737. [77] Blevins, R.C., "Sloshing in Tanks, Basins and Harbours," Formulas for Natural Frequencies and Mode Shapes, Van Nostrand Reinhold, New York, 1979, pp. 364-385. [78] Chester, W., "Resonant Oscillations of Water Waves. Part I: Theory," Proceedings of the Royal Society of London, Vol. A306, 1968, pp. 5-22. [79] Ockendon, J.R., and Ockendon, H., "Resonant Surface Waves" Journal of Fluid Mechanics, Vol. 59, 1973, pp. 397 - 413. [80] Cox, E.A., and Mortell, M.P., "The Evolution of Waver-Wave Oscillations" Journal of Fluid Mechanics, Vol. 162, 1986, pp. 99 - 116. [81] Miles, J., "Resonantly Forced, Nonlinear Gravity Waves in a Shallow Rectangular Tank," Wave Motion, Vol. 7, 1985, pp. 291-297. [82] Ockendon, H., Ockendon, J.R., and Johnson, A.D., "Resonant Sloshing in Shallow Water," Journal of Fluid Mechanics, Vol. 167, 1986, pp. 465-479. [83] Verhagen, J.H.G., and Wijngaarden, L., "Nonlinear Oscillations of Fluid in a Container," Journal of Fluid Mechanics, Vol. 22, Part 6. 1956, pp. 737-754. [84] Miles, J., "Nonlinear Surface Waves in Closed Basins", Journal of Fluid ics, Vol. 75, Part 3, 1976, pp. 419-448. Mechan- [85] Miles, J., "Internally Resonant Surface Waves in a Circular Cylinder," Journal of Fluid Mechanics, Vol. 149, 1984, pp. 1-14. [86] Miles, J., "Resonantly Forced Surface Waves in a Circular Cylinder," Journal of Fluid Mechanics, Vol. 149, 1984, pp. 15-31. [87] Su, T.C., and Wang, Y., "Numerical Simulation of Three-Dimensional Large Amplitude Liquid Sloshing in Cylindrical Tanks Subjected to Arbitrary Excitations," ASME Transaction, Flow-Structure Vibration and Sloshing, Editors: D.C. Ma, J. Tani, and S.S. Chen, PVP-Vol. 191, 1990, pp 127-148. [88] Amano, K., Koizumi, M . , and Yamakawa, M . , "Three-Dimensional Analysis Method for Potential Flow with a Moving Liquid Surface Using a Boundary Element Method," ASME Transaction, Sloshing and Fluid-Structure Vibration, 183 Editors: D.C. Ma, J. Tani, S.S. Chen, and W.K. Lin, PVP-Vol. 157, 1989, pp. 127-132. [89] Shiojiri, H., and Hagiwara, Y., "Development of a Computational Method for Nonlinear Sloshing by BEM," ASME Transaction, Flow-Structure Vibration and Sloshing, Editors: D.C. Ma, et. al., PVP-Vol. 191, 1989, pp. 149-154. [90] Nakayama, T., and Washizu, K., "The Boundary Element Method Applied to the Analysis of Two-Dimensional Nonlinear Sloshing Problems," International Journal for Numerical Methods in Engineering, Vol. 17, 1981, pp. 1631-1646. [91] Komatsu, K., "Nonlinear Sloshing Analysis of Liquid in Tanks with Arbitrary Geometries", International Journal of Non-Linear Mechanics, Vol. 22, 1987, pp. 193-207. [92] Harlow, F.H., and Welch, J.E., "Numerical Calculation of Time-Dependent Viscous Incompressible Flow, " Physics of Fluids, Vol. 8, 1965, pp. 2182 - 2189. [93] Hirt, C.W., Nichols, B.D., and Romero, N.C., "SOLA-A Numerical Solution Algorithm for Transient Fluid Flows," Los Alamos Scientific Laboratory Report LA-5852, 1975. [94] Hirt, C.W., and Nichols, B.D., "Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries," Journal of Computational Physics, Vol. 39, 