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UBC Theses and Dissertations

A study of the interactions of heaving cylinders Mikkelsen, Jon 1989

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A STUDY OF THE INTERACTIONS OF HEAVING CYLINDERS b y J o n M i k k e l s e n B . A . S c . ( M e c h . ) U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1985 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPL IED SCIENCE i n , FACULTY OF GRADUATE STUDIES D e p a r t m e n t o f M e c h a n i c a l E n g i n e e r i n g We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF B R I T I S H COLUMBIA 31 M a r c h , 1989 © J o n M i k k e l s e n , 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mechanical Engineering The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 r i l 28, 1989 ABSTRACT T h i s t h e s i s d e a l s w i t h t h e b e h a v i o u r o f h y d r o d y n a m i c c o e f f i c i e n t s o f v e r t i c a l , s u r f a c e p i e r c i n g , c i r c u l a r c y l i n d e r s s u b j e c t e d t o h e a v e m o t i o n . T h r e e t y p e s o f e x p e r i m e n t s a r e i n v e s t i g a t e d ; i ) The d e t e r m i n a t i o n o f t h e added mass and damp ing c o e f f i c i e n t s o f a compound c y l i n d e r i n heave m o t i o n . T h i s t e s t i s a r e p e a t o f a p r e v i o u s l y p e r f o r m e d e x p e r i m e n t b u t t h e a c c u r a c y o f t h e r e s u l t s a r e i m p r o v e d . i i ) The s i d e f o r c e s i n d u c e d on a s i m p l e c y l i n d e r due t o a n e a r b y , g e o m e t r i c a l l y s i m i l a r c y l i n d e r s u b j e c t e d t o s i n u s o i d a l h e a v e m o t i o n a r e d e t e r m i n e d and show good r e s u l t s w i t h t h e n u m e r i c a l m o d e l . i i i ) The d e t e r m i n a t i o n o f t h e i n d u c e d h e a v e added mass and damping c o e f f i c i e n t s a l s o due t o a n e a r b y , g e o m e t r i c a l l y s i m i l a r c y l i n d e r s u b j e c t e d t o s i n u s o i d a l heave m o t i o n . These r e s u l t s show a r e a s o n a b l e amount o f ag reemen t w i t h t h e p r e d i c t e d v a l u e s . I n e a c h e x p e r i m e n t , t h e e f f e c t o f v a r y i n g t h e d i s p l a c e m e n t , a m p l i t u d e and t h e f r e q u e n c y o f o s c i l l a t i o n a r e i n v e s t i g a t e d . I n c a s e s i i ) and i i i ) , t h e c y l i n d e r s e p a r a t i o n was v a r i e d and t h e e x p e r i m e n t s were p e r f o r m e d i n a deep w a t e r as w e l l as a s h a l l o w w a t e r s c e n a r i o . The r e s u l t s o f e