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Dynamic response of moored floating breakwaters Fraser, G. Alex 1979

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DYNAMIC RESPONSE OF MOORED FLOATING BREAKWATERS by GLEN ALEXANDER FRASER B.A.Sc, University of B r i t i s h Columbia, 1975 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of C i v i l Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA January 19 79 @ Glen Alexander Fraser, 1979 In presenting this thesis in p a r t i a l f u l f i l l m e n t of the requirements for an advance-d degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the -Head of my Department or by his representatives. It i s understood that copying or publication of this thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of C i v i l Engineering The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date ABSTRACT The problem of finding the response of a moored f l o a t i n g breakwater i s examined with p a r t i c u l a r reference to the e f f e c t of the moorings on breakwater motions. A computer program has been developed i n which hydrodynamic c o e f f i c i e n t s are calculated using the f i n i t e element method. The complete breakwater-mooring system i s modelled using the techniques of plane-frame s t r u c t u r a l analysis. The importance of the second-order d r i f t force i n the mooring analysis i s noted. Comparison i s made between the motions of an unrestrained f l o a t i n g body and those of a body restrained by slack moorings. A simple approximation for high-frequency motions i s proposed. TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF FIGURES v NOMENCLATURE v i ACKNOWLEDGEMENT i x CHAPTER 1 INTRODUCTION 1 1.1 I n t r o d u c t i o n 1 1.2 Survey of A n a l y s i s Methods 2 1.3 D e s c r i p t i o n of Method 4 CHAPTER 2 HYDRODYNAMIC ANALYSIS 7 2.1 I n t r o d u c t i o n 7 2.2 P o t e n t i a l Flow 8 2.3 E x c i t i n g F o r c e s , Added Mass, and Damping 12 2.4 Equations of Motion 15 2.5 D r i f t Force 17 2.6 F i n i t e Element S o l u t i o n 18 CHAPTER 3 MOORING ANALYSIS 21 3.1 I n t r o d u c t i o n 21 3.2 S t a t i c A n a l y s i s 22 3.3 Dynamic A n a l y s i s 25 CHAPTER 4 MOTION OF THE UNRESTRAINED BODY 34 4.1 Added Mass and Damping 34 4.2 Motion of the U n r e s t r a i n e d Body 37 4. 3 D r i f t Force 40 i i i page CHAPTER 5 EFFECT OF MOORINGS 41 5.1 I n t r o d u c t i o n 41 5.2 D i m e n s i o n a l A n a l y s i s 42 5.3 P r o p e r t i e s o f B r e a k w a t e r 44 5.4 E f f e c t o f M o o r i n g s 46 CHAPTER 6 CONCLUSIONS AND FURTHER STUDIES 5 2 6.1 C o n c l u s i o n s . . ,52 6.2 F u r t h e r S t u d i e s 5 3 BIBLIOGRAPHY 55 APPENDIX A THE QUADRATIC ISOPARAMETRIC ELEMENT 5 8 APPENDIX B FREE CABLE VIBRATIONS 62 i v LIST OF FIGURES Page F i g . 1 DEFINITION SKETCH - FLUID REGION 6 4 F i g . 2 FINITE ELEMENT MESH 65 F i g . 3 DEFINITION SKETCH - MOORING SYSTEM 66 F i g . 4 CABLE DEFINITIONS 67 F i g . 5 DYNAMIC STRUCTURAL MODEL 6 8 F i g . 6 CABLE ELEMENT 69 F i g . 7 COMPUTER PROGRAM FLOW CHART..... 70 F i g . 8 TYPICAL COMPUTER PLOT ' 71 F i g . 9 ADDED MASS AND DAMPING - CIRCULAR CYLINDER 72 F i g . 10 ADDED MASS AND DAMPING - RECTANGULAR CYLINDER 7 3 F i g . 11 EXCITING FORCES - CIRCULAR CYLINDER 74 F i g . 12 AMPLITUDES OF MOTION - CIRCULAR CYLINDER 75 F i g . 13 TRANSMISSION COEFFICIENT - CIRCULAR CYLINDER • 76 F i g . 14 DRIFT FORCE - CIRCULAR CYLINDER 77 F i g . 15 GENERAL ARRANGEMENT - RECTANGULAR BREAKWATER 7 8 F i g . 16 CONTRIBUTION OF MODES TO TRANSMITTED WAVE 79 F i g . 17 SLACK-MOORED TRANSMISSION COEFFICIENT 80 F i g . 18 SWAY AMPLITUDE AND PHASE ANGLE 81 F i g . 19 HEAVE AMPLITUDE AND PHASE ANGLE 8 2 F i g . 20 ROLL AMPLITUDE AND PHASE ANGLE 8 3 F i g . 21 EFFECT OF WAVE STEEPNESS.. 84 F i g . 22 EFFECT OF CABLE TAUTNESS 85 F i g . 23 SEAWARD CABLE TENSION ' 86 F i g . 24 SHOREWARD CABLE TENSION 87 F i g . A - l QUADRATIC ISOPARAMETRIC ELEMENT 88 F i g . B - l NATURAL FREQUENCIES AND MODE SHAPES , 89 v NOMENCLATURE A = cross-sectional area of mooring l i n e [A] = matrix of added mass c o e f f i c i e n t s a. , = added mass c o e f f i c i e n t B = beam of breakwater [B] = matrix of damping c o e f f i c i e n t s or structure damping matrix [b] = cable element damping matrix b. .^ = damping c o e f f i c i e n t C = unstretched length of mooring l i n e C D = drag c o e f f i c i e n t of mooring l i n e c = wave c e l e r i t y c^ = wave group v e l o c i t y D = draught of breakwater D = diameter of mooring l i n e c ^ d = water depth or cable drag E = e l a s t i c modulus of mooring l i n e F = force or drag force per unit length of mooring l i n e F^ = ex c i t i n g force on body F.^ = force in j di r e c t i o n due to unit amplitude motion in k di r e c t i o n F c = Coulomb damping force F D = d r i f t force (F) = cable element nodal force vector f ., = complex form of F., jk ^ jk g = acceleration due to gravity H = horizontal force in mooring l i n e h = wave height or r i s e in mooring l i n e .,h ,h t = incident, r e f l e c t e d , and transmitted wave heights I Q = polar mass moment of i n e r t i a per unit length of breakwater about z-axis I = moment of i n e r t i a of waterplane area per unit z length of breakwater about z-axis i_,_2 = unit vectors in x and y directions K c = Keulegan-Carpenter number K = r e f l e c t i o n and transmission c o e f f i c i e n t s r ' t [K] = hydrostatic or structure s t i f f n e s s matrix [K.] = f i n i t e element matrices x k = wave number (k = 2 T T / A ) v i cable element s t i f f n e s s matrix span of mooring l i n e or length of cable element l o c a l coordinate on perimiter of f i n i t e element mass matrix mass per unit length of breakwater or mooring l i n e v i r t u a l mass per unit length of mooring l i n e cable element mass matrix vector of interpolation functions unit outward normal from f l u i d region d i r e c t i o n cosines of n rate of energy transfer per unit width of wave pressure th pressure due to unit amplitude motion i n i mode Reynold's number' position vector of point on body surface f l u i d boundary or spacing of anchors prescribed v e l o c i t y boundary or body surface radiation boundary free surface transformed f i n i t e element coordinate period of o s c i l l a t i o n or tension in mooring l i n e time or transformed f i n i t e element coordinate 9 maximum horizontal f l u i d v e l o c i t y f l u i d v e l o c i t y i n x and y directions l a t e r a l v e l o c i t y of cable element normal v e l o c i t y underwater volume per unit length of breakwater rate of energy d i s s i p a t i o n i n moorings buoyant weight per unit length of mooring lin e complex e x c i t i n g forces Cartesian coordinates x-coordinate of centroid of waterplane area y-coordinate of centre of buoyancy y-coordinate of centre of gravity length of breakwater associated with one set of moorings elongation of mooring l i n e or log decrement l a t e r a l displacement of mooring l i n e vector of cable element nodal displacements v i i n — s u r f a c e e l e v a t i o n .6 - a n g l e o f m o o r i n g l i n e w i t h s e a b e d X = w a v e l e n g t h h = complex a m p l i t u d e s o f body m o t i o n p = d e n s i t y o f f l u i d p c = d e n s i t y o f m o o r i n g l i n e m a t e r i a l $ = v e l o c i t y p o t e n t i a l <}> - complex v e l o c i t y p o t e n t i a l *0 = i n c i d e n t wave p o t e n t i a l *R - s c a t t e r e d wave p o t e n t i a l * i = f o r c e d m o t i o n p o t e n t i a l s (<J>) = v e c t o r o f n o d a l v a l u e s o f <j> a = domain o f f l u i d r e g i o n CO - a n g u l a r f r e q u e n c y o f o s c i l l a t i o n v i i i ACKNOWLEDGEMENT , The author wishes to thank Drs. M. De St. Q. Isaacson, M. D. Olson, and R. F. Hooley for t h e i r advice and assistance in the preparation of this thesis. F i n a n c i a l support was provided by a National Research Council of Canada Postgraduate Scholarship. i x CHAPTER .1 INTRODUCTION 1.1 I n t r o d u c t i o n D i f f e r e n t t y p e s o f m a r i n e a c t i v i t i e s r e q u i r e d i f f e r e n t l e v e l s o f p r o t e c t i o n f r o m w a v e s . A h a r b o u r may a d e q u a t e l y p r o t e c t l a r g e s h i p s f r o m o c e a n w a v e s , y e t t h e waves g e n e r a t e d i n s i d e t h e h a r b o u r by w i n d and s h i p movements may be o b j e c t i o n a b l e t o s m a l l c r a f t . As a «result i t i s o f t e n n e c e s s a r y t o p r o v i d e a d d i t i o n a l ' p r o t e c t i o n a r o u n d s m a l l c r a f t m a r i n a s . T h i s p r o t e c t i o n n e e d n o t be e f f e c t i v e a g a i n s t l o n g w a v e l e n g t h o c e a n waves as o n l y s h o r t e r w a v e l e n g t h s a r e p r e s e n t i n s i d e t h e h a r b o u r . R u b b l e mound b r e a k w a t e r s c a n be u s e d b u t t h e y c a n be e x p e n s i v e , p a r t i c u l a r l y i f t h e w a t e r i s d e e p . A l s o , any s o l i d b r e a k w a t e r i n t e r f e r e s w i t h t h e s l o w c i r c u l a t i o n o f w a t e r i n t h e h a r b o u r w h i c h may c a u s e s e d i m e n t a t i o n , e r o s i o n , and p o l l u t i o n p r o b l e m s . The a d v a n t a g e o f a s o l i d b r e a k w a t e r i s t h a t i t r e f l e c t s o r a b s o r b s e s s e n t i a l l y a l l o f t h e i n c i d e n t w a v e s , r e g a r d l e s s o f w a v e l e n g t h . I n c o n t r a s t , a f l o a t i n g b r e a k w a t e r a l l o w s s l o w c i r c u l a t i o n o f w a t e r u n d e r n e a t h t h e body o f t h e b r e a k w a t e r and t h e c o s t i s n e a r l y i n d e p e n d e n t o f w a t e r d e p t h . H o w e v e r , a f l o a t i n g b r e a k w a t e r may o r may n o t be e f f e c t i v e i n r e d u c i n g t h e h e i g h t o f waves i n t h e a r e a i t i s s u p p o s e d t o p r o t e c t . The success of f l o a t i n g breakwaters has been mixed, p a r t i a l l y due to problems in predicting the forces on the breakwater and moorings and i n predicting the effectiveness of the breakwater i n reducing wave heights. The purpose of t h i s thesis i s to present a method of analysing a prismatic f l o a t i n g breakwater including the effects of slack mooring l i n e s in order to investigate how the moorings a f f e c t the motions and e f f i c i e n c y of the breakwater. 1.2 Survey of Analysis Methods The analysis procedures for f l o a t i n g breakwaters are extensions of the study of ship motions. The usual method of analysis i s to solve for f l u i d v e l o c i t i e s according to l i n e a r p o t e n t i a l theory. Exact solutions are available only for a few (2) (3) special geometries such as v e r t i c a l b a r r i e r s (Dean , U r s e l l ). There are several numerical methods for analysing flow around horizontal cylinders of a r b i t r a r y cross-section. The (4) e a r l i e s t numerical method i s that of U r s e l l in which the p o t e n t i a l f i e l d i s represented by a source function and multi-poles at the o r i g i n of the coordinate system. The method can be used for c i r c u l a r cylinders and shapes that can be obtained from a c i r c l e by conformal mapping. The most common numerical method i s the method of i n t e g r a l equations using Green's functions. The Green's functions, f i r s t obtained by J o h n ^ , give the p o t e n t i a l at any point in the f l u i d r e s u l t i n g from a point source on the body surface. The boundary condition on the body surface can then be developed into a l i n e i n t e g r a l equation which can be solved numerically to obtain the d i s t r i b u t i o n of source strengths along the body contour. The 3 Green's functions then give the p o t e n t i a l for any point in the f l u i d region. A more recent numerical method i s the use of f i n i t e elements in which the f l u i d surrounding the body i s divided into a number of regions or elements. The method i s discussed by Newton ( 6^, B a i ( 7 ^ , Chen and M e i ^ 8 ) , and Bettess and (9) Zienkiewicz . The governing d i f f e r e n t i a l equation i s expressed in v a r i a t i o n a l form as the minimum of some functional. The pote n t i a l f i e l d i s defined by the p o t e n t i a l at a f i n i t e number of nodes on the element boundaries and by int e r p o l a t i n g functions within the elements. Minimizing the functional with respect to the nodal values of the p o t e n t i a l gives a set of algebraic equations that can be solved to find the pot e n t i a l f i e l d . The f i n i t e element method produces a much larger set of equations than the i n t e g r a l equation method for the same degree of accuracy. However, the matrix to be solved i n the f i n i t e element method i s symmetric and banded which allows the use of very e f f i c i e n t solution procedures. Extensions of the f i n i t e element method are the hybrid element method of Chen and Mei, and the use of i n f i n i t e elements developed by Bettess and Zienkiewicz. Both methods reduce the size of the f i n i t e element region and thus the order of the matrix to be solved. Other numerical methods include a v a r i a t i o n a l method (Mei and B l a c k ^ ^ ) , matched expansions ( T a k a n o ^ ) , and (12) f i n i t e differences (Nichols and H i r t ) . The fundamental difference between the analysis of ship motions and the analysis of f l o a t i n g breakwaters i s that the o s c i l l a t o r y mooring forces s i g n i f i c a n t l y a f f e c t the body motions 4 (13) in the l a t t e r case. Yamamoto and Yoshida " have represented (14) the mooring system by lin e a r springs. Adee and Martin also model the moorings by lin e a r springs but suggest that for ordinary slack moorings, the mooring s t i f f n e s s may be neglected. (15) Remery and van Oortmerssen suggest that the non-linear behavior of the moorings i s important. They state that the t o t a l load on the mooring i s made up of o s c i l l a t o r y wave forces, which in the l i n e a r analysis are proportional to the wave height, and steady wind, current, and wave d r i f t forces which are proportional to the square of the wind v e l o c i t y , current v e l o c i t y , and wave height respectively. In th i s thesis, the mooring analysis suggested by Remery and van Oortmerssen w i l l be applied to the problem of finding the motions of a f l o a t i n g breakwater. 