1981, pp. 201-225. [95] Lepelletier, T.G., and Raichlen, F., "Nonlinear Oscillations in Rectangular Tanks," Transaction of the ASME, Journal of Engineering Mechanics, Vol. 114, No.l, 1988, pp. 1- 23. [96] Hayama, S., Aruga, K., and Watanabe, T., "Nonlinear Responses of Sloshing in Rectangular Tanks," Transaction of the JSME, Vol. 26, No. 219, 1989, pp. 1641 - 1648. [97] Shimizu, T., and Hayama, S., "Nonlinear Response of Sloshing Based on the Shallow Water Wave Theory," JSME International Journal, Vol. 30, No. 263, 1987, pp. 806-813. [98] Sun, L.M., Fujino, Y., Pacheco, M., and Chaiseri, P., "Modelling of Tuned Liquid Damper (TLD)," Journal of Wind Engineering and Industrial Aerodynamics, Vol. 41-44, 1992 , pp. 1883-1894. [99] Stoker, J.J., "Long Waves in Shallow Water," Water Waves, Interscience Publishers, Inc., New York, 1957, pp. 291-326. 184 [100] Weiyan, T., "Stability Analysis and Boundary Procedures," Shallow Water Hydrodynamics - Mathematical Theory and Numerical Solution for a Two-Dimensional System of Shallow Water Equations, Elsevier Science Publishing Company, Inc., New York, U.S.A., 1992, pp. 388 - 413. [101] Hartlen, R.T., and Currie, I.G., "Lift-Oscillator Model of Vortex-Induced Vibration," ASCE Journal of the Engineering Mechanics Division, Vol. 96, 1970, pp. 577-591. [102] Iwan, W.D., and Blevins, R.D., "A Model for Vortex Induced Oscillation of Structures," Transaction of the ASME, Journal of Applied Mechanics, Vol. 41, 1974, pp. 581-586. [103] Moretti, P.M., "Flow-Induced Vibrations in Arrays of Cylinders," Annual Review of Fluid Mechanics, Vol. 25, 1993, pp 99-114. [104] Zdravkovich, M.M., and Pridden, D.L., "Interference Between Two Circular Cylinders; Series of Unexpected Discontinuities," Journal of Industrial Aerodynamics, Vol. 2, 1977, pp. 255-270. [105] Strykowski, P.J., and Sreenivasan, K.R., "Control of Vortex Shedding Behind Bluff Bodies", Fifth Symposium on Turbulent Shear Flows, 1984, Cornell University, Ithaca, New York, U.S.A., 1985, Springer-Verlag, New York, pp. 1 11. [106] Washe, D.E., and Cowdrey, C.F., "A Brief Study of the Effect of Shrouds on Buffet Amplitudes of Chimney Stacks," National Physical Laboratory, Teddington, U.K., Maritime Sci.Tech. Memo 2-72, 1972. [107] Vickery, B.J., and Watkins, R.D., "Flow Induced Vibrations of Cylindrical Structures," Proceedings of the 1st Australian Conference on Hydraulics and Fluid Mechanics, Nedlands, Australia, 1962, Editor: R. Silvester, Pergamon Press, pp. 213-241. [108] Wong, H.Y., "Vortex-Induced Wake Buffeting and its Suppression," Journal of Wind Engineering and Industrial Aerodynamics, Vol. 6, 1980, pp. 49 - 57. [109] Zdravkovich, M.M., "Review of Interference-Induced Oscillations in Flow Past Two Parallel Circular Cylinders in Various Arrangements," Journal of Wind Engineering and Industrial Aerodynamics, Vol. 28, 1988, pp. 183-200. [110] Zdravkovich, M.M., "Reduction of Effectiveness of Means for Suppressing WindInduced Oscillation," Engineering Structures, Vol. 6, 1984, pp. 344-350. 