a c h e x p e r i m e n t a r e compared t o t h e t h e o r e t i c a l p r e d i c t i o n s o f t h e M a t c h i n g T e c h n i q u e . The M a t c h i n g T e c h n i q u e u s e s c o n t i n u i t y o f p r e s s u r e s and v e l o c i t i e s b e t w e e n a d j a c e n t r e g i o n s i n t h e f l o w f i e l d t o s o l v e f o r t h e v e l o c i t y p o t e n t i a l s and h e n c e t h e 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 i TABLE OF CONTENTS ABSTRACT i i L I S T OF FIGURES i x L I S T OF TABLES x i v ACKNOWLEDGEMENTS x v 1. INTRODUCTION 1 2 . EXPERIMENTAL WORK 9 2 . 1 . HYDRODYNAMIC COEFFICIENTS OF A COMPOUND CYLINDER IN HEAVE MOTION 10 2 . 2 INDUCED SIDE FORCES ON A CYLINDER FROM A SECOND CYLINDER IN HEAVE MOTION 12 2 . 3 INDUCED HEAVE HYDRODYNAMIC COEFFICIENTS ON A SINGLE CYLINDER FROM A SECOND CYLINDER IN HEAVE MOTION 14 3 . BASIC THEORY 16 3 .1 GENERAL CASE OF A BODY IN A FLUID DOMAIN 16 3 . 1 . 1 GOVERNING EQUATION 16 i i i 3 . 1 . 2 BOUNDARY CONDITIONS 18 3 . 1 . 3 DETERMINATION OF HYDRODYNAMIC COEFFICIENTS 21 3 . 2 DIMENSIONAL ANALYSIS OF PROBLEM 23 4 . THEORETICAL MODELS 27 4 . 1 MATCHING TECHNIQUE FOR PREDICTION OF HYDRODYNAMIC COEFFICIENTS OF A T R I P L E CYLINDER 28 4 . 1 . 1 DEFINIT ION OF FLOWFIELD 28 4 . 1 . 2 DEFINIT ION OF POTENTIALS 28 4 . 1 . 2 . 1 REGION 1 30 4 . 1 . 2 . 2 REGION 2 31 4 . 1 . 2 . 3 REGION 3 32 4 . 1 . 2 . 4 REGION 4 34 4 . 1 . 3 SOLVING FOR THE UNKNOWN COEFFICIENTS 35 4 . 1 . 4 CALCULATION OF THE ADDED MASS AND DAMPING COEFFICIENTS 36 4 . 2 MATCHING TECHNIQUE FOR PREDICTION OF HEAVE MOTION INDUCED SIDE FORCES ON A SECOND VERTICAL CYLINDER 38 4 . 2 . 1 DEFINIT ION OF THE POTENTIAL 39 4 . 2 . 1 . 1 EXTERIOR POTENTIAL 41 4 . 2 . 1 . 2 INTERIOR POTENTIAL 44 4 . 2 . 2 MATCHING CONDITIONS 45 4 . 2 . 3 CALCULATION OF HEAVE INDUCED SIDE FORCE 50 i v 4 . 3 MATCHING TECHNIQUE FOR PREDICTION OF HEAVE MOTION HEAVE HYDRODYNAMIC COEFFICIENTS ON A SECOND VERTICAL CYLINDER 53 4 . 3 . 1 CALCULATION OF HEAVE INDUCED HYDRODYNAMIC COEFFICIENTS 53 5. PRESENTATION AND ANALYSIS OF RESULTS 56 5 . 1 SAMPLE DATA PLOTS 58 5 . 2 HYDRODYNAMIC COEFFICIENTS OF A COMPOUND CYLINDER IN HEAVE MOTION 61 5 . 2 . 1 ADDED MASS COEFFICIENTS OF A COMPOUND CYLINDER IN HEAVE MOTION 61 5 . 2 . 2 DAMPING COEFFICIENTS OF A COMPOUND CYLINDER IN HEAVE MOTION 63 5 . 3 INDUCED SURGE FORCES ON A CYLINDER DUE TO A SECOND CYLINDER IN HEAVE MOTION 66 5 . 3 . 1 INDUCED SURGE FORCE 67 5 . 3 . 1 . 1 DEEP WATER 67 5 . 3 . 1 . 2 SHALLOW WATER 73 5 . 4 INDUCED HEAVE HYDODYNAMIC COEFFICIENTS ON A CYLINDER DUE TO SECOND CYLINDER IN HEAVE MOTION 79 5 . 4 . 1 DEEP WATER 79 5 . 