1.3 Description of Method The analysis procedure which i s described i n d e t a i l in Chapters 2 and 3 may be divided into two parts. The f i r s t i s the hydrodynamic analysis of an unrestrained f l o a t i n g body. This analysis of f l u i d motion in the region surrounding the body y i e l d s the e x c i t a t i o n and response c h a r a c t e r i s t i c s of the body when subject to an incident wave t r a i n . These body c h a r a c t e r i s t i c s are combined with the mooring system c h a r a c t e r i s t i c s in the second part of the analysis, which i s the s t r u c t u r a l analysis of the combined body-mooring system. The s t r u c t u r a l analysis y i e l d s the motions of the body-mooring system, from which the desired wave amplitudes, pressures, stresses, and body motions may be calculated. For the hydrodynamic analysis, a region of f l u i d surrounding the body i s is o l a t e d . F l u i d motions within the region are described according to po t e n t i a l theory by a dif f e r e n t i a l - e q u a t i o n and boundary conditions. The d i f f e r e n t i a l equation i s solved by the f i n i t e element method to give the f l u i d v e l o c i t y p o t e n t i a l i n terms of the incident wave and body motion amplitudes. The Bernoulli equation i s used to f i n d the res u l t i n g pressures on the body. The pressures are then integrated over the body surface to f i n d the ex c i t i n g forces due to the incident wave and the forces opposing motion of the body. The opposing forces can be resolved into two components, one r e s i s t i n g acceleration and another r e s i s t i n g v e l o c i t y . The f i r s t involves the added mass of the body and the second the damping. The added mass and damping c o e f f i c i e n t s and ex c i t i n g forces found from the f i n i t e element analysis are then combined with the true mass of the body and the hydrostatic s t i f f n e s s to y i e l d three coupled li n e a r equations of motion for the body. Solution of these equations gives the body motion amplitudes and phase angles. The transmitted and r e f l e c t e d wave heights can then be calculated from the body amplitudes and the i r respective v e l o c i t y p o t e n t i a l f i e l d s . F i n a l l y , the steady d r i f t force on the body can be calculated using the p r i n c i p l e of momentum conservation and retaining the second-order terms in the Bernoulli equation. For the s t r u c t u r a l analysis, the response of the system i s assumed to be of two parts. The f i r s t i s a s t a t i c response to the steady d r i f t force and any steady wind and current forces. Here, non-linear equations for a catenary are used to fi n d the equilibrium position of the body and moorings. The mooring 6 lines are then modelled as a series of straight bar elements, each with a mass, added mass and damping. The equations of motion for these bar elements are combined with the equations of motion for the unrestrained body and solved to find the amplitudes of body motion. As for the unrestrained body, the transmitted and reflected wave amplitudes and the steady d r i f t force are then calculated. Since the new d r i f t force is in general different from that used to find the equilibrium con-figuration of the mooring lines, the procedure is repeated until convergence of the d r i f t force is achieved. Finally, the forces in the mooring lines are calculated, completing the analysis. CHAPTER 2 HYDRODYNAMIC ANALYSIS 2.1 I n t r o d u c t i o n The p r o b l e m o f f i n d i n g t h e m o t i o n o f an u n r e s t r a i n e d f l o a t i n g b o d y i n t h e p r e s e n c e o f s u r f a c e waves h a s b e e n d e a l t w i t h b y many a u t h o r s . Newman^^' , G a r r i s o n , a n d Wehausen^"*" h a v e d i s c r i b e d t h e g e n e r a l t h r e e - d i m e n s i o n a l l i n e a r b o u n d a r y -v a l u e p r o b l e m t o be s o l v e d i n o r d e r t o o b t a i n f l u i d v e l o c i t i e s and h e n c e p r e s s u r e s on t h e b o d y and t h e b o d y m o t i o n s . Adee a nd M a r t i n a n d Yamamoto and Y o s h i d a ^ h a v e d e s c r i b e d t h e t w o - d i m e n s i o n a l p r o b l e m w i t h p a r t i c u l a r r e f e r e n c e t o f l o a t i n g b r e a k w a t e r s . I n t h i s c h a p t e r t h e f o r m u l a t i o n o f t h e t w o - d i m e n s i o n a l b o u n d a r y - v a l u e p r o b l e m f o r p o t e n t i a l f l o w w i t h a f r e e s u r f a c e and t h e p r o c e d u r e f o r g e n e r a t i n g t h e e q u a t i o n s o f m o t i o n o f t h e bo d y w i l l be r e v i e w e d . As w e l l , t h e f i n i t e e l e m e n t m e t h o d o f s o l v i n g t h e b o u n d a r y - v a l u e p r o b l e m w i l l be d e s c r i b e d . 8 2.2 Potential Flow The region of f l u i d that w i l l be analysed i s shown i n Figure 1. The breakwater i s assumed to be i n f i n i t e l y long i n the d i r e c t i o n p a r a l l e l to the wave crests so that the f l u i d flow i s two-dimensional. The f l u i d surrounding the breakwater i s assumed to be i n v i s c i d and incompressible and the f l u i d motion i r r o t a t i o n a l so that pot e n t i a l flow theory applies. The x-axis i s at the s t i l l water l e v e l and i s posi t i v e i n the d i r e c t i o n of wave propogation. The y-axis" is. p o s i t i v e upward through the centre of buoyancy of the body when in i t s equilibrium position i n s t i l l water. The vel o c i t y potential <Mx,y,t) i s defined such that 9$/9x=u and 9<I>/9y=v where t i s time and u and v are the f l u i d v e l o c i t i e s in the x and y directions respectively. I t i s further assumed that the f l u i d motions are simple harmonic so that the space and time variables may be separated $ (x,y,t)=Re[(j> (x,y)e i a ) t] 2.1 where co i s the angular frequency of motion. For the above d e f i n i t i o n of cf> the requirements of incompressibility and z e r o - v o r t i c i t y r e s u l t in Laplace's equation ¥ ± + li£ = 0 2.2 9x 2 9 y 2 as the governing d i f f e r e n t i a l equation to be solved in the f l u i d region. The boundary conditions must be expressed i n terms of the v e l o c i t y p o t e n t i a l . On the prescribed ve l o c i t y boundary ( S ^ , the normal v e l o c i t y of the boundary (V ) must equal the normal ve l o c i t y of the f l u i d at the boundary. This y i e l d s the boundary condition J £ = V on Sn 2.3 ^n : n 1 where n i s a unit normal outward from the f l u i d region. For a fixed boundary, the boundary condition i s then |& = 0 2.4 8n On the free surface, the pressure i s zero r e l a t i v e to atmospheric pressure and the normal v e l o c i t y of the free surface must equal the normal v e l o c i t y of a p a r t i c l e at the free surface. The Bernoulli equation may be written as p + | p (u 2 + v 2) + pgy + p | |.= 0 2.5 where p i s the pressure, p i s the density of f l u i d and g the acceleration due to gravity. Putting p=0 in the Bernoulli equation and neglecting the v e l o c i t y squared terms as second order gives the zero pressure condition g r i + | | = 0 at y = 0 2.6 where n i s the surface elevation measured from y=0. The. ve l o c i t y squared terms are neglected as they can be shown to be of the same order as the wave steepness squared. The wave steepness i s assumed to be small. For the lin e a r analysis the free surface boundary condition i s applied at y=0 rather than at the true free surface y=n. The li n e a r i z e d kinematic boundary condition, assuming small slope of the surface p r o f i l e i s Combining Equations 2.6 and 2.7 and eliminating n gives the free surface boundary condition 10 a t y = 0 2.8 On the radiation boundary, the poten t i a l must have the form of a wave t r a v e l l i n g outward from the region. .The po t e n t i a l for a plane wave t r a v e l l i n g i n the +x di r e c t i o n i s $ = R e { i 2 h cosh(k(y+d)) cicot-ikx } 2 _ 9 2co cosh(kd) where h i s the wave height, k i s the wave number 2T T/A, X the wavelength, and d the depth of water. Not a l l the variables on the r i g h t side of Equation 2.9 are independent as the dispersion r e l a t i o n , r e l a t i n g wave number to wave frequency may be applied. The dispersion r e l a t i o n i s to2= gk tanh(kd) 2.10 In order to s a t i s f y Equation 2.9 on the po s i t i v e radiation boundary, $ must be periodic i n (tot-kx) . This i s equivalent to requiring U L = _ ^ i i 2 11 Applying Equation 2.1 to Equation.2.11 gives for the p o s i t i v e radiation boundary M = - M A 2.12 3x c y where c i s the wave c e l e r i t y (c=o)/k) . The same procedure for a wave t r a v e l l i n g i n the -x di r e c t i o n gives for the negative radiation boundary i l = M * 2.13 •3x c y For v e r t i c a l boundaries, both Equation 2.12 and 2.13 can be replaced by 3 i = - i f i U 2.14 3n c T where n i s a unit outward normal from the f l u i d region. The degrees of freedom assigned to the body are indicated in Figure 1. j=l i s horizontal displacement (sway), j=2 i s v e r t i c a l displacement (heave), and j=3 i s rotation about the or i g i n of the coordinate system ( r o l l ) . The.displacement from the equilibrium position at any time i s given by Re{•£je l a ) t} th where i s the complex amplitude i n the j mode. I t i s assumed that a l l amplitudes are small i n comparison to the size of the body. The v e l o c i t y potential in the region surrounding the body i s assumed to be of the form 3 a> = <J>„ + + I 5i<j>4 2.15 j = l 3 3 where <$>0 i s the incident wave p o t e n t i a l , $ i s the pot e n t i a l of the scattered wave from the fixed body, and <j> 1 , <$>2r and <j> 3 are the forced motion potentials due to unit amplitude sway, heave, and r o l l respectively i n calm water. The incident wave po t e n t i a l from Equation 2.9 i s 1 ^ i cosh(k(y+d)) -ikx ~ , c *° = — c o s h (id) c 2 - 1 6 where the subscript i indicates the incident wave. By th i s d e f i n i t i o n a wave crest passes x=0 at oyt=0. The sum of the normal v e l o c i t i e s of the incident and scattered waves at the surface of the fixed body must be zero. This results i n the boundary condition K - 2.17 8n 8n on the body surface S^. The scattered wave poten t i a l can then be found by solving the boundary-value problem with -'3<t>0/3n as the prescribed v e l o c i t y on the body surface. The forced motion potentials can be found in a s i m i l a r manner. Defining r = xi_ + yj_ as the p o s i t i o n vector of a point on the body surface and n = n^i_ + n2J_ as a unit normal vector into the body, a unit amplitude displacement i n each mode y i e l d s the following prescribed v e l o c i t i e s for the body surface, assuming small amplitudes of motion sway V n = i a j n 1 heave V = icon,, 2.18 n 2 r o l l V n = ico (r x n) In the l i n e a r analysis, the boundary condition i s applied at the equilibrium p o s i t i o n of the body surface rather than at the instantaneous position. 