185 [Ill] Reinhold, T.A., Tieleman, H.W. and Maher, F.J., "Interaction of Square Prisms in Two Flow Fields," Journal of Industrial Aerodynamics, Vol. 2, 1977, pp. 223-241. [112] Bokaian, A., and Geoola, F., "Hydroelastic Instabilities of Square Cylinders," Journal of Sound and Vibrations, Vol. 92, No. 1, 1984, pp. 117-141. [113] Bokaian, A., and Geoola, F., "Proximity-Induced Galloping of Two Interfering Circular Cylinders," Journal of Fluid Mechanics, Vol. 140, 1984, pp. 417-449. [114] Blessman, J., and Riera, J.D., "Interaction Effects in Neighboring Tall Buildings," Proceedings of the Fifth International Conference on Wind Engineering, Colorado, U.S.A., 1979, Editor: J. Cermak, pp. 381 - 396. [115] Ruscheweyh, H.P., "Aeroelastic Interference Effects Between Slender Structures," Journal of Wind Engineering and Industrial Aerodynamics, Vol. 14, 1983, pp. 129-140. [116] Su, T.C., Lian, Q.X., and Lin, Y.K., "Vibrations of a Pair of Elastically Supported Tall Building Models in a Uniform Stream," Journal of Wind Engineering and Industrial Aerodynamics, Vol. 36, 1990, pp. 1115-1124. [117] Sakamoto, H., Haniu, H., and Obata, Y., "Fluctuating Forces Acting on Two Square Prism in a Tandem Arrangement," Journal of Wind Engineering and Industrial Aerodynamics, [118] Lamb, H., Hydrodynamics, 571; pp. 579 - 581. Vol. 26, 1987, pp. 85-103. Dover Press, New York, 1945, 6th Edition, pp. 562 - [119] Bearman, P.W., Gartshore, I.S., Maull, D.J., and Parkinson, G.V., "Experiments on Flow-Induced Vibration of a Square-Section Cylinder," Journal of Fluids and Structures, 1987, Vol. 1, pp. 19-34. [120] Sun, L.M., and Fujino, Y., "Effectiveness of Multiple Tuned Liquid Dampers", Private Communications, June, 1991. [121] Modi, V.J., and El-Sherbiny, S., "On the Wall Confinement Effects in the Industrial Aerodynamic Studies", International Symposium on Vibration Problems in Industry, Keswick, England, April 1973, Paper No. 116, pp. 1 - 20. [122] Phillips, O.M., The Dynamics of the Upper Ocean, Cambridge University Press, New York, 1977, pp. 46 - 49. 186 [123] Pilkey, W.D., and Wunderlich, W., Mechanics of Structures: Variational and Computational Methods, CRC Press, Inc., Boca Raton, Florida, 1994, pp. 469491. [124] Wood, W.L., Practical 1990, pp. 325-353. [125] Time Handbook of Mathematical Stepping Schemes, Clarendon Press, New York, Functions with Formulas, Graphs, and Mathematical Tables, Editors: M. Abramowitz and I. Stegun, National Bureau of Standards, Applied Mathematics Series, Vol. 55, 1964, pp. 136. 187 APPENDICES 188 I. C A L I B R A T I O N O F I N S T R U M E N T A T I O N U S E D IN T H E T E S T P R O G R A M 1.1 Force Balance CO a E 189 1.2 L V D T used to Measure Structural Displacement During W i n d Tunnel • Tests CO .'"V — T—1 r 1 - 1 r —i——i—;—f——r \p '• " . ' 1 1 1 1 J— 1 M ; o o — — M ,C O CO e. CM \ CT _ CO CVJ - CT) LO / / / / / \ \ cu \ \ .. \ / \ \ o / o / o . — \ \ CM - co 6 _ . •• - • - CD co. \ . \ ; .' . O \ O o d -rj 2 _ _ -+J \ K m . 