4 . 1 . 1 ADDED MASS COEFFICIENT 79 5 . 4 . 1 . 2 DAMPING COEFFICIENT 83 5 . 4 . 2 SHALLOW WATER 85 v 5 . 4 . 2 . 1 ADDED MASS COEFFICIENT 85 5 . 4 . 2 . 2 DAMPING COEFFICIENT 89 5 . 5 COMPARISON OF EXPERIMENTAL VALUES AND THEORETICAL PREDICTIONS 91 5 . 5 . 1 L IMITS IN THE MATCHING TECHNIQUE THEORETICAL MODEL 91 5 . 5 . 2 L IMITS IN EXPERIMENTAL RESULTS 95 5 . 5 . 2 . 1 ANALYSIS OF EXPERIMENTAL ERROR 97 5 . 6 U T I L I T Y OF RESULTS 101 CONCLUSION 103 RECOMMENDATIONS 109 BIBLIOGRAPHY I l l 6 . APPENDIX A . EXPERIMENTAL S E T - U P 113 6 . 1 EXPERIMENTAL F A C I L I T I E S 113 6 . 1 . 1 TOWING TANK 113 6 . 2 EXPERIMENTAL EQUIPMENT 116 6 . 2 . 1 MOTION GENERATION SYSTEM 116 6 . 2 . 2 SECONDARY CYLINDER FRAME 117 6 . 2 . 3 DATA COLLECTION EQUIPMENT 118 6 . 2 . 3 . 1 ST41B™ SIGNAL CONDITIONER 118 6 . 2 . 3 . 1 . 1 SIGNAL CONDITIONER PHASE LAG TESTS 121 v i 6 . 2 . 3 . 2 MINC 11 MINI COMPUTER 123 6 . 2 . 4 CYLINDER MODELS 124 6 . 2 . 5 INSTRUMENTATION USED 126 6 . 2 . 5 . 1 LOAD CELL DYNAMOMETER 126 6 . 2 . 5 . 1 . 1 DYNAMOMETER ORIENTATION 127 6 . 2 . 5 . 1 . 2 DYNAMOMETER CALIBRATION 128 6 . 2 . 5 . 2 UNIVERSAL SHEAR BEAMS™ 129 TM 6 . 2 . 5 . 2 . 1 UNIVERSAL SHEAR BEAM CALIBRATION 130 6 . 2 . 5 . 3 YO-YO POSITION TRANSDUCER 130 7 . APPENDIX B. SOFTWARE USED IN THE EXPERIMENTS 132 7 . 1 DATA ACQUISIT ION SYSTEM 133 7 . 1 . 1 ADCAL PROGRAM 133 7 . 1 . 2 ADMAIN PROGRAM 134 7 . 1 . 3 ADMUX PROGRAM 135 7 . 1 . 4 ADVIEW PROGRAM 136 7 . 2 DATA ANALYSIS SOFTWARE 136 7 . 2 . 1 QW PROGRAM 137 7 . 2 . 1 . 1 DEMUX PROGRAM 139 7 . 2 . 1 . 2 TREND SUBROUTINE 140 7 . 2 . 1 . 3 F I L T E R SUBROUTINE 141 7 . 2 . 1 . 4 FOURT SUBROUTINE 142 7 . 2 . 1 . 5 FFT SUBROUTINE 144 7 . 2 . 1 . 6 REALTIME SUBROUTINE 145 . . . 7 B IG SUBROUTINE 6 v i i 7 . 2 . 1 . 8 COEF SUBROUTINE 147 8. APPENDIX C . FIGURES 149 9 . APPENDIX D. PHOTOGRAPHS 157 10. APPENDIX E . GRAPHICAL REPRESENTATION OF RESULTS 168 11. APPENDIX F . BESSEL FUNCTIONS AND RELATED FORMULAE 215 v i i i L I S T OF FIGURES FIGURE 8 .1 COORDINATE SYSTEM AND DEFINIT ION OF MOTIONS 150 FIGURE 8 . 2 GEOMETRY OF T R I P L E CYLINDER FLUID DOMAIN USED FOR MATCHING TECNIQUE THEORY 151 FIGURE 8 . 3 GEOMETRY OF SINGLE CYLINDER FLUID DOMAIN USED FOR MATCHING TECHNIQUE THEORY 152 FIGURE 8 . 4 DEFIN IT ION OF TWIN CYLINDER COORDINATES USED FOR MATCHING TECHNIQUE THEORY 153 FIGURE 8 . 5 GEOMETRY OF SINGLE CYLINDER MODEL 154 FIGURE 8 . 6 GEOMETRY OF T R I P L E CYLINDER MODEL 155 FIGURE 8 . 7 FLOWCHART OF THE DATA ANALYSIS SOFTWARE 156 FIGURE 9 . 1 GULF CANADA'S ' K U L L A C ' 158 FIGURE 9 . 2 EXTERIOR VIEW OF THE OCEAN ENGINEERING CENTRE 159 FIGURE 9 . 