2.3 E x c i t i n g Forces, Added Mass, and Damping The p o t e n t i a l flow theory of Section 2.2 i s based on the assumption of i r r o t a t i o n a l flow, hence flow separation and the formation of vortices are not included in the theory. An i n d i c a t i o n of flow separation and thus the a p p l i c a b i l i t y of p o t e n t i a l theory i n finding the forces on the breakwater i s the Keulegan-Carpenter number (K ). The Keulegan-Carpenter number i s proportional to the r a t i o between the amplitude of f l u i d p a r t i c l e horizontal displacement and a t y p i c a l dimension of the body K = U T/B 2.19 c m where U i s the maximum p a r t i c l e horizontal v e l o c i t y , T i s the m period of o s c i l l a t i o n and B i s the beam of the breakwater. For the range of frequencies to be studied, K c w i l l usually be less than three. In th i s range i t can be assumed that i n e r t i a e f f e c t s predominate over drag e f f e c t s , therefore wave forces w i l l be calculated from the pote n t i a l flow solution. While the p o t e n t i a l theory cannot s t r i c t l y be applied to flow past a rectangular body (19) due to the formation of vortices at the corners, Bearman et a l have found that the presence of these vortices has l i t t l e e f f e c t on forces on a square-section cylinder with sides p a r a l l e l to the incident flow at low values of K . Their results suggest that p o t e n t i a l flow theory can be used to fi n d the forces on a rectangular section f l o a t i n g breakwater. The Bernoulli equation (Eqn. 2.5) i s l i n e a r i z e d by neglecting the velocity squared terms as second order. Also, the hydrostatic pressure gy i s omitted so as to give the r e l a t i o n between the pote n t i a l f i e l d and the dynamic pressure p as P " "P3t or p = -ReUojp^e 1^} 2.20 In the l i n e a r i z e d solution, the ex c i t i n g forces on the body are independent of body motion, thus the exciting forces are the integrals over the equilibrium body surface of the pressure due to the incident and scattered waves only. I f the e x c i t i n g force i s of. the form Fj = Re{X je l a ) t} j = l,2,3 2.21 then n . i=l,2 X i = - I W P / s , + V ( r x n ) d s i=3 2.22 where i s the equilibrium body surface. The forces r e s i s t i n g motion of the body can be found by applying the Bernoulli equation to the forced motion po t e n t i a l s . If F. i s the force in the j d i r e c t i o n due to a unit amplitude motion i n the k d i r e c t i o n and p k i s the pressure due to a unit amplitude motion in the k d i r e c t i o n , then F j k = / P k ^ x n ) d s j = 3 / 2 2 ' 2 3 b l I f F ., = Re[f e 1 W t ] 2. 24 jk jk then the force can be expressed in terms of the forced motion potentials by f j k = pco/ lm(d»k) >ds - ip.J Re(* k) < n ) d s ] = \' 2 b l b l 2.25 The force can also be expressed in "terms of components opposing acceleration and v e l o c i t y of the body Vjk = " ajA " bj'A 2-26 where a^ k i s the added mass c o e f f i c i e n t and b_-k i s the damping c o e f f i c i e n t . For a unit amplitude displacement f j k = 0 ) 2 a j k " i w b j k 2 - 2 7 Comparing Equations 2.25 and 2.2 7 y i e l d s n. j=l,2 a ., = 2-f ImC*. ) ( : )ds . , 2.28 3k odJ c Yk' r x n 3 = 3 b l b j k - pj H e ( 4 , k ) ( r n j n ) d s 2 2.29 b l An alternate way of c a l c u l a t i n g the exciting forces i s by (1 f>) the use of Haskind's re l a t i o n s . By applying Green's Theorem, i t i s possible to eliminate the scattered wave poten t i a l from the expression for ex c i t i n g forces. This y i e l d s 8$. 34>0 x j = -PJ ( * 0 _ 3 K " ^ j ^ T i ) d s J = l,2,3 2.30 S l 2.4 Equations of Motion The t o t a l or v i r t u a l mass of the body to be used in the equations of motion i s the sum of the actual mass of the body and the added mass defined by Equation 2.28. Since the o r i g i n of the coordinate system i s not necessarily at the centre of gravity of the body, the mass matrix may contain off-diagonal terms. The mass matrix for the body i s m 0 -my [M] - 0 m 0 2. 31 -mya 0 lo where m i s the mass per unit length of the body, lo i s the polar mass moment of i n e r t i a about the o r i g i n per unit length, and (0,y ) i s the coordinate of the centre of gravity. The hydrostatic s t i f f n e s s matrix can be determined by giving the body a small displacement in each of the three modes in turn and ca l c u l a t i n g the forces required to maintain equilibrium. For small displacements the s t i f f n e s s matrix i s [ K ] = 0 0 0 pgB PgBx 0 pgBx. pgVL zz V 2. 32 where i s the centroid of waterplane area, V i s the underwater volume, I i s the moment of i n e r t i a of the waterplane area ' zz * about the z-axis, and (-OfY^) i s the coordinate of the centre of buoyancy, v and I are calculated for a unit length of breakwater. The mass and s t i f f n e s s matrices, added mass and damping c o e f f i c i e n t s , and ex c i t i n g forces can be assembled into the equations of motion of the body. If [A] i s the matrix of added mass c o e f f i c i e n t s and [B] i s the matrix of damping c o e f f i c i e n t s , then the equations of motion are {-OJ 2([MJ + [ A ] ) + ioj[Bj + [K]}(£) = (X) 2.33 which can be solved for the complex amplitudes The amplitudes of motion can then be used to find the transmitted and r e f l e c t e d wave heights. The wave height i s related to the v e l o c i t y p o t e n t i a l by h = ^ A (X,0) 2. 34 g For the transmitted wave, 3 $ = <j)0 + <f>R + E K^A 2.35 j = l 3 3 and i s evaluated at the po s i t i v e radiation boundary. For the re f l e c t e d wave 3 4> = * R + I ?.<!). 2.36 j = l J J and i s evaluated at the negative radiation boundary. The transmission c o e f f i c i e n t i s defined as the transmitted wave height divided by the incident wave height From conservation of energy, the rate of energy transfer into the f l u i d region must equal the rate of energy transfer out of the f l u i d region. Since the energy in a wave i s proportional to the square of the wave height, the incident, transmitted, and re f l e c t e d wave heights are related by 2 2 2 h. = h + h 2.38 1 r t Si m i l a r l y , the transmission c o e f f i c i e n t and the r e f l e c t i o n c o e f f i c i e n t K , defined as the r e f l e c t e d wave height divided by the incident wave height, are related by K 2 + K 2 = 1 2.39 r t 2 . 5 D r i f t Force According to the previous l i n e a r theory, the unrestrained body w i l l maintain the same mean position'and undergo only simple harmonic motion in the presence of surface waves. In -fact, the body w i l l tend to d r i f t in the d i r e c t i o n of propogation of the waves due to the presence of higher order forces. This d r i f t force i s important when ca l c u l a t i n g the equilibrium configuration of the moorings. Maruo'^' has calculated the second order d r i f t force (21) for a two-dimensional body i n deep water. Longuet-Higgins has extended th i s theory to include shallow water and the absorption of energy by the body. The d r i f t force i s calculated f. by considering conservation of momentum of the body and surrounding f l u i d , including second-order terms. The d r i f t force i s the force required on the body to conserve momentum of the combined body-fluid system. The d r i f t force can also be 18 considered as the force required to maintain the body i n a constant mean position. While the d r i f t force varies slowly with time, a time average of the force i s calculated and the force i s assumed to be steady for further calculations. According to Longuet-Higgins, the d r i f t force i s related to wave heights by F = §5(1 + • 2*^ ,) (h. 2 + h 2 - h 2) 2.40 D 16 smh(2kd) I r t Equation 2.38 need not apply when ca l c u l a t i n g the d r i f t force as energy may be dissipated through the breakwater or mooring system. I f no energy i s dissipated by the breakwater or moorings, Equation 2.40 may be reduced to v - £2(1 + 2 k d )h 2 2 41 FD ~ 8 U + sinh(2kd) , h r ^ ' 4 1 2.6 F i n i t e Element Solution The f i n i t e element method has been used extensively for solving problems i n s t r u c t u r a l mechanics. The method can also be used to solve a variety of steady-state and transient f i e l d problems, one of which i s the flow of an i d e a l f l u i d . The application of f i n i t e elements to f l u i d problems i s detailed • » • i • • ( 2 2 ) c i • ^ (23) , T^ . (6) , . ,. in Zienkiewicz , Segerlmd , and Newton , and w i l l only be described b r i e f l y here. The d i f f e r e n t i a l equation to be solved i s Laplace's equation The boundary conditions can be expressed in the form | ^ + q + a 4 = 0 2.43 3 n ^ v On the various boundaries, the values of q and a are: fixed boundary q = 0 a = 0 prescribed v e l o c i t y boundary q = -V a = 0 free surface q = 0 n a = -to 2/g radiation boundary q = 0 a = ioj/c (22) From the calculus of variations or Galerkin's method , the solution to the above d i f f e r e n t i a l equation and boundary conditions i s the function that minimizes the functional " = J h^Z+ (|^) 2)dxdy + / (.q* + ±a$2)ai 2.44 ft2 3 x 3 y S 2 where 0, i s the domain of the f l u i d and S i s the entire boundary of the f l u i d . In the f i n i t e element method, the domain i s divided up into a f i n i t e number of regions or elements. Figure 2 shows a t y p i c a l f i n i t e element mesh for a rectangular breakwater. Each element has nodes on the boundary where the value of the variable <j> i s matched to adjacent elements. Within any element the T pot e n t i a l 4> can be expressed as (N) (<Me where (N) i s a column vector of interpolation functions and (<J>) i s a column vector of nodal values of $. The p a r t i c u l a r quadratic isoparametric element used here i s described in Appendix A. For an element, minimizing TT with respect to the nodal values of 4> gives the element equation 2. 45 2e 3e The subscript e denotes integration over one element only. i s the body surface, S 2 i s the radiation boundary and i s the free surface. The element equations are assembled using standard f i n i t e element techniques to produce the global equations. 20 I n s e r t i n g the a p p r o p r i a t e values of q and a, the g l o b a l equations are { J 0 [ W W + ] d x d ^ + i ~ ^ < N ) ( N ) d£ " S2 - / — (N) (N) Td£ } (<j>) = / V (N)d£ 2.46 S g q n b 3 S l This can be w r i t t e n as the m a t r i x e q u a t i o n {-oo2 [K 3] + ioo[K2] + [K^Ho)) = (P) 2.47 where [K ] = 1 / (N) (N) Td£ 2.49 c „ b2 [K_] = - f (N) (N) Td£ 2.50 3 g S (p) = f V (N)d£ 2.51 S n  b l The m a t r i c e s [ K ^ ] , ! ^ ] , and [K-^l are symmetric and banded. A Cholesky s o l u t i o n technique capable of h a n d l i n g complex numbers i s employed to solve f o r (<}>). F i g u r e 2 shows the f i n i t e element mesh f o r the lower f r e q u e n c i e s o f a r e c t a n g u l a r breakwater. The mesh has 317 nodes, 88 elements, and a h a l f bandwidth of 32. The mesh i s f i n e s t near the body and along the f r e e s u r f a c e where the h i g h e s t v e l o c i t y g r a d i e n t s are expected. The q u a d r a t i c i s o p a r a m e t r i c element has a l i n e a r v a r i a t i o n of v e l o c i t y . About e i g h t elements per wavelength are r e q u i r e d on the f r e e s u r f a c e to a c c u r a t e l y p r e d i c t wave h e i g h t s and phase r e l a t i o n s a t the r a d i a t i o n boundary. As a r e s u l t the mesh shown i s only good f o r B/X<0.5. 21 CHAPTER 3 MOORING ANALYSIS 3.1 I n t r o d u c t i o n F o r t h e m o o r i n g a n a l y s i s i t i s assumed t h a t t h e f o r c e s on t h e b r e a k w a t e r a r e o f two p a r t s ; a s t e a d y d r i f t f o r c e , w h i c h may i n c l u d e s t e a d y w i n d and c u r r e n t f o r c e s , and t i m e - v a r y i n g f o r c e s w i t h t h e same f r e q u e n c y as t h e i n c i d e n t wave- S i m i l a r l y , t h e r e s p o n s e o f t h e b r e a k w a t e r i s assumed t o c o n s i s t o f a s t a t i c r e s p o n s e t o t h e d r i f t f o r c e p l u s a d y n a m i c r e s p o n s e t o t h e t i m e - v a r y i n g f o r c e s . The s t a t i c r e s p o n s e i s f o u n d b y a p p l y i n g t h e e q u i l i b r i u m e q u a t i o n s f o r a c a t e n a r y c a b l e . T h i s a n a l y s i s y i e l d s t h e e q u i l i b r i u m c o n f i g u r a t i o n f o r t h e d y n a m i c a n a l y s i s . ' I n t h e d y n a m i c a n a l y s i s t h e m o o r i n g c a b l e s a r e m o d e l l e d b y a s e r i e s o f s t r a i g h t , p i n - e n d e d b a r s and t h e s t i f f n e s s , m a s s , and d a m p i n g m a t r i c e s f o r t h e c o m b i n e d m o o r i n g - b r e a k w a t e r s y s t e m a r e a s s e m b l e d . S o l u t i o n o f t h e s e e q u a t i o n s g i v e s t h e m o t i o n s o f t h e b r e a k w a t e r . S i n c e t h e d r i f t f o r c e a s s o c i a t e d w i t h t h e s e m o t i o n s i s i n g e n e r a l n o t t h e same as t h e d r i f t f o r c e assumed i n t h e s t a t i c a n a l y s i s , t h e p r o c e d u r e i s r e p e a t e d u n t i l c o n v e r g e n c e on t h e d r i f t f o r c e i s a c h i e v e d . 