0 co ^ <u \ II II \ X _ O - \ CO CO CD - CT p n • dv d + - g CO / / + cr _ / * T— .60 \ :A co o . eb • - o o \ " . T3 o o ^ 0). . V 2 a 00 \ - - , • co £ o o CO •a i—i 2 ,-0 •2 "S CU 6- 1 CO d tt) CM JO P ^. d CM 4> • M 190 II. LISTING OF P R O G R A M S D E V E L O P E D F O R T H E S T U D Y II.1 logdec.m: Logarithmic Decrement from Digitized Data function[]=logdec() %n=10; r e - load buffer.rn.at; p=pk(buffer); zeta=(log(p(l)/p(n))/(n-l))/(sqrt(4*pi 2+(log(p(l)/p(n))/(n-^ A loadsdc.dat b=[b;zeta]; save sdc.dat b delete buffer.mat 191 II.2 getfreq.m: Amplitude, Frequency and Phase of the Sloshing Force function[rct]=getfreq() % demux data data, dat 1 , load data.dat; data=data-mean(data); %z-abs(fft(data(l:4094)))/4096 4150; z=abs(fft(data)); z=z./length(z); .- ..; [y,i]=max(z(l:240)); [y2j]=max(z(241:1000)); yp=z(i+l); ypp=z(i+2); yppp=z(i+3); ym=z(i-l); ymm=z(i-2); ymmrri=z(i-3); f=100/length(z)*i; f2=100/length(z)*j; ret=[ymmm,ymm,yrn,y,yp,ypp,yppp,f,y2,f2]; delete data.dat savtun2(ret) - function[]=phase() . .'demux data ref.dat 0 !demux data sig.dat 1 load ref.dat load sig.dat ref=ref-mean(ref); zr=fft(ref(l :4094))/4094;. 192 sig=sig-mean(sig); zs=fft(sig(l:4()94))/4094; za=abs(fft(sig(l :4094)))/4094; [y,i]=max(za(l:100)); f=50/4094*i; ph=atan(imag(zs(i))/real(zs(i)) )-atan(imag(zr(i))/real(zr(i))); dat=[ph,za(i-l),za(i),za(i+l),fj; •• \ savd500(dat); delete ref.dat delete sig.dat delete data %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%/o%/o%%% 0 % % % % % % • . • ; • : , : ' function[dat]=peaks() load inp.dat l=length(inp); avg=mean(inp(4000:l)); ., inp=inp-avg; base=mean(inp(l: 100)); % user controlled ~ N tells how many of first peaks and valley to seek % — srchlim tells in what bw to look for extrema N=14; . srchlim=i50; first=300; ,% guess at the bw thatfirstpeak lies in fitbw=40; . . . pad=30; % % returns an array of N peaks, N valleys, the init ampl (base), avg % of oscillation and N-l relative logarithmic decrements % July 30/91--Mae Seto % May 22/94 --Mae Seto %%%FIND I N I T I A L , M A X A N D M I N P E A K S 193 0 inp=inp'; % % [posp(j),pospi(j)]=max(inp( 1 :first)) x=(pospi(j)-fitbw:pospiG)+fitbw); coeff=polyfit(x,inp(x),3); % third order fit 1st max pospfit(j)=max(polyval(coeff,x)); inpi=pospi(j)+pad; l=length(inp); % now locate the minimum in shortened array [negpfj ),negpin]=min(inp(inpi: inpi+srchlim)); negpi(j)=negpin+inpi; % keep running count on index x=(negpi(j )-fitbw: negpi(j )+fitb w); coeff=polyfit(x,inp(x),3); negpfit(j )=min(polyval(coeff,x)); inpi=negpi(j)+pad; l=length(inp); %%%%%%%%%%%%%%%%%%%%%%FIND R E S T forj=2:N; [posp(j),pospin]=max(inp(inpi:inpi+srchlim)); pospi(j)=pospin+inpi; x=(po spi(j )-fitbw: pospilj )+fitb w); coeff=polyfit(x,inp(x),2); % third orderfit,2nd max pospfit(j)=max(polyval(coeff,x)); inpi=pospiG)+pad; l=length(inp); %%%%%%%%%%%%%%%%%%%%% [negp(j),negpin]=min(inp(inpi:inpi+srchlim)); negpiO )=negpin+inpi; x=(negpi(j)-fitbw:negpi(j)+fitbw); coeff=polyfit(x,inp(x),2); negpfit(j)=min(polyval(coeff,x)); 194 OF P E A K S % inp(negpi(j)+pad:l); inpi=negpi(j)+pad; l=length(inp); end %forj=l:N; % pospfitn(j)=pospfit(j)/pospfit(l); % negpfitn(j )=negpfit(j )/negpfit( 1); % end %%%%%%%%%%% D E T E R M I N E % LOGARITHMIC DECREMENTS