3.2 S t a t i c Analysis Figure 3 shows a t y p i c a l breakwater mooring system. The moorings consist of pairs of chains or cables at r i g h t angles to the axis of the breakwater spaced at a distance Z along the breakwater. The anchors are spaced a distance S apart. I t i s assumed that the water depth i s constant between the anchors. much less than the hydrostatic s t i f f n e s s of the breakwater in heave or r o l l . Therefore, i t i s assumed for the s t a t i c analysis that the breakwater i s fixed i n the heave and r o l l degrees of freedom and that the d r i f t force causes a displacement in the sway mode only. The purpose of the s t a t i c analysis i s then to find the horizontal position of the breakwater with respect to the anchors that results^ i n equilibrium between the horizontal forces in the mooring cables and the d r i f t force. Two possible configurations for a mooring cable are given in Figure 4. The cable has a buoyant weight of wc per unit length, a cross-sectional area A, and an e l a s t i c modulus E. The unstretched i n i t i a l length of the cable i s C, the r i s e i s h, and the span L. The horizontal force at the upper end of the cable to maintain equilibrium i s H. The subscripts 1 and 2 refer to the seaward and shoreward cables respectively. Figure 4(a) shows a cable that i s raised above the sea (24) bed for i t s entire length. Equilibrium for the cable y i e l d s For slack moorings, the s t i f f n e s s of the moorings i s 2H w L {(C + A ) 2 - h 2 } 7 = w •sinh ( 2H 3.1 c where A i s the e l a s t i c elongation of the cable given by A = + 2w L H c sinh ( - T J — ) ) w L. 3.2 Equation 3.1 can be written as w L A = [ (iiisinhCjij-) ) + h 1 " C 3 ' 3 c Equating the r i g h t hand sides of Equations 3.2 and 3.3 gives an equation which can be solved i t e r a t i v e l y to give the horizontal force H for a given span L. The slope of the cable at the anchor can be calculated from w w L tan6 = ^ f l h coth C^-) - (C + A ) ] 3.4 If tan6 < 0, the cable l i e s along the sea bed for part of i t s length as shown in Figure 4(b). The raised span L g i s given by L s = ^ o s h ( - i r + x ) 3 - 5 c The i n i t i a l length of the catenary i s . C = C - (L - L ) 3.6 s s Equations 3.2 and 3.3 can be applied by replacing A , L, and C with A , L , and C respectively. The same i t e r a t i v e solution s s s i s used but a new raised span must be calculated for each t r i a l value of H. The t o t a l elongation of the cable, including the portion on the sea bed i s A = A + §-(L - L ) 3.7 s AE s Since the horizontal force cannot be expressed e x p l i c i t l y as a function of the span an i t e r a t i v e procedure must be used to find the equilibrium position of the breakwater. A t r i a l value for the span of the seaward cable (L.^ ) i s chosen and the span of the shoreward cable (L 2) i s calculated. If L 2 + h 2 < C2' the shoreward cable hangs straight down and i s neglected in further calculations. The horizontal force i n each cable i s calculated and horizontal equilibrium EF = F DZ + H 2 - H 1 3.8 i s checked. The span of the seaward cable i s then adjusted according to the sign of EF and new horizontal cable forces are calculated. The procedure i s repeated u n t i l EF = 0 within the desired tolerance. Once horizontal equilibrium of the breakwater i s s a t i s f i e d , the location of any point on the catenary can be determined. The equation of the catenary i s w x y = — cosh(-§- + A) + B 3.9 J w H c where _.. w h w L A = s i n h " 1 ! C w L ) " SIT 3 - 1 0 2H sinh ( 2 §~> B = - — c o s h (A) 3.11 w c x and y are in the l o c a l coordinate system for the cable. In this system the o r i g i n i s at the anchor and the coordinate of the upper end of the cable i s (L,h). In the case where the cable l i e s on the sea bed, the o r i g i n i s at the point of tangency with the sea bed and L i s replaced by L . The tension i n the cable can be calculated from T = H (1 + (p.) V 3.12 d X 25 3.3 Dynamic Analysis The dynamic analysis of the combined mooring-breakwater system uses the matrix s t i f f n e s s method to assemble the equations of motion of the system. The cables are modelled by a series of s traight, pin-ended bars and the breakwater i s modelled by a r i g i d frame connecting the cables to the breakwater degrees of freedom. The f i r s t step i n the dynamic analysis i s to d i s c r e t i z e the system. Figure 5 shows a t y p i c a l d i s c r e t i z e d model. For c l a r i t y , the figure shows four bar elements per cable while i n practice, eight or more elements are used. The s t a t i c analysis gives the equations of the catenaries in terms of l o c a l coordinates for the cables. For n segments per cable, the coordinates of n+1 nodes per cable must be calculated. One node i s at the anchor and another i s at the point of attachment to the breakwater. If the cable l i e s p a rtly on the sea bed, a node i s placed at the point of tangency. The remaining intermediate nodes are placed such that there i s an equal change i n slope between adjacent segments. The coordinates of these nodes are then transformed into the global coordinate system. The global x-axis i s at the s t i l l water le v e l and pos i t i v e i n the d i r e c t i o n of wave propogation. The global y-axis i s at the centre of buoyancy of the body when in equilibrium with the d r i f t force and p o s i t i v e upward. The average i n i t i a l tension i n each segment i s calculated by averaging the tensions at the ends of the element. A description of the matrix s t i f f n e s s method may be found in Refs. (22) and (25). The assembly of the element matrices into the structure matrix and the solution of the matrix equation i s 26 standard for plane-frame problems, therefore only the cable element w i l l be described here. S t i f f n e s s Matrix The structure shown i n Figure 5 i s unstable unless a s t a b i l i t y function i s included in the element s t i f f n e s s matrix. The s t a b i l i t y function represents the resistance to l a t e r a l displacement provided by the i n i t i a l tension i n the cable. Figure 6(a) shows a cable element i n the l o c a l coordinate system. The element has a length L, cross-sectional area A, e l a s t i c modulus E, and i n i t i a l tension T. The four degrees of freedom associated with each element are shown. The force-displacement r e l a t i o n for the element i s of the form [k] (6) = (F) 3.13 where [k] i s the element s t i f f n e s s matrix, (6) i s the vector of nodal displacements, and (F) i s the vector of nodal forces. Applying a displacement to each of the degrees of freedom i n turn and c a l c u l a t i n g the forces required for equilibrium gives the element s t i f f n e s s matrix [k]. A unit a x i a l elongation (Fig. 6(b)) yie l d s the a x i a l s t i f f n e s s terms k n = _ k 3 l = A E / L k„, = k =0 21 4 i A unit l a t e r a l displacement (Fig. 6(c)) yiel d s the s t a b i l i t y terms k22 = _ k42 = T / L ^ q z - 3.15 k12 = k14 = ° Repeating for displacements i n degrees of freedom 3 and 4 gives the complete element s t i f f n e s s matrix IkJ = ± AE Q -AE 0 0 T 0 -T -AE 0 AE 0 0 -T 0 T 27 3.16 I t i s assumed h e r e t h a t t h e a x i a l t e n s i o n i s c o n s t a n t f o r an e l e m e n t . I n f a c t , t h e t e n s i o n w i l l v a r y w i t h t i m e when t h e d y n a m i c l o a d s a r e a p p l i e d , b u t t h i s i s n e g l e c t e d f o r t h e l i n e a r a n a l y s i s . Mass M a t r i x The e l e m e n t mass m a t r i x i s f o u n d i n t h e same way as t h e e l e m e n t s t i f f n e s s m a t r i x , e x c e p t t h a t an a c c e l e r a t i o n r a t h e r t h a n a d i s p l a c e m e n t i s a p p l i e d a t e a c h d e g r e e o f f r e e d o m . T h e r e a r e two m a s s e s a s s o c i a t e d w i t h t h e e l e m e n t . F o r a x i a l a c c e l e r a t i o n , t h e a c t u a l mass o f t h e e l e m e n t i s u s e d . F o r l a t e r a l a c c e l e r a t i o n , t h e v i r t u a l mass, w h i c h i s t h e sum o f t h e a c t u a l and a d d e d m a s s e s i s u s e d . The a d d e d mass f o r t h e c a b l e s i s assumed t o be i n d e p e n -d e n t o f f r e q u e n c y . F o r a c i r c u l a r c r o s s - s e c t i o n t h e a d d e d mass i s t a k e n as t h e mass o f f l u i d d i s p l a c e d b y t h e c a b l e . F o r an a c t u a l mass p e r u n i t l e n g t h m and a v i r t u a l mass p e r u n i t l e n g t h m', t h e e l e m e n t mass m a t r i x i s m/3 0 m/6 0 [m] = L 0 m'/3 0 m'/6 m/6 0 m/3 0 0 m'/6 0 m'/3 3.17 A c h e c k on t h e ab o v e mass and s t i f f n e s s m a t r i c e s i n w h i c h t h e n a t u r a l f r e q u e n c i e s and mode s h a p e s c a l c u l a t e d by t h e m a t r i x s t i f f n e s s method a r e c o m p a r e d t o an a n a l y t i c s o l u t i o n i s c o n t a i n e d i n A p p e n d i x B. 28 Damping Matrix The damping force on the cable i s made up of the drag force of the f l u i d and the st r u c t u r a l damping of the cable i t s e l f . The drag force per unit length of cable i s given by F = ^pC D v | v | 3.18 where v i s the normal v e l o c i t y of the cable, C D i s the drag c o e f f i c i e n t , and D c i s the diameter of the cable. The drag c o e f f i c i e n t i s a function of Reynold's Number (Re) and the Keulegan-Carpenter Number (K ). Unlike the breakwater body motions for which the Keulegan-Carpenter Number was s u f f i c i e n t l y small that drag could be neglected, the cable displacements are ( 2 6 ) large compared to the diameter fo the cable. Sarpkaya has found that for a c i r c u l a r cylinder with K c > 3 0 , C D i s independent of K c and approximately equal to 1.2. C D i s also nearly independent of Reynold's number for the high values of Re at which the cables operate. The drag force i s proportional to ve l o c i t y squared while for l i n e a r equations of motion the damping must be proportional to v e l o c i t y . The drag force can be l i n e a r i z e d for simple harmonic motion. For an amplitude of motion 6, the drag force can be written as F = -|pC DD c(to6) 2cos (tot) | cos (cot) | 3.19 Expanding cos (cot) | cos (cot) | as a Fourier series and retaining only the l i n e a r term gives F - C^D coSv 3.20 3 TT D c which i s a lin e a r function of v e l o c i t y . Integrating Equation 3.19 and the li n e a r approximation Equation 3.20 over one cycle y i e l d s the work done by each force for a given amplitude of displacement. The work calculated from the l i n e a r approximation i s about 1.08 times the work calculated from Equation 3.19. The s t r u c t u r a l damping can be assumed to be Coulomb or f r i c t i o n damping due to interaction between the various elements (27) of the cable or chain. Ramberg and G r i f f i n have measured the log decrement of the free vibrations of a slack cable i n a i r . The Coulomb damping force can be calculated from the log decrement by oo 2m6 f where F i s the Coulomb damping force, oo i s the natural c c 3 ' n frequency of vibration, A i s the log decrement and 6 f i s the amplitude of vibration at which A i s measured. For the 5/8" diameter cable studied by Ramberg and G r i f f i n , the damping force at low cable tensions i s of the - 2 order 10 l b / f t . For the same cable, with an amplitude of motion of 30 times the cable diameter and a frequency of 3 rad/sec, the average f l u i d damping force from Equation 3.20 i s of order 1 l b / f t . Thus the s t r u c t u r a l damping of the cable i s neglected when ca l c u l a t i n g the damping matrix of the cable element. Since the amplitude 6 i s not known i n Equation 3.20, the dynamic analysis i s done f i r s t with no damping and the r e s u l t i n g l a t e r a l amplitudes aire used to calculate the damping force for the next i t e r a t i o n . The procedure i s repeated u n t i l convergence on 6 i s achieved. Calculated values of 6/D indicate that K i s ' c c in the range 20 to 100. The element damping matrix can be found by applying a unit 30 [b] = L 3.22 v e l o c i t y at each degree of freedom and u s i n g Equation 3.20 to f i n d the a s s o c i a t e d f o r c e s . The r e s u l t i n g damping m a t r i x i s ~0 0 0 0~ 0 d/3- 0 d/6 0 0 0 0 0 d/6 0 d/3 where d = 3 ? C D D c a 6 3 ' 2 3 and <5 i s the average l a t e r a l amplitude o f the element. Energy Balance The cable damping i s the o n l y means of d i s s i p a t i n g energy i n the system. The r a t e of energy d i s s i p a t i o n i s 2TT WD 2TT CO / J F Dvdtd£ 3.24 0 0 Using Equation 3.20 and r e p l a c i n g v by co6coscot g i v e s Vtf = -Tr-c^pC^D f 6 3 die D 3TT h D C J 3.25 0 For an element, assuming a l i n e a r v a r i a t i o n of displacements between the ends o f the element / 6 3d£ = j(0 2 3 + -6 2 2 6 „ + 6 2 S , 2 + 6, 3) 0 3. 26 where 6 2 and 6 4 are the l a t e r a l displacements of the ends of the element. For two c a b l e s of n segments each 2n 2 , 3 „ r . n L (S 3 _L JC 2 * j _ , s 2 j _ x 3 D . L n 3TT w = y n L i = l p C D D c ? ( - 6 2 + 6 2 6 k + & 2 & " + & k ]i 3. 27 This r a t e of energy d i s s i p a t i o n must equal the d i f f e r e n c e between the r a t e of energy t r a n s f e r i n t o the f l u i d r e g i o n and the r a t e of energy t r a n s f e r out of the f l u i d r e g i o n . The r a t e of energy 3 1 t r a n s f e r p e r u n i t w i d t h o f wave i s P = P^L-c 3 . 2 8 8 g wh e r e i s t h e g r o u p v e l o c i t y c = i CI + — :— e T ^ T - H - r ) c 3.29 g 2 s m h ( 2 k d ) F o r c o n s e r v a t i o n o f e n e r g y 2 Zpgc h g i CI - K - K ) - W „ = 0 3.30 8 v t r ' D F o r an u n r e s t r a i n e d b o d y , W D = 0 and t h e ab o v e e q u a t i o n s i m p l i f i e s t o E q u a t i o n 2.39. R e p r e s e n t a t i o n o f B r e a k w a t e r The b o d y o f t h e b r e a k w a t e r i s r e p r e s e n t e d b y r i g i d beam e l e m e n t s c o n n e c t i n g t h e p o i n t s o f a t t a c h m e n t o f t h e c a b l e s t o a node a t t h e o r i g i n o f t h e c o o r d i n a t e s y s t e m . T h i s i s n e c e s s a r y b e c a u s e t h e mass, a d d e d mass, d a m p i n g , s t i f f n e s s , a n d e x c i t i n g f o r c e m a t r i c e s f r o m t h e u n r e s t r a i n e d b o d y a n a l y s i s " a r e e x p r e s s e d f o r t h r e e d e g r e e s o f f r e e d o m a t t h e o r i g i n . The r i g i d beam e l e m e n t s a r e f i x e d t o t h e node a t t h e o r i g i n a n d p i n - e n d e d a t t h e c a b l e a t t a c h m e n t n o d e s . T h e s e beam e l e m e n t s e n s u r e t h e c o r r e c t r i g i d - b o d y d i s p l a c e m e n t r e l a t i o n s h i p b e t w e e n d i s p l a c e m e n t s a t t h e o r i g i n node and d i s p l a c e m e n t s a t t h e u p p e r e n d s o f t h e c a b l e s . T h e r e i s no mass o r d a m p i n g a s s o c i a t e d w i t h t h e s e e l e m e n t s . The s t i f f n e s s m a t r i x f o r beam e l e m e n t s may be f o u n d i n R e f . 25. J 32 Assembly of Structure Matrices The element matrices [m], l b ] , and [k] are assembled into the structure matrices [M], IB], and IK] using the usual techniques of matrix s t i f f n e s s analysis. Degrees of freedom are assigned to the nodes as shown i n Figure 5. The element matrices are transformed into the global coordinate system and added into the structure matrices. The mass, added mass, damping, and s t i f f n e s s matrices from the unrestrained analysis (Eqn. 2.33) are added at the degrees of freedom of the o r i g i n node. The load vector (X) i s expanded to match the increased number of degrees of freedom. The only non-zero entries in the load vector are associated with the o r i g i n node degrees of freedom as i t i s assumed that the incident wave forces do not act d i r e c t l y on the cables. F i n a l l y , the assumption of simple harmonic motion i s applied in order to express v e l o c i t i e s and accelerations in terms of displacements. The r e s u l t i s the matrix equation {-GJ2 [M] + icu[B] + [K]} U ) = (X) 3.31 The matrices [M] , [B] , and [K] are b'anded, symmetric, and of order 4n + 3. In addition [K] i s p o s i t i v e d e f i n i t e . The equations are solved by the same complex Cholesky solution used for the f l u i d f i n i t e elements. The r e s u l t i n g displacement vector (?) contains both the cable nodal amplitudes and the breakwater amplitudes. The transmission c o e f f i c i e n t and a new d r i f t force are calculated from the breakwater amplitudes i n the same way as for the unrestrained body. Figure 7 i s a flow chart for the complete solution. The solution for one frequency using a 317 node f i n i t e element mesh and 8 elements per cable takes approximately 4 seconds CPU time 33 on an Amdahl 470 V/6 Model II computer. Of this approximately 3 seconds are required for the f i n i t e element solution and one second for the mooring solution. A t y p i c a l computer p l o t showing both the equilibrium and instantaneous position of the breakwater and cables i s given in Figure 8. 34 CHAPTER 4_ MOTION OF THE UNRESTRAINED BODY 4.1 A d d e d Mass and Damping The i m p o r t a n t r e s u l t s o f t h e h y d r o d y n a m i c a n a l y s i s o f t h e u n r e s t r a i n e d body p e r t a i n t o t h e a d d e d mass and d a m p i n g c o e f f i c i e n t s , t h e e x c i t i n g f o r c e s , t h e t r a n s m i s s i o n c o e f f i c i e n t , and t h e d r i f t f o r c e . The a d d e d mass, d a m p i n g , and e x c i t i n g f o r c e s a p p e a r i n t h e e q u a t i o n s o f m o t i o n o f t h e c o m b i n e d b o d y -m o o r i n g s y s t e m w h i l e t h e t r a n s m i s s i o n c o e f f i c i e n t a n d d r i f t f o r c e s e r v e as t h e b a s i s f o r d e t e r m i n i n g t h e e f f e c t s o f t h e m o o r i n g s y s t e m . The a d d e d mass and d a m p i n g c o e f f i c i e n t s and e x c i t i n g f o r c e s f o r s e v e r a l s i m p l e g e o m e t r i e s h a v e b e e n c a l c u l a t e d a c c o r d i n g t o p o t e n t i a l t h e o r y and e x p e r i m e n t a l l y v e r i f i e d by (29) V u g t s . The p o t e n t i a l t h e o r y r e s u l t s w e r e c a l c u l a t e d b y t h e (4) m ethod o f U r s e l l . S i n c e V u g t s was i n t e r e s t e d p r i m a r i l y i n s h i p m o t i o n s , i n f i n i t e w a t e r d e p t h was assumed and t h e f r e q u e n c y was d e s c r i b e d by t h e d i m e n s i o n l e s s p a r a m e t e r c o V (B/2g) w h e r e B i s t h e beam o f t h e b o d y . F o r t h e c a s e o f a f l o a t i n g b r e a k w a t e r , i n t e r m e d i a t e and s h a l l o w d e p t h s m u s t be c o n s i d e r e d w h i c h means t h e f r e q u e n c y i s no l o n g e r s u f f i c i e n t t o d e s c r i b e t h e w a v e l e n g t h o f t h e i n c i d e n t wave. A more u s e f u l d i m e n s i o n l e s s p a r a m e t e r f o r describing frequency i s the beam to wavelength r a t i o B/X. In this case, a family of added mass and damping curves are produced, each corresponding to a d i f f e r e n t water depth r a t i o B/d. In deep water B/X = (1/TT) (WB/2g) 2 . The s u i t a b i l i t y of the f i n i t e element method i n predicting the motions of the breakwater depends on both the agreement of f i n i t e element results with other p o t e n t i a l flow solution methods and the agreement of pote n t i a l theory results with experiment. Figures 9 through 11 compare results for various added mass and damping c o e f f i c i e n t s and ex c i t i n g forces for a c i r c u l a r cylinder and a rectangular cylinder with a beam to draught r a t i o of 4:1. The s o l i d curves correspond to the results of Vugts. The added mass and damping curves are-calculated according to p o t e n t i a l theory by the method of U r s e l l . .The ex c i t i n g forces are calculated from the damping c o e f f i c i e n t s by the relationship given by (6) Newman IX.I2 = 2pgc b.. 4.1 where c^ i s the group v e l o c i t y of the wave t r a i n . This i s v a l i d only for bodies that are symmetric about x = 0. The f i n i t e element results are presented as points and are seen to correspond closely to the pote n t i a l theory results of Vugts. Two f i n i t e element meshes were required to cover the range of frequencies shown on the graphs. For 0.1 < B/X < 0.5, the 317 node mesh shown i n Figure 2 was used. This was adapted to the c i r c u l a r cylinder by changing only the elements immediately adjacent to the body. The free surface elements of the mesh have a width of B/4, therefore for the shortest wave, there are eight surface elements per wavelength. The mesh extends a distance 2.5B from the body, t h e r e f o r e f o r the l o n g e s t wave, the mesh extends 1 / 4 A from the body. For 0.5 < B/X < 1.75, a 315 node mesh was used. The f r e e s u r f a c e elements have a width of B/10, t h e r e f o r e f o r the s h o r t e s t wave there are 5.7 s u r f a c e elements per wavelength. The mesh extends a d i s t a n c e o f B away from the body, t h e r e f o r e f o r the l o n g e s t wave, the mesh extends 1/2X from the body. Using these two meshes y i e l d s added mass and damping c o e f f i c i e n t s t h a t are g e n e r a l l y w i t h i n 5% of the p o t e n t i a l theory values of Vugts. The e f f e c t o f mesh s i z e on the accuracy o f the r e s u l t s has not been s t u d i e d . The experimental values of Vugts agree c l o s e l y w i t h the p o t e n t i a l theory r e s u l t s with a few e x c e p t i o n s . At very low fr e q u e n c i e s (B/X < 0.02) the experimental values are c o n s i d e r a b l y d i f f e r e n t from the p o t e n t i a l theory r e s u l t s . A frequency of B/X = 0.02 and a wave steepness of H/X = 0.01 corresponds to a Keulegan-Carpenter number of 16 which i s i n the range where v i s c o u s e f f e c t s are important. B/X = 0.02 i s below the range of fr e q u e n c i e s o f i n t e r e s t as the t r a n s m i s s i o n c o e f f i c i e n t i s n e a r l y u n i t y f o r such long waves. Other d i s c r e p a n c i e s occur i n the damping c o e f f i c i e n t i n r o l l and i n the coupled modes i n v o l v i n g r o l l . The experimental damping c o e f f i c i e n t i s sometimes much g r e a t e r than t h a t p r e d i c t e d by p o t e n t i a l theory. As a r e s u l t , p r e d i c t e d r o l l amplitudes may be too l a r g e , p a r t i c u l a r l y near the resonant frequency i n r o l l . A dee^ 3 (^ suggests u s i n g twice the p o t e n t i a l theory r o l l damping i n the equations o f motion to r e p r e s e n t the v i s c o u s e f f e c t s . 37 4.2 Motion of the U n r e s t r a i n e d Body ^ The combination of added mass and damping c o e f f i c i e n t s and e x c i t i n g f o r c e s with the body mass and h y d r o s t a t i c s t i f f n e s s g i v e s the equations of motion f o r the u n r e s t r a i n e d body (Eqn. 2.33). These are s o l v e d to gi v e the. amplitudes of body motion. F i g u r e 12 shows the amplitudes of motion i n the heave and sway modes f o r a c i r c u l a r c y l i n d e r . The mass of the c y l i n d e r i s pV and the centre o f mass i s assumed to be at the s t i l l water l e v e l . F i g u r e 13 shows the corresponding t r a n s m i s s i o n c o e f f i c i e n t . Both amplitudes and the t r a n s m i s s i o n c o e f f i c i e n t are presented as f u n c t i o n s o f the beam to wavelength r a t i o (B/A). For a c i r c u l a r c y l i n d e r , o s c i l l a t i o n s i n the r o l l mode do not cr e a t e any di s t u r b a n c e i n the f l u i d so the amplitude of the t r a n s m i t t e d wave w i l l depend only on the heave and sway amplitudes and phase angles. At low f r e q u e n c i e s (B/A < 0.1) the sway and heave motions are i n phase w i t h , and have the same amplitude as, the h o r i z o n t a l and v e r t i c a l p a r t i c l e e x c u r s i o n s r e s p e c t i v e l y . For a deep water wave-, the h o r i z o n t a l and v e r t i c a l amplitudes f o r a p a r t i c l e at the f r e e s u r f a c e w i l l be equal to the wave amplitude. Thus the breakwater f o l l o w s the same c i r c u l a r o r b i t as the water p a r t i c l e s around i t and d i s t u r b a n c e of the wave t r a i n i s at a minimum. T h i s 1 r e s u l t s i n a t r a n s m i s s i o n c o e f f i c i e n t near u n i t y ( F i g . 13). Because the i n c i d e n t wave i s not g r e a t l y d i s t u r b e d by the break-water, the f i n i t e element s o l u t i o n i s not r e q u i r e d and a much (16) simpler approximation such as the Froude-K'rylov hypotheses can be used to f i n d the f o r c e s on the body. 38 As the frequency of the incident wave increases, the v e r t i c a l amplitude of body motion increases to a maximum, then decreases while the horizontal amplitude steadily decreases. In addition, a phase angle i s introduced between the body displacements and the p a r t i c l e displacements of the incident wave. The disturbance caused to the incident wave by the breakwater increases with frequency and i n general the transmission coef-f i c i e n t decreases. It i s in this frequency range that the f i n i t e element method i s useful i n finding the p o t e n t i a l f i e l d s due to wave scattering and body motion. At high frequencies (B /x > 1.0) the heave amplitude approaches zero and the predominant motion i s sway. Since the short waves attenuate quickly with depth, very l i t t l e of the incident wave passes under the body, thus the transmitted wave is produced mainly by the sway motion. This can be checked by considering the incident wave to be completely r e f l e c t e d and the transmitted wave to be completely generated by the horizontal motion of the body. Linear p o t e n t i a l theory gives the dynamic pressure on a v e r t i c a l wall with a r e f l e c t i o n c o e f f i c i e n t of 1.0 as , cosh(k(y + d) ) . 0 p = pgh. , - . x cosojt 4.2 r M ^ l cosh(kd) For a wall extending from y = 0 to y = -D, where D i s the draught of the breakwater, and for deep water waves, integrating the pressure over the depth gives the force on the wall as P g 2 h i _ k D F = — 5 — ^ ( 1 " e )cosojt 4.3 CO Assuming only one degree of freedom (sway) for the breakwater, the equation of motion becomes 39 (-.u2 (.m + C I - Q ) + i w h 1 1 ) ? 1 = F 4.4 Neglecting damping, the sway amplitude i s then . 