forj=2:N, % decmax(j -1 )=log(pospfit(j -1 )/pospfit(j)); % decmin(j -1 )=log(negpfit(j -1 )/negpfit(j)); % end %%%%%%%%%% O U T P U T D A T A %%% FIT A C U R V E TO THE P E A K S - obtain the coeff of the 3rd order fit % pcoeff=polyfit(pospi,pospfitn,3); % ncoeff=polyfit(negpi,negpfitn,3); m=5; % etarlp=l/(2*pi)*(pcoeff(2)+2*m*pcoeff(3^ m 2+pcoeff(4)*m 3); etarlp= 1 /(2 *pi *m) * log(pospfit( 1 )/pospfit(m)); % etarln=l/(2*pi)*(ncoeff(2)+2*m*ncoeff(3)+3*m 2*ncoeff(4))/(ncoeff(l)+ncoeff(2)*m+ncoeff(3)* m 2+ncoeff(4)*m 3); etarln= 11(2 * pi * m) * log(negpfit( 1 )/negpfit(m)); % dat=[posp, negp]; A A A A A dat=[etarlp,etarln] % dat=[pospfit,negpfit] savt(dat); % delete inp.dat; clear etarln etalp 195 II.3 whtor.m: Determination of Toroidal Geometry function[riRo,result,L,Rc]=solvtor(f, V, Do) % cgs units: f=freq (Hz), V=volume of liquid in mL, Do is outer diameter in cm % to invoke from Matlab type in [riRo, res, L,Rc]=solvtor(f, V/Do) Ro=Do/2; index=l; for ri=l:0.25:fix(Ro), lambda=Ro*2*pi*Psqrt(pi*(Ro 2 - ri 2)/981A/); A A % ' , Yl=bessely(l,lambda*ri/Ro); Y12=bessely(0,lambda*ri/Ro)-l/(lambda*ri/Ro)*bessely(l,lambda*ri/Ro); Y1 p=bessely(0,lambda)-1/lambda*bessely( 1,lambda); % Jl=besselj(l, lambda*ri/Ro); J12=bcsselj(0,lambda*ri/Ro)-l/(lambda*ri/Ro)*bessclj(l,lambda*ri/Ro); Jlp=besselj(0,lambda)-1/Iambda*besselj(l,lambda); riRo(index)=ri/Ro; result(index)=Y 1 p* JI 2-J1 p* Y12; index=index+l; % result should be zero if it is close to the solution end % now solve for a square damper with the same frequency and liquid volume L=sqrt( sqrt(981*V)/(2*f)); % solve for circular cylinder that is resonant at the same frequency and volume Rc=sqrt( 1.8412/(2*pi*f)*sqrt(981 * V/pi)); % dat=[result]; function[result]=geom(V,Do) f=1.5; • ; % V is in mL, and lengths in cm, f is in Hz, Do is in cm 196 save param.mat ri=fzero('tor,0.5*Do/2, 0.01) riRo=ri/(Do/2) ht=V/pi/( (Do/2) 2 - ri 2) Ro=Do/2; , A A % frequency f sloshing (shallow water form) is calculated as a check on results f=sqrt( (2*Ro/(Ro+ri))*981/Ro *tanh (2*Ro/(Ro+ri)*ht/Ro) )/(2*pi) % now solve for a square damper with the same frequency and liquid volume L=sqrt(sqrt(981*V)/(2*f)) hs=V/L 2 : A % solve for circular cylinder that is resonant at the same frequency and volume Rc=sqrt( 1.8412/(2*pi*f)*sqrt(981*V/pi)) hc=V/(pi*Rc 2) result=ri; A % listing should include tor.m % the liquid height is important in that if the oscillation amplitude is % too large then the surface is not always wetted — avoid this %%%%%%%%%%%%%%%%% /o%%% /o /o /o% /o%% 0 0 0 0 0 function[result]=tor(ri) % cgs units: f=freq (Hz), V=volume of liquid in mL, Do is outer diameter in cm % to invoke from Matlab type in [riRo, res, L,Rc]=solvtor(f, V, Do) load param.mat % f=0.58; % V=730; %Do=30; . Ro=Do/2; lambda=Ro*2*pi*f*sqrt(pi*(Ro 2 - ri 2)/98 W ) ; A A if imag(lambda) •== 0 % Yl=bessely(l,lambda*ri/Ro); 197 Y12=bessely(0,lambda*-ri^ Y1 p=bessely(0 Jambda)-1/lambda*bessely( 1,lambda); Jl=besselj(l, lambda*ri/Ro); % J12=besselj(0,lambda*ri/Ro)-l/(lambda*ri/Ro)*besselj(l,lambda*ri/Ro); J1 p=besselj (0,lambda)-1/lambda*besselj (1 ,lambda); result=Ylp*J12-Jlp*Y12; % result should be zero if it is close to the solution else % disp('no solution') end 198 II.4 Damper Geometries (Circular, Toroidal and Rectangular) for Given Liquid Frequency and Volume solvior.m: function[result]=whtor(f,V) % to invoke: [a]=whtor(0.58,730) % responds with Ro, ri, riRo, h' for valid answers % V in ml, fin Hz % calls tordiamain.m which uses tordia.m i=i; str=14.4; endr=16.5; for Ro=str:0.1:endr [x(i),y(i),z(i),a(i)]=tordiamain(f,V,Ro); i=i+l; end .- x=x'; y=y'; z=z'; a=a'; % result in tabular form as Ro, ri, riRo, h (liquid;height) result=[x,y,z,a]; disp('Ro, ri, riRo, h') %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% •%%%%%%%% " • ' function[result]=tordia(Ri) % called by tordianiain.ril which is called by whtor.m % f=0.58; % V=730; %Ro=20cm; . . . load param.mat result=sqrt(Ro 2-Ri 2)*(Ro+Ri)-l/(pi*f)*sqrt(981*V/pi); A A % answer is in cm % ht=V/pi/(Ro 2-Ri 2) A A 199 functionfROjrijriROjhl^ordiamainCtVjRo) % called by whtor.m, uses tordia.m save param.mat ri=fzero('tordia',0.6*Ro,0.01); riRo=ri/Ro; h=V/pi/(Ro 2-ri 2); A A 200 III. S H A L L O W W A T E R M O D E L F O R N U T A T I O N D A M P E R III.l Expression for dw/dt From eq. (4.11b), w du dv dx dy tanh K,(h + z), therefore 1 r d (du\ dw d (dy\\ , , , Substituting in eq. (4.14a) for du/dt and in eq. (4.14b) for dv/dt gives dw ^ = 1 r d ( dr) « l ^ l ^ 1 3 2 2 ^ + + 9 ^ A p dw \ d (dn ^ ) dz + d ^ \ g 1 d y ' 9 2 + ^ a U 3 + f V 9 w ^ ^ , \1 *JJ x tanh K ( / I + z), i /5 77 2 r a 7?\ i / 2 d o a d »\ n dw . 2 , a 2 n dw , i x tanhre(/i+ z), x tanhre(/i-f z), V p is, of course, zero. V of the integrand is also zero. Therefore, 2 2 dw 1 /d r] 2 d r}\ 2 201 , „ III.2 Nondimensional Form of Nonlinear Shallow Water Wave Equations Momentum Equations in x and y Directions L E T : v T z x* = y, y* = , z* = p t*=iM;n = V t r i n I -4= -: ? v* - y/gnai K/I — , and V o"l tanh /c/i c= \ I Then: du c du* dt n dt* dx ~ hdrf 9 d , c du* — (u ) = dx i dx* 2 K K dx dx 2 ' ( _ (gh\ Idx* ~ W drf J ~dx* ] 2 J 5<-I = > (g J dt*' dr) _ 9 h\du* v^Vgho-! S \ \ *i) )a dx* ' 2U x > = ( T ) ^ t a n h ^ l + r,)] = ^ ^ t a n h ^ l gh'dtfdW = K¥o^dx^ tanh[Kh{1 + „•)], + r ) ) ] - Rewriting the constant coefficients on the left hand side in terms of previously defined nondimensional variables results in 202 gh h , ,, __tanh/cfc(Hr»7 / /c/ 2 ^dv* d r]* 2 = dx* dx* 2 gh h? tanh khll + n*) ~l 7 12" I I Kh 2 tanh Kh dn* d n* 2 * tanh Kh dx* dx* ' 2 'gh\ h? tanh /c/i tanh Kh(l + 77*) drf d rf 2 I JI Kh 2 fgh\ _ tanh Kh dx* dx* ' 2 t 9 r / * 5 ^ 2 2 Thus the dissipation term becomes — - a i i = acu = a = a = —aw — u , o"i gh 1 \fgha\ u , Integrated Continuity Equation Similarly, the terms in the integrated continuity equation can be nondimensionalized as dn f\/ghai\ drf dt~ \ )&t*' . . tanh/c/i h — tanhK[h + 77) x — - x —, l dx* K tanh/eft. hi d_ u dx I — tanh K(JI + 77) c d l dx" [u*xhar], Ighh d a\ 1 ox* Multiplying all terms in the continuity equation by l/^gho~\ gives for this term, dn a dt cTi {xu) + {xv) Tx Yy 203 Rewriting the equation so that