2 ^ h i U - e - k D ) ' ^ l 1 = (m + a n ) 4 * 5 The dashed lin e of Figure 12 shows the sway amplitude for a c i r c u l a r cylinder calculated from Equation 4.5. The amplitude of the transmitted wave can be estimated by (16) the wavemaker theory . The height of wave generated by the horizontal motion of a v e r t i c a l wall i s A 0 v h = iSL / u(y) e K ydy 4.6 where the horizontal v e l o c i t y of the wall i s U(y)coscot. Taking U(y) = OJ|£JJ for -D <_ y _< 0 and U(y) = 0 elsewhere, the trans-mission c o e f f i c i e n t becomes K = i P ^ l f j ^ ) 4.7 t u H (m + a i ; L) Equation 4.7 assumes that none of the incident wave i s transmitted beneath the breakwater. This would require a draught of break-water of at least one-half the wavelength, however i t appears that the approximation i s good even for smaller r a t i o s of D / X . The dashed lin e of Figure 13 indicates the transmission coef-f i c i e n t for a c i r c u l a r cylinder as calculated from Equation 4.7. For B/X > 1.0 the approximate solution corresponds closely to the f i n i t e element solution. There can be an advantage to using the approximation i n that the only f i n i t e element r e s u l t required i s the added mass c o e f f i c i e n t . For some shapes th i s c o e f f i c i e n t may be known, or i t can be estimated by extrapolating the results for lower frequencies. Also, the added mass can be accurately calculated using a f i n i t e element gri d that i s too coarse to 40 allow c a l c u l a t i o n of the wave height on the radiation boundary. The wave height depends on the phase r e l a t i o n among the various p o t e n t i a l f i e l d s at the boundary and a fine grid i s required to accurately predict these phase r e l a t i o n s . 4. 3^  D r i f t Force The f i n a l r e s u l t of the hydrodynamic analysis i s the d r i f t force. According to Maruo^ 2 0^, the d r i f t force on an unrestrained body i s proportional to the square of the r e f l e c t e d wave amplitude. From Equation 2.39, the d r i f t "force i s thus proportional to 2 (1 - ) . For long wavelengths, where the entire wave i s trans-mitted, the d r i f t force should be near zero. For short wave-lengths, where nearly a l l the wave i s r e f l e c t e d , the d r i f t force 1 2 should approach -^pg (h^/2) . Figure 14 shows the d r i f t force, 1 2 normalized by div i d i n g by ^pg(h^/2) for an unrestrained c i r c u l a r cylinder. The d r i f t force i s proportional to the square of the incident wave height while the o s c i l l a t o r y forces are li n e a r functions of the wave height. Thus the d r i f t force i s expected to be more important for large wave heights than for small. 41 CHAPTER 5 EFFECT GE MOORINGS 5.1 I n t r o d u c t i o n The s o l u t i o n of the u n r e s t r a i n e d motion problem gi v e s the d e s i r e d p r e s s u r e s , motions, and t r a n s m i s s i o n c o e f f i c i e n t f o r a f r e e l y f l o a t i n g body. When c o n s i d e r i n g a f l o a t i n g breakwater, the body of the breakwater i s r e s t r a i n e d by some type of mooring system and i t i s d e s i r a b l e to know the e f f e c t of the mooring system on the body motions and the f o r c e s i n the mooring l i n e s . (13) Yamamoto and Yoshida have c o n s i d e r e d the problem by mod e l l i n g the mooring l i n e s as l i n e a r s p r i n g s and i n c l u d i n g the s p r i n g s t i f f n e s s i n the three equations of motion of the unres-t r a i n e d body. This method has a l s o been suggested by Adee and (14) Mart i n . The method i s u s e f u l when c o n s i d e r i n g t a u t mooring l i n e s s i n c e the fo r c e - d i s p l a c e m e n t r e l a t i o n f o r t a u t cables can be w e l l r e p r e s e n t e d by a l i n e a r s p r i n g . A l s o f o r t a u t c a b l e s , the i n i t i a l t e n s i o n i n the cables i s very much l a r g e r than the d r i f t f o r c e . Thus the cable s t i f f n e s s i s n e a r l y independent of wave h e i g h t and frequency. In p r a c t i c e , s l a c k moorings are g e n e r a l l y used t o reduce anchor f o r c e s and allow f o r changes i n water depth due to t i d e s . Adee and M a r t i n have used the r e s u l t s of the u n r e s t r a i n e d a n a l y s i s as an approximation of a breakwater with slack moorings. In this chapter, a t y p i c a l breakwater section with slack moorings w i l l be analysed using the procedures of Chapters 2 and 3. F i r s t , a dimensional analysis w i l l be carried out to determine the important parameters, then the e f f e c t of varying some of these parameters w i l l be demonstrated. 5.2 Dimensional Analysis The inclusion of mooring lines greatly increases the number of variables needed to describe the behavior of the K system. For a given body shape the unrestrained motion may be defined i n terms of h^,d, and X (which specify the incident wave t r a i n ) , p, g, and B. In addition, the dispersion r e l a t i o n oo2 = gk tanh(kd) 5.1 i s required to relate the wavelength to the frequency of the wave. A dimensional analysis gives for any chosen dependent variable, say h^ _, E7 = ^ t ' X ' A ^ 5 ' 2 i For deep water (B/d 0) and small wave heights (h^/A 0) , h t/h^ asymptotically approaches a f i n i t e value, therefore Equation 5.2 reduces to £ = f n ( f ) 5.3 l Thus for deep water and lin e a r waves, the transmission c o e f f i c i e n t of an unrestrained body depends only on the geometry of the body and the beam to wavelength r a t i o . Similar functional dependence exists for other dependent variables such as amplitudes of motion and forces on the body. In order to completely describe the motions of a breakwater restrained by mooring cables, the additional variables S, Z, and Dc, p c , C, and E for each cable must be s p e c i f i e d . p c i s the unit weight of'the cable. For s i m p l i c i t y , t h i s analysis w i l l assume that both cables are i d e n t i c a l and inextensible and thus only the f i v e additional variables S, Z, D c, p c , and C are required. Combining these with the variables for unrestrained body motion and performing a dimensional analysis y i e l d s \ _ . r B ^ B S C Z ^ c ^ c . h. - t r U X ' X 'd'd'd'B'B 'p ; 3 - 4 1 Again, the wave frequency need not enter Equation 5.4 as the dispersion r e l a t i o n (Eqn 5.1) relates frequency to the wavelength. The r a t i o h./h. i s the transmission c o e f f i c i e n t K,. As r I t this i s a measure of the e f f i c i e n c y of the breakwater, i t i s chosen here as the representative dependent variable and the. e f f e c t of other parameters on K t w i l l be investigated. B/X i s a measure of the frequency of the incident wave. The p l o t of K vs. B/X shows the range of frequencies for which the breakwater i s e f f e c t i v e . h^/X i s the wave steepness. For a linear analysis such as that of the unrestrained body, the transmission c o e f f i c i e n t i s independent of the wave steepness. In the mooring analysis, the d r i f t force, i n i t i a l configuration of the cables, and the cable damping are a l l non-linear, therefore the transmission c o e f f i c i e n t w i l l depend on the wave steepness. B/d i s a measure of the depth of water. In the hydrodynamic analysis, the added mass and damping c o e f f i c i e n t s and e x c i t i n g forces depend on the B/d r a t i o . In p a r t i c u l a r , the added mass in (31) heave increases for shallow water. Bai , and Yamamoto and Yoshida have found that for depth to draught r a t i o s (d/D) greater than s i x , i n f i n i t e depth can be assumed. In the mooring analysis, B/d along with S/d are needed to describe the geometry of the moorings. The r a t i o C/d i s the scope of the mooring l i n e s . For a given B/d and S/d, the r a t i o C/d i s a measure of the tautness of the cables. The aspect r a t i o Z/B i s the length of breakwater associated with one set of mooring l i n e s . The mooring lines are described by a dimensionless size parameter Dc/B and th e i r s p e c i f i c gravity P c/p. 5.3 Properties of Breakwater The dimensional analysis indicates there are eight independent parameters a f f e c t i n g the motion of a moored breakwater For this study only one geometry and three independent parameters, B/X, h^/X, and C/d w i l l be considered. The general arrangement of the breakwater to be analysed i s shown in Figure 15. The breakwater body i s rectangular i n section with a beam to draught r a t i o (B/D) of 4.0. - The mass of the body i s the underwater volume times the density of water. The mass i s assumed to be evenly destributed throughout the underwater volume when ca l c u l a t i n g the centre of gravity and the polar moment of i n e r t i a of the body. Thus the moment of i n e r t i a of the body about the o r i g i n i s yfa'P 6'* a n d t h e c e n t r e o f gravity i s at (0,-B/8). The additional v e r t i c a l force due to the mooring cables has been neglected so v e r t i c a l equilibrium i s not s a t i s f i e d However, the v e r t i c a l component of the mooring lin e tensions i s generally less than five percent of the weight of the body for 45 the breakwater analyzed here. The added mass and damping c o e f f i c i e n t s i n heave and sway for th i s body are shown i n Figure 10. The frequency range to be studied i s 0.05 < B/X < 1.75. Most v a r i a t i o n i n K t occurs within t h i s range. The range of wave steepness considered is' 0.01 <_ h^/X < 0.14. For hu/X = 0.01, the d r i f t force i s nearly zero. Also, the l a t e r a l amplitude of the cables, and thus the drag i s near zero. Therefore, the results for h^/X = 0.01 should correspond to those of a t o t a l l y l i n e a r analysis. h^/A = 0.14 corresponds approximately to the steepest wave i n deep water. The depth r a t i o B/d i s set at 0.2 so that for most of the frequency range studied (B/X > 0.1) the wavelength to depth r a t i o i s less than two and deep water can be assumed. This allows comparison of the hydrodynamic c o e f f i c i e n t s found by the f i n i t e element method with the results of Vugts. A range of C/d values w i l l be studied to show the e f f e c t of taut or slack moorings. The remaining parameters are a r b i t r a r i l y selected to r e f l e c t t y p i c a l conditions for f l o a t i n g breakwaters. These parameters are: Z/B = 4.0, S/d = 5.5, Dc/B = 0.0187, and P c/p = 6.8. For the above parameters and with C/d = 3.0, the t o t a l cable weight, assuming a s o l i d c i r c u l a r cross-section, i s 6.4% of the weight of the breakwater. 46 5.4 E f f e c t of Moorings From Equation 2.35, the ve l o c i t y p o t e n t i a l of the trans-mitted wave i s 3 * = <j> + cf) + I ? . o > . 5.5 u j=l J J The transmission c o e f f i c i e n t can be written as 3 5.6 The v e l o c i t y potentials are functions of the incident wave and the breakwater geometry, therefore any change i n the transmission c o e f f i c i e n t due to the moorings i s the resu l t of a change i n the complex amplitudes £j. Figure 16 shows the transmission c o e f f i c i e n t for the rectangular breakwater with f i v e d i f f e r e n t r e s t r a i n t conditions; free,- fixed, sway motion only, heave motion only, and r o l l motion only. As for the unrestrained c i r c u l a r cylinder discussed i n Chapter 4, the response of the rectangular breakwater may be divided into three frequency ranges; low frequency (B/X < 0.1), intermediate frequency (0.1 < B/X < 1.0), and high frequency (B/X > 1.0). At low frequencies, the transmission c o e f f i c i e n t i s near unity regardless of the r e s t r a i n t condition. In terms of Equation 5.6, the incident wave pot e n t i a l O>Q predominates at low frequencies. At intermediate frequencies, the transmission c o e f f i c i e n t i s highly dependent on the r e s t r a i n t condition. The transmission c o e f f i c i e n t i s generally highest for the freely f l o a t i n g body and lowest for the fixed body. For the freely f l o a t i n g body there i s 47 a condition of complete transmission at about B/A = 0.38. The location of this maximum i s extremely sensitive to changes i n the location of the centre of gravity of the body. The r o l l motion only condition also shows a l o c a l maximum in the trans-mission c o e f f i c i e n t near B./A = 0.38. For the fixed, sway only, and heave only conditions, the transmission c o e f f i c i e n t decreases monotonically with increasing B/A. At high frequencies only the free and sway only conditions have a non-zero transmission c o e f f i c i e n t . This supports the approximation of Section 4.2 that for high frequencies, a l l the transmitted wave i s produced by motion i n the sway mode. Figure 17 shows the transmission c o e f f i c i e n t for the breakwater with a t y p i c a l slack mooring. The parameters are C/d = 3.0 and h^/A = 0.1. The fixed and free transmission c o e f f i c i e n t s are shown for reference. Figures 18 through 20 show the amplitude and phase angle for each mode in both the free and slack moored conditions. The phase angle 6 i s defined such that the displacement at any time i s given by | | cos (cot - 6 ) . At low frequencies the moorings do not change the amplitude or phase angle of the motions. The body of the break-water follows the same path as the surrounding water p a r t i c l e s , just as for the unrestrained c i r c u l a r cylinder. At intermediate frequencies, the moorings have l i t t l e e f f e c t on the heave amplitude or phase angle. The moorings s i g n i f i c a n t l y reduce the r o l l amplitude and change the amplitude and phase angle of the sway motion. These changes can be explained by comparing the unrestrained and slack-moored equations of motion (Eqn. 2.33 and Eqn. 3.31). In the heave mode, the 48 u n r e s t r a i n e d body has c o n s i d e r a b l e damping and h y d r o s t a t i c s t i f f n e s s . The s t i f f n e s s and damping i n t h e m o o r i n g s i s s m a l l by c o m p a r i s o n , t h u s t h e m o o r i n g s have l i t t l e e f f e c t on t h e heave m o t i o n . T h e r e i s a s l i g h t r e d u c t i o n i n a m p l i t u d e n e a r t h e h e a v e r e s o n a n t f r e q u e n c y . I n t h e r o l l mode, t h e u n r e s t r a i n e d body has c o n s i d e r a b l e s t i f f n e s s b u t v e r y l i t t l e damping. T h i s l e a d s t o t h e r e s o n a n t r e s p o n s e shown i n F i g u r e 20. The m o o r i n g s i n t r o d u c e some damping i n t o t h e s y s t e m , r e s u l t i n g i n a l a r g e r e d u c t i o n i n r o l l a m p l i t u d e n e a r t h e r e s o n a n t f r e q u e n c y . T h e r e i s a l s o a s l i g h t change i n p h a s e a n g l e a t t h e r e s o n a n t f r e q u e n c y due t o t h e e x t r a damping. I f t h e m o o r i n g s were m o d e l l e d as l i n e a r s p r i n g s , t h e r e s o n a n t r o l l f r e q u e n c y w o u l d change b u t t h e r e w o u l d be no r e d u c t i o n i n maximum r o l l a m p l i t u d e . I n t h e sway mode, t h e u n r e s t r a i n e d body has mass and damping, b u t no s t i f f n e s s . The m o o r i n g s i n t r o d u c e some h o r i z o n t a l s t i f f n e s s r e s u l t i n g i n a l a r g e change i n p h a s e a n g l e ( F i g . 