the case of % — 1) i- - the small wave amplitude e condition is emphasized, drt ~^ + dt a du — o-\ .dx dv d , . d , For the liquid sloshing resonance condition, a = o~i, therefore 577 dt fdu \dx dv\ dy) £-0( du dx dv\ y J 204 a d a\ ^ d X - l ) u + Ty { x - 1 ) v = 0. III.3 Free Surface and Kinematic Boundary Conditions In problems of water waves, it is reasonable to assume incompressibility condition, i.e. p is constant. The body force F = — pgk, where g is the acceleration due to gravity and k is the unit vector in the positive z direction. Thus the fundamental equations for water wave motion are: ^ at + (U-V)U = - - V p p f f k; V - U = 0. They represent nonlinear partial differential equations with four unknowns u, v,w, and p. The objective is to solve this set of equations with appropriate initial and boundary conditions. Furthermore, the motion may be taken as irrotational, which physically means that the individual fluid particle does not rotate. Mathematically, this implies i i ) z V x U = 0 where u> is the vorticity. Thus, there exists a single valued velocity potential <f> so that U = V^>. Now, the continuity equation reduces to the Laplace equation, vV = o, i.e. 4> is a harmonic function. Using the identity (U • V ) U = \W 2 — U x u> with ui = 0 and U = V^>, the momentum equation can be rewritten in the form 4>t + ;kv<£) + - + gz 2 2 0. p This can be integrated with respect to the space variable to give: h + k^4>) + - + gz = c(t). 2 2 p Without loss of generality, C(t) can be taken as zero since a function of time added to the pressure field has no effect on the motion. This is Bernoulli's equation. 205 Consider the case of a body of water with air above it so that S is the interface between them. The surface S may be represented by an equation S(x,y,z,t) = 0. At the surface of the water, the kinematic and free surface boundary conditions must be satisfied. The kinematic condition is derived from the fact that the normal velocity of the surface must equal to the velocity of the fluid normal to the surface. Recognizing that the normal velocity of the surface is given by — (ds/dt)/\VS\, and the normal velocity of the fluid is U • n, where n = V 5 / | V 5 | is the unit normal vector, dS , dS — + (U • V)S = — = 0. dt K ' dT This means that any fluid particle originally on the surface will remain there. It is convenient to represent the free surface by the equation z = r](x,y,t). Then we have for S the equation S - n(x,y,t) - z = 0, where z is independent of other variables. Hence, with U = Vc6, the total derivative at 5 becomes dn d<f> dn dt dx dx d<f> dn d<j> dy dy dz This is the kinematic free surface condtion. Since the free surface is exposed to the atmospheric pressure, p , the dynamic a free-surface boundary condtion is imposed by the requirement that the difference of pressure on two sides of the interface S is balanced by the effects of surface tension. If the surface tension is neglected, 4t + 9V=\{V<l>) = 02 206
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- An investigation on the suppression of flow induced...