18). The a m p l i t u d e i n sway i s a l s o r e d u c e d . The r e s u l t o f t h e above changes i s a g e n e r a l r e d u c t i o n i n t h e t r a n s m i s s i o n c o e f f i c i e n t . The f u l l t r a n s m i s s i o n c o n d i t i o n a t B/X = 0.38 i s e l i m i n a t e d due t o t h e r e d u c t i o n i n r e s o n a n t r o l l a m p l i t u d e . The g r e a t e s t change i n K f c due t o t h e m o o r i n g s o c c u r s n e a r t h e m i d d l e o f t h e f r e q u e n c y r a n g e . N e a r B/X = 0.1 and B/X =1.0 t h e m o o r i n g s have l i t t l e e f f e c t . A t h i g h f r e q u e n c i e s , t h e s l a c k - m o o r e d t r a n s m i s s i o n c o e f f i c i e n t a p p r o a c h e s t h a t o f t h e u n r e s t r a i n e d body, d e s p i t e t h e a d d i t i o n a l h o r i z o n t a l s t i f f n e s s p r o v i d e d by t h e m o o r i n g s . A t h i g h f r e q u e n c i e s t h e mass terms i n t h e e q u a t i o n s o f m o t i o n p r e d o m i n a t e . S i n c e t h e b r e a k w a t e r body mass i s much g r e a t e r t h a n that of the moorings, the slack-moored and unrestrained motions are nearly i d e n t i c a l . Thus the short wavelength approximation developed for the unrestrained body can also be applied to a slack-moored body. Figure 21 shows the transmission c o e f f i c i e n t for the slack-moored breakwater for four d i f f e r e n t values of wave steepness hu/A. The wave steepness determines the r e l a t i v e magnitudes of the steady d r i f t force and the time dependent forces. The d r i f t force i s proportional to the square of the wave height while the time dependent forces increase l i n e a r l y with wave height. For hu/A = 0.01, the d r i f t force i s e s s e n t i a l l y zero. As the d r i f t force increases, the span of the seaward cable increases .as does i t s horizontal s t i f f n e s s 8H^/8L^. As we l l , the span and horizontal s t i f f n e s s of the shoreward cable decreases. Taking derivatives of the cable equilibrium equation (Eqn. 3.1), i t can be shown that both 9H/8L and 3 2H/8L 2 are always p o s i t i v e . For i d e n t i c a l cables, the net r e s u l t of an increase in the d r i f t force w i l l be an increase in the horizontal s t i f f n e s s of the combined system. The e f f e c t of the increase i n mooring s t i f f n e s s i s a decrease in the transmission c o e f f i c i e n t in the intermediate frequency range. In the high and low frequency ranges, the transmission c o e f f i c i e n t i s nearly independent of the wave steepness since at these frequencies the mooring s t i f f n e s s has l i t t l e e f f e c t on K,.. In Figure 21, K. i s seen to decrease with t t increasing hu/A in the intermediate frequency range only. A similar pattern appears in Figure 22 with the trans-mission c o e f f i c i e n t decreasing with increasing cable tautness. 50 For C/d = 3.33, the shoreward cable i s completely slack and i s neglected except for waves in the low frequency range where the d r i f t force i s near zero. As the cable tautness increases (decreasing C/d) the horizontal and v e r t i c a l s t i f f n e s s of the mooring increases and the transmission c o e f f i c i e n t i s reduced. Again the e f f e c t i s limited to the intermediate frequency range. The tension at the upper end of the seaward and shoreward cables for h^/A =0.1 and C/d = 3.0 i s given i n Figures 23 and 24. The dashed l i n e indicates the cable tension in s t i l l water. At low frequencies the r e f l e c t i o n c o e f f i c i e n t and thus the d r i f t force i s near zero. As a r e s u l t the equilibrium cable tensions are both near t h e i r value for s t i l l water. As the d r i f t force increases, the equilibrium tension in the seaward cable increases while the shoreward cable tension decreases. At high frequencies, the r e f l e c t i o n c o e f f i c i e n t approaches unity, but since the tensions are given for constant wave steepness, the height of the r e f l e c t e d wave and the d r i f t force decrease. As a r e s u l t , the equilibuium tensions approach th e i r s t i l l water value. The dynamic cable tension i s greatest at B/A - 0.35. This corresponds to the greatest reduction in Kfc due to the moorings. At higher frequencies, the dynamic cable tensions approach zero as the amplitudes of body motion approach zero. In the dynamic model i t was assumed that the dynamic tension was small compared to the equilibrium tension. In Figure 23 the dynamic tension i s for some frequencies larger than the equilibrium tension r e s u l t i n g in a negative minimum cable tension. A better approximation would be to calculate the s t a b i l i t y terms in the s t i f f n e s s matrix according to the actual tension in the cable rather than the e q u i l i b r i u m t e n s i o n , however t h i s w o u l d r e q u i r e a more complex n o n - l i n e a r d y n a mic a n a l y s i s . The above r e s u l t s a r e f o r t h e s p e c i f i c b r e a k w a t e r d e s c r i b i n S e c t i o n 5 .3 . Many f a c t o r s o t h e r t h a n t h e wave s t e e p n e s s and m o o r i n g t a u t n e s s i n f l u e n c e t h e t r a n s m i s s i o n c o e f f i c i e n t . T h e s e i n c l u d e t h e shape and l o c a t i o n o f c e n t r e o f g r a v i t y o f t h e b r e a k w a t e r body, a s p e c t r a t i o , a n c h o r s p a c i n g r a t i o , and w e i g h t o f t h e m o o r i n g l i n e s . T y p i c a l v a l u e s were s e l e c t e d f o r t h e b r e a k w a t e r a n a l y z e d h e r e , b u t t h e above r e s u l t s may n o t be g e n e r a l i f o t h e r v a l u e s a r e s e l e c t e d . 52 CHAPTER 6_ CONCLUSIONS AND FURTHER STUDIES 6.1 C o n c l u s i o n s A method o f a n a l y z i n g s l a c k - m o o r e d f l o a t i n g b r e a k w a t e r s has been p r e s e n t e d . The f i n i t e e l e m e n t method, p r e v i o u s l y u s e d t o f i n d h y d r o d y n a m i c c o e f f i c i e n t s i n s h i p m o t i o n p r o b l e m s , c a n be u s e d t o f i n d t h e v e l o c i t y p o t e n t i a l f i e l d a r o u n d a f l o a t i n g b r e a k w a t e r a c c o r d i n g t o l i n e a r t h e o r y . The f i n i t e e l e m e n t method i s n o t e f f i c i e n t f o r s h o r t w a v e l e n g t h s due t o t h e l a r g e number o f e l e m e n t s r e q u i r e d , however, a s i m p l e a p p r o x i m a t i o n f o r t h e t r a n s m i s s i o n c o e f f i c i e n t may be u s e d f o r s h o r t w a v e l e n g t h s . The m o o r i n g a n a l y s i s may be d i v i d e d i n t o a n o n - l i n e a r s t a t i c a n a l y s i s and a l i n e a r d y n a m ic a n a l y s i s . The s e c o n d - o r d e r d r i f t f o r c e on t h e b r e a k w a t e r i s i m p o r t a n t i n t h e s t a t i c a n a l y s i s . The l i n e a r dynamic a n a l y s i s can be a c h i e v e d u s i n g s t a n d a r d p l a n e -frame a n a l y s i s t e c h n i q u e s w i t h t h e a d d i t i o n o f s p e c i a l mass and damping m a t r i c e s f o r t h e c a b l e e l e m e n t s . The t r a n s m i s s i o n c o e f f i c i e n t c a l c u l a t e d f r o m t h e combined b o d y - m o o r i n g e q u a t i o n s o f m o t i o n may be q u i t e d i f f e r e n t f r o m t h a t o f t h e u n r e s t r a i n e d body. F o r t h e b r e a k w a t e r a n a l y z e d , t h e s i g n i f i c a n t changes i n t r a n s m i s s i o n c o e f f i c i e n t were c o n f i n e d t o f r e q u e n c i e s where t h e beam t o w a v e l e n g t h r a t i o was b e t w e e n 53 0.1 and 1.0, which covers the usual range of design conditions. At lower frequencies the transmission c o e f f i c i e n t was nearly unity, regardless of the type of mooring, while at higher frequencies the transmission c o e f f i c i e n t for both the unrestrained and slack-moored breakwaters could be found from the same short wavelength approximation. Very low transmission c o e f f i c i e n t s could not be achieved at reasonably small beam to wavelength rati o s unless motion i n the sway mode was eliminated. Changes i n the transmission c o e f f i c i e n t due to the moorings can be related to changes in the sway amplitude and phase angle, and the r o l l amplitude. Heave amplitude and phase angle are not s i g n i f i c a n t l y affected by the moorings. The combined body-mooring analysis i s non-linear so the transmission c o e f f i c i e n t i s dependent on wave steepness. The transmission c o e f f i c i e n t tends to decrease with increasing wave steepness due to an increase i n the s t i f f n e s s of the moorings. Increasing the tautness of the moorings also tends to decrease the transmission c o e f f i c i e n t . 6.2 Further Studies Further studies i n slack-moored f l o a t i n g breakwaters could be made in several areas. As shown in Section 5.2, there are many parameters that influence the transmission c o e f f i c i e n t . Only a few have been considered here. The e f f e c t of body geometry, centre of gravity, cable density, anchor spacing, and other factors could be examined. The computer program developed here can be used to investigate t h e o r e t i c a l l y such factors, and comparison could be made with experimental r e s u l t s . The p r e s e n t a n a l y s i s i s f o r a m o n o c h r o m a t i c wave. F o r d e s i g n p u r p o s e s i t i s d e s i r a b l e t o o b t a i n a t r a n s m i t t e d wave s p e c t r u m and e x t r e m e m o o r i n g f o r c e s f o r a g i v e n i n c i d e n t wave s p e c t r u m . The n o n - l i n e a r n a t u r e o f t h e r e s p o n s e means t h a t t h e t r a n s f e r f u n c t i o n a p p r o a c h can n o t be u s e d . The i n c i d e n t wave has been assumed t o be n o r m a l t o t h e b r e a k w a t e r . An o b l i q u e wave w o u l d c r e a t e a s i m p l e h a r m o n i c v a r i a t i o n o f wave f o r c e s a l o n g t h e a x i s o f t h e b r e a k w a t e r . T h i s c o u l d r e d u c e t h e a m p l i t u d e o f m o t i o n o f t h e b r e a k w a t e r . The f i n i t e e l e m e n t s o l u t i o n f o r an o b l i q u e wave i n t e r a c t i n g w i t h a h o r i z o n t a l f r e e l y - f l o a t i n g c y l i n d e r has b e e n p r e s e n t e d (32) by B a i . T h i s c o u l d be e x t e n d e d t o t h e c o m p l e t e b r e a k w a t e r a n a l y s i s . 55 BIBLIOGRAPHY 1 Adee, Bruce H. "Operational Experience with Floating Breakwaters", Marine Technology, v o l . 14, no. 4, Oct. 1977, pp. 379-386. 2 Dean, W. R. "On the r e f l e c t i o n of surface waves by a submerged plane b a r r i e r " , Proc. Cambridge P h i l . S o c , 41, 1945, pp. 231-238. 3 U r s e l l , F. "E f f e c t of a fixed b a r r i e r on surface waves in deep water", Proc. Cambridge P h i l . S o c , 43, 1947, pp. 374-382. 4 U r s e l l , F. "On the heaving motion of a c i r c u l a r cylinder on the surface of a f l u i d " , Q. J. Mech. Math., vo l . 2, 1949, pp. 218-231. 5 John, F. "On the motion of f l o a t i n g bodies", Comm. Pure Applied Math., v o l . 2, 1949, pp. 13-57. 6 Newton, R. E. " F i n i t e Element Analysis of Two-dimensional Added Mass and Damping", F i n i t e Elements in F l u i d s , ed. Gallagher, R. H., Oden, J. T., Taylor, C., and Zienkiewicz, 0. C., New York, Wiley, 1975, v o l . 1, pp. 219-232. 7 Bai, K. J. "A V a r i a t i o n a l Method in Potential Flows with a Free Surface", Report No. NA 72-2, University of C a l i f o r n i a , Berkeley, College of Engineering, 1972. 8 Chen, H. S. and Mei, C. C. " O s c i l l a t i o n s and wave forces in a man made harbour in the open sea", P r o c 10th Symp. Naval Hydrodynamics, Cambridge, Mass., 1974, pp. 573-596. 