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
An investigation on the suppression of flow induced vibrations of bluff bodies Seto, Mae L. 1995
pdf
Page Metadata
Item Metadata
Title | An investigation on the suppression of flow induced vibrations of bluff bodies |
Creator |
Seto, Mae L. |
Date Issued | 1995 |
Description | Wind induced oscillations of bridges, tall buildings, smokestacks, transmission line conductors and similar bluff bodies have been of interest to scientists and engineers for a long time. Unchecked, the vortex resonance and galloping type of vibrations can cause damage to structures. Some examples include the original Tacoma Narrows Bridge, the dramatic cracking of industrial chimneys, and the destruction of building components as well as the structure itself. Apart from catastrophic destruction, this class of low frequency vibrations are known to cause undesireable working conditions leading to nausea, dizziness, disorientation and vertigo, particularly for those working at relatively greater heights as in tall buildings and air traffic control towers. Flow induced vibrations are of special relevance to engineers today because of the tendency to build taller structures and longer span bridges with ever lighter building materials. The thesis studies vortex induced and galloping type of instabilities associated with structural geometries of fundamental importance. Of particular interest is the effectiveness of energy dissipation to suppress the oscillations. To that end, a comprehensive study focuses on the design of nutation dampers and assesses their effectiveness in arresting wind induced instabilities. To begin with, a parametric study of the damper, in conjunction with frequency response tests, is used to identify important system variables contributing to significant energy dissipation. The results show that optimum combinations of the damper parameters such as the geometry, liquid height, surface seeding, and compartmenting can lead to an efficient damper, particularly if the operating conditions are conducive to wavebreaking. Among the dampers tested, the circular cylindrical geometry proved to be the most efficient. The addition of floating particles further improved the performance by around 30 %. Next, a numerical model, based on the nonlinear shallow water equations of motion, is developed to predict dissipation characteristics. The numerical results are also animated to provide better visual appreciation of the free surface wave dynamics. The agreement between numerical and experimental results is quite good considering the complex character of the flow. This is followed by construction of a fluid-structure interaction model through coupling of the nonlinear shallow water equations to a single degree of freedom structure undergoing vortex resonance. The agreement between wind tunnel experiments and the model is surprisingly good considering the nonlinear character of the fluid dynamics and structural interactions with it. The numerical algorithm should serve as a valuable tool in designing this class of dampers for practical applications. Finally, wind tunnel tests with two-dimensional models substantiate, rather dramatically, the effectiveness of the nutation dampers in arresting both vortex resonance and galloping types of instabilities. The dampers continue to be effective even for the case when the structure is located in the wake of other structures, the situation frequently encountered in practice. A visualization study, for flow within the damper and suppression of structural instabilities during the wind tunnel tests, complemented the experimental and numerical investigations. Both still photographs as well as a video were taken. Each phase of the study represents innovative contributions and the results obtained are of far-reaching consequence for a class of structures currently under design and those planned for the future. The thesis ends with some concluding comments and recommendations on rewarding avenues of research to pursue in the next phase of the work. |
Extent | 16351855 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-02-18 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080841 |
URI | http://hdl.handle.net/2429/4758 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 1996-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
Download
- Media
- 831-ubc_1996-091600.pdf [ 15.59MB ]
- Metadata
- JSON: 831-1.0080841.json
- JSON-LD: 831-1.0080841-ld.json
- RDF/XML (Pretty): 831-1.0080841-rdf.xml
- RDF/JSON: 831-1.0080841-rdf.json
- Turtle: 831-1.0080841-turtle.txt
- N-Triples: 831-1.0080841-rdf-ntriples.txt
- Original Record: 831-1.0080841-source.json
- Full Text
- 831-1.0080841-fulltext.txt
- Citation
- 831-1.0080841.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0080841/manifest