9 Bettess, P. and Zienkiewicz, 0. C. " D i f f r a c t i o n and Refraction of Surface Waves Using F i n i t e and I n f i n i t e Elements", Int. Journal for Numerical Methods in Engineering, v o l . 11, 1977, pp. 1272-1290. 10 Mei, Chiang C. and Black, Jared L. "Scattering of Surface Waves by Rectangular Obstacles in Waters of F i n i t e Depth", J. F l u i d Mech., v o l . 38, 1969, pp. 499-511. 11 Takano, K. "Effects d'un obstacle parallelepipedique sur l a propogation de l a houle", Houille Blanche, 15, p. 247. 12 Nichols, B. D. and H i r t , C. W. "Numerical Calculation of Wave Forces on Structures", Proc. 15th Coastal Engineering Conference, 1975, p. 2254. 56 13 Yamamoto, Tokuo and Yoshida, Akinori. " E l a s t i c Moorings of Floating Breakwaters", 7th Int. Harbour Congress, Antwerp, 1978. 14 Adee, Bruce H. and Martin, W. "Theoretical Analysis of Floating Breakwater Performance", Proc. 1974 Floating Breakwater Conf., University of Rhode Island, Marine Technical Report Series #24, pp. 21-39. 15 Remery, G. F. M. and van Oortmerssen, G. "The Mean Wave, Wind, and Current Forces on Offshore Structures and t h e i r Role i n the Design of Mooring Systems", 5th OTC, Paper OTC 1741, 1973. 16 Newman, J.N., Marine Hydrodynamics, Cambridge, Mass., MIT Press, 1977, pp. 237-311. 17 Garrison, C. J. "Hydrodynamics of large objects i n the sea, Part I, Hydrodynamic analysis", J. Hydronautics, vol . 8, no. 1, 1974, pp. 5-12. 18 Wehausen, John V. "The Motion of Floating Bodies", Ann. Review of F l u i d Mechanics, v o l . 3, 1971, pp. 237-268. 19 Bearman, P. W., Graham, J. M. R. and Singh, S. "Forces on cylinders in harmonically o s c i l l a t i n g flow", Symp. on Mechanics of Wave-incuced Forces on Cylinders, B r i s t o l , 1978. 20 Maruo, Hajime. "The D r i f t of a Body Floating on Waves", J. Ship Research, v o l . 4, 1960, pp. 1-10. 21 Longuet-Higgins, M. "The mean forces exerted by waves on f l o a t i n g or submerged bodies with applications to sand bars and wave power machines", Proc. Royal Soc. London A. v o l . 352, 1977, pp. 463-480. 22 Zienkiewicz, 0. C , The F i n i t e Element Method in Engineering Science, 2nd ed., London, McGraw-H i l l , 1971 23 Segerlind, L. J., Applied F i n i t e Element Analysis, New York, Wiley, 1976. 24 Dean, D. L. " S t a t i c and Dynamic Analysis of Guy Cables", Trans. ASCE, v o l . 127, part I I , 1962, pp. 382-402. 25 Coates, R. C., Coutie, M. G. and Kong, F. K., Structural Analysis, New York, Wiley, 1972. 26 Sarpkaya, T. "Forces on Cylinders and Spheres in Sinusoidally O s c i l l a t i n g F l u i d " , Trans. ASME J. Appl. Mech., v o l . 42, March 1975, pp. 32-37. 57 27 Ramberg, Steven E. and G r i f f e n , Owen M. "Free Vibrations of Taut and Slack Marine Cables, Journal of the Structural D i v i s i o n , ASCE, vol. 103, no. ST11, Nov. 1977, pp. 2079-2092. 28 Irvine, H. M. and Caughey, T. K. "The l i n e a r theory of free vibrations of a suspended cable", Proc. Royal Soc. Lond. A. vol 341, 1974, pp. 299-315. 29 Vugts, J. H. "The hydrodynamic c o e f f i c i e n t s for swaying, heaving and r o l l i n g cylinders i n a free surface", •International Shipbuilding Progress, 15, no. 167, July 1968, pp. 251-276. 30 Adee, Bruce H. "Floating Breakwater Performance", Proc. 15th Coastal Engineering Conference, 1975, p. 2777. 31 Bai, K. J. "Added Mass of 2-D Cylinders in Water of F i n i t e Depth", J. F l u i d Mechanics, v o l . 81, 1977, p. 85. 32 Bai, K. J. " D i f f r a c t i o n of Oblique Waves by an I n f i n i t e Cylinder", J. F l u i d Mechanics, v o l . 68, 1975, p. 513. 58 APPENDIX A THE QUADRATIC ISOPARAMETRIC ELEMENT The q u a d r a t i c i s o p a r a m e t r i c e l e m e n t u s e d t o s o l v e t h e b o u n d a r y - v a l u e p r o b l e m can r e p r e s e n t a q u a d r a t i c g e o m e t r i c b o u n d a r y shape and a q u a d r a t i c v a r i a t i o n i n t h e v e l o c i t y p o t e n t i a l (<J)) . S i n c e t h e v e l o c i t y o f t h e f l u i d i s t h e f i r s t d e r i v a t i v e o f t h e p o t e n t i a l , a l i n e a r v a r i a t i o n o f v e l o c i t y a c r o s s an e l e m e n t can be r e p r e s e n t e d . The v e l o c i t y p o t e n t i a l i s c o n t i n u o u s a c r o s s e l e m e n t b o u n d a r i e s w h i l e t h e v e l o c i t y i s n o t . The e l e m e n t i s (22) d e s c r i b e d i n d e t a i l by Z i e n k i e w i c z F i g u r e A - l shows b o t h t h e g l o b a l and t r a n s f o r m e d e l e m e n t s . The e l e m e n t i n t h e g l o b a l x-y s y s t e m , w h i c h i s i n g e n e r a l any q u a d r i l a t e r a l w i t h s t r a i g h t o r p a r a b o l i c e d g e s , i s t r a n s f o r m e d i n t o t h e much s i m p l e r e l e m e n t i n t h e s - t c o o r d i n a t e s y s t e m t o f a c i l i t a t e i n t e g r a t i o n . The t r a n s f o r m a t i o n s a r e 8 X = £ N . x, 1=1 8 y = l N.y. A l i = l 1 1 8 cf, = E N cf,. i = l 1 where (x.,y.) i s t h e c o o r d i n a t e o f t h e i node i n t h e x-y coordinate system and *^ i s the value of <J> at the i quadratic shape functions are th node, The N l = - (1 - s) (1 - t) (.1 + s + t)/4 N 2 = - (1 + s) (.1 - t) (.1 - s + t)/4 N 3 = - (1 + s) (1 + t) Cl - s - t)/4 N4 = - (1 - s) Cl + t) Cl + s - t)/4 N 5 = (1 - s 2) (1 - t l / 2 N6 = (1 - t 2 ) (1 + s)/2 N 7 = (1 - s 2) (1 + t)/2 N8 = (1 - t 2 ) (1 - s)/2 A2 From Equation 2.4 8 the element matrix to be calculated i s C k ] = / [W + ^ V ] d x d y e A3 The shape functions are functions of the l o c a l coordinates s and t. Since x and y can be expressed as functions of s and t "3N ." l ~3N ." l 3x 3s 3s 3s 3N. l 3x IX _3t 3t 3x 3N . 3y Inverting 3N. I 3x 3N . 3y [J] [J] -1 3N 3x 3N. 3y 3N , 3s 3N 3t A4 A5 Thus at any point i n the element the desired column vectors (3N/3x) and (3N/3y) can be calculated. Integration over the domain of the element i s done numerically using Gaussian quadrature. The expression [f (s,t)] = C^> (g^) + (^y) ( ^ J A6 i s evaluated at nine points i n the region and the sum 3 3 [k] = E Z W.W. If (s. ,t.) ] | J| A7 i=l i = l 1 3 1 3 i s calculated where s l = S2 = t l = -0 . 7745967 *2 = 0. 0 S3 = fc3 = + 0 .7745967 W l = w3 = 0. 5555556 w2 = 0. 8888889 A8 The boundary integrals are calculated in a sim i l a r manner. On the prescribed v e l o c i t y boundary, the boundary i n t e g r a l i s e Js, n le The prescribed v e l o c i t y i s of the form V = V i + V j A9 — x— y— therefore i t i s necessary to fi n d the normal to the element boundary in order to calculate the normal v e l o c i t y . Considering the boundary containing nodes 1,5, and 2, an outward normal to the boundary i s 8 3N. 8 8N, n = E —r—y . i - E —^—x.j A10 i = l i = l The normal v e l o c i t y i s then V • n V = — I — r — - A l l n n Again numerical integration i s used and the expression 61 Ig(.s)J = V n ( N ) A12 i s c a l c u l a t e d a t t h r e e i n t e g r a t i o n p o i n t s on t h e b o u n d a r y . The b o u n d a r y i n t e g r a l i s t h e n 3 (P) = I W. [ g ( s . ) ] I N l A13 e i = l 1 1 W. and s. a r e as p r e v i o u s l y d e f i n e d , i i ^ J On t h e f r e e s u r f a c e and r a d i a t i o n b o u n d a r i e s , t h e r T i n t e g r a l J g(N) (N) d£. must be c a l c u l a t e d . T h i s i s done i n t h e same way as t h e p r e s c r i b e d v e l o c i t y b o u n d a r y b u t w i t h t h e T m a t r i x (N)(N) r e p l a c i n g t h e v e c t o r V (N). n 62 APPENDIX B FREE CABLE VIBRATIONS A theory of l i n e a r vibrations of a parabolic cable i n a i r (28) has been developed by Irvine and Caughey . Their theory applies to both extensible and inextensible cables with sag to (27) span r a t i o s of 1:8 or less. Ramberg and G r i f f m have found that the Irvine and Caughey theory can be applied to marine cables i f allowance i s made for f l u i d damping and added mass. According to Irvine and Caughey, the behavior of a suspended cable i s described by the dimensionless parameter 2 ,8d,2 L L HL V AE ' where d i s the sag, L the span, H the horizontal force in the cable, A the cross-sectional area of the cable, and E the modulus of e l a s t i c i t y . L g i s an e f f e c t i v e cable length which for parabolic cables may be approximated by L - L (1 + 8 (j-) 2 ) B2 e L The dimensionless frequency of vibration i s described by the parameter a ( 3 D n = % L ( g ) 2 B3 where con i s a natural frequency and m i s the mass per unit length o f t h e c a b l e . F o r a n t i - s y m m e t r i c i n - p l a n e modes, t h e n a t u r a l f r e q u e n c i e s a r e g i v e n by (BL) = 2nTT n = l , 2 , 3 . ... B4 F o r s y m m e t r i c i n - p l a n e modes, t h e n a t u r a l f r e q u e n c i e s a r e g i v e n by t h e t r a n s c e n d e n t a l e q u a t i o n t a n ( ^ L ) = (±BL) - ' Cp-) (|gL) 3 B5 F o r an i n e x t e n s i b l e c a b l e , A 2 i s v e r y l a r g e and t h e a p p r o x i m a t e n a t u r a l f r e q u e n c i e s f o r s y m m e t r i c i n - p l a n e modes a r e g i v e n by (B'L) 2.86TT (BL) 2 4.92TT B6 (BL) (2n + 1)TT n=3,4,5 The c a b l e model o f C h a p t e r 3 was c h e c k e d a g a i n s t t h e r e s u l t s o f I r v i n e and Caughey. A c a b l e w i t h a s a g t o span r a t i o o f 1:10 was s e l e c t e d . The c a b l e was d i v i d e d i n t o t e n e q u a l l e n g t h segments and t h e mass and s t i f f n e s s m a t r i c e s were a s s e m b l e d . The added mass and damping were s e t t o z e r o . The r e s u l t i n g e q u a t i o n {-to n 2 [MJ + [K]} (?) = (0) B7 was s o l v e d f o r t h e n a t u r a l f r e q u e n c i e s co and t h e mode s h a p e s (?) u s i n g a l i b r a r y e i g e n v a l u e r o u t i n e . The r e s u l t s a r e shown i n F i g u r e B - l . The n o n - d i m e n s i o n a l f r e q u e n c y BL i s shown a l o n g w i t h t h e r e s u l t s o f I r v i n e and Caughey i n b r a c k e t s . F o r t h e f i r s t f o u r modes, t h e maximum e r r o r i n BL i s 9%. The e r r o r g e n e r a l l y i n c r e a s e s as t h e number o f e l e m e n t s p e r w a v e l e n g t h . d e c r e a s e s . The maximum 9% e r r o r c o r r e s p o n d s t o f o u r e l e m e n t s p e r w a v e l e n g t h . n e g a t i v e r a d i a t i o n b o u n d a r y p o s i t i v e r a d i a t i o n b o u n d a r y f i x e d b o u n d a r y DEFINITION SKETCH - FLUID REGION FIGURE 1 F I N I T E ELEMENT MESH FIGURE 2 FIGURE 3 (a) e > 0 CABLE DEFINITIONS FIGURE 4 DYNAMIC STRUCTURAL MODEL FIGURE 5 0 0 CABLE ELEMENT FIGURE 6 CO H CO >< hi < 2 u 2 >-i P O p m | READ DATA CALC. INCIDENT WAVE PROPERTIES CALC. ELEMENT MATRIX CALC. BOUNDARY MATRICES GO TO NEXT ELEMENT ADD INTO GLOBAL MATRICES no y e s SOLVE SCATTERED AND FORCED MOTION POTENTIALS CALC. ADDED MASS AND DAMPING COEFF. AND EXCITING FORCES ASSEMBLE AND SOLVE EQUATIONS OF MOTION CALC. TRANSMISSION COEFFICIENT 1 CALC. DRIFT FORCE CALC. EQUILIBRIUM POSITION OF CABLES DISCRETIZE CABLE SYSTEM ASSUME NO DAMPING IN CABLES X CALC. ELEMENT MATRICES | GO TO NEXT ELEMENT ADD INTO STRUCTURE MATRICES no CALC. NEW DAMPING yes SOLVE FOR BODY AND CABLE AMPLITUDES no - DO CABLE AMPLITUDES EQUAL ASSUMED VALUES? CHOOSE NEW VALUE FOR DRIFT FORCE yes CALC. DRIFT FORCE | no )OES DRIFT FORCE "EQUAL ASSUMED VALUE? yes CALC. TRANSMISSION COEFFICIENT COMPUTER PROGRAM FLOW CHART FIGURE 7 PLOT SYSTEM co H CO < o 2 H « o o | CALC. STRESSES IN CABLE~S~1 o e q u i l i b r i u m p o s i t i o n w i t h d r i f t f o r c e i n s t a n t a n e o u s p o s i t i o n B/A = 0. 35 h i/A = 0.1 C/d = 2. 83 tot = 70° K t = 0. 24 TYPICAL COMPUTER PLOT FIGURE 8 a PV 1. 0 0.9 -0 . 8 -p V 0.5 O 1.6 2.0 ADDED MASS V u g t s O f i n i t e e l e m e n t 1. 6 2.0 DAMPING ADDED MASS AND DAMPING - CIRCULAR CYLINDER FIGURE 9 ADDED MASS AND DAMPING - RECTANGULAR CYLINDER (B/D FIGURE 10 0 0.4 0.8 1.2 1.6 2.0 B /A HORIZONTAL 1.0 0 0.4 0.8 1.2 1.6 2.0 B /A VERTICAL EXCITING FORCES - CIRCULAR CYLINDER FIGURE 11 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 B/X TRANSMISSION COEFFICIENT - CIRCULAR CYLINDER FIGURE 13 11 0 0.4 0.8 1.2 1.6 2.0 B/X DRIFT FORCE - CIRCULAR CYLINDER FIGURE 14 GENERAL ARRANGEMENT - RECTANGULAR BREAKWATER FIGURE 15 B/A CONTRIBUTION OF MODES TO TRANSMITTED WAVE FIGURE 16 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 B/A SLACK-MOORED TRANSMISSION COEFFICIENT FIGURE 17 oo o 81 SWAY AMPLITUDE AND PHASE ANGLE FIGURE 18 TT 3TT/4 -TT/2 -0 . 4 0 . 8 1 . 2 1 . 6 2 . 0 B / A H E A V E A M P L I T U D E A N D P H A S E A N G L E F I G U R E 1 9 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 B/X EFFECT OF WAVE STEEPNESS FIGURE 21 oo 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 B/A EFFECT OF CABLE TAUTNESS FIGURE 22 oo (-1,1) ® © (-1,-1) e-© (1,1) (1,-1) te) © TRANSFORMED COORDINATE SYSTEM _© © © GLOBAL COORDINATE SYSTEM QUADRATIC ISOPARAMETRIC ELEMENT FIGURE A - l L 0.13x10 (inextensible) . , H Le, = 0.13xl0 2 (extensible) d = 1 0 AE ' 3 L = 13.08 6L = 9.56 (12.57) (9.46) 6L = 16.79 (15.44) 3 L = 11.80 (12.57) INEXTENSIBLE EXTENSIBLE NATURAL FREQUENCIES AND MODE SHAPES FIGURE B-l 

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