3 INTERIOR VIEW OF THE OCEAN ENGINEERING CENTRE 160 FIGURE 9 . 4 OVERVIEW OF TOWING CARRIAGE 161 FIGURE 9 . 5 OVERHEAD HOIST 162 FIGURE 9 . 6 HYDRAULIC POWER UNIT 163 FIGURE 9 . 7 SINGLE CYLINDER MODEL WITH MOTION GENERATION CONNECTION 164 FIGURE 9 . 8 T R I P L E CYLINDER MODEL 165 FIGURE 9 . 9 DYNAMOMETER AND ADAPTER BLOCK DETAIL 166 FIGURE 9 . 1 0 DYNAMOMETER WITHOUT PROTECTIVE PLATES 167 i x FIGURE 1 0 . 1 UNFILTERED DISPLACEMENT TRACE FOR T Y P I C A L HYDRODYNAMIC TEST 169 FIGURE 1 0 . 2 F ILTERED DISPLACEMENT TRACE FOR T Y P I C A L HYDRODYNAMIC TEST 170 FIGURE 1 0 . 3 F ILTERED DISPLACEMENT SPECTRUM 171 FIGURE 1 0 . 4 UNFILTERED SURGE CHANNEL TRACE FOR T Y P I C A L HYDRODYNAMIC TEST 172 FIGURE 1 0 . 5 F ILTERED SURGE CHANNEL TRACE FOR T Y P I C A L HYDRODYNAMIC TEST 173 FIGURE 1 0 . 6 SUPERPOSITION OF FILTERED AND UNFILTERED SURGE FORCE TRACE 174 FIGURE 1 0 . 7 UNFILTERED SURGE FORCE SPECTRUM 175 FIGURE 1 0 . 8 HEAVE ADDED MASS COEFFICIENT VERSUS FREQUENCY T R I P L E CYLINDER RESULTS, DRAFT 90cm. MIKKELSEN (1988) 176 FIGURE 1 0 . 9 HEAVE ADDED MASS COEFFICIENT VERSUS FREQUENCY T R I P L E CYLINDER RESULTS, DRAFT 90cm. VENUGOPAL (1984) 177 FIGURE 1 0 . 1 0 HEAVE ADDED MASS COEFFICIENT VERSUS FREQUENCY T R I P L E CYLINDER RESULTS, DRAFT 120cm. MIKKELSEN (1988) 178 FIGURE 1 0 . 1 1 HEAVE ADDED MASS COEFFICIENT VERSUS FREQUENCY T R I P L E CYLINDER RESULTS, DRAFT 120cm. VENUGOPAL (1984) 179 x FIGURE 1 0 . 1 2 HEAVE DAMPING COEFFICIENT VERSUS FREQUENCY T R I P L E CYLINDER RESULTS, DRAFT 90cm. MIKKELSEN (1988) 180 FIGURE 1 0 . 1 3 HEAVE DAMPING COEFFICIENT VERSUS FREQUENCY T R I P L E CYLINDER RESULTS, DRAFT 90cm. VENUGOPAL (1984) 181 FIGURE 1 0 . 1 4 HEAVE DAMPING COEFFICIENT VERSUS FREQUENCY T R I P L E CYLINDER RESULTS, DRAFT 120cm. MIKKELSEN (1988) 182 FIGURE 1 0 . 1 5 HEAVE DAMPING COEFFICIENT VERSUS FREQUENCY T R I P L E CYLINDER RESULTS, DRAFT 120cm. VENUGOPAL (1984) 183 FIGURE 1 0 . 1 6 INDUCED SURGE FORCE VERSUS FREQUENCY DEEP WATER, B=2 .05 184 FIGURE 1 0 . 1 7 INDUCED SURGE FORCE VERSUS FREQUENCY DEEP WATER, B=2.48 185 FIGURE 1 0 . 1 8 INDUCED SURGE FORCE VERSUS FREQUENCY DEEP WATER, B=3.00 186 FIGURE 1 0 . 1 9 INDUCED SURGE FORCE VERSUS FREQUENCY DEEP WATER, B=3.47 187 FIGURE 1 0 . 2 0 INDUCED SURGE FORCE VERSUS FREQUENCY DEEP WATER, B=4.00 188 FIGURE 1 0 . 2 1 INDUCED SURGE FORCE VERSUS FREQUENCY SHALLOW WATER, B=2.08 189 FIGURE 1 0 . 2 2 INDUCED SURGE FORCE VERSUS FREQUENCY SHALLOW WATER, B=2.47 190 x i FIGURE 1 0 . 2 3 INDUCED SURGE FORCE VERSUS FREQUENCY SHALLOW WATER, B=3.00 191 FIGURE 1 0 . 2 4 INDUCED SURGE FORCE VERSUS FREQUENCY SHALLOW WATER, B=3.48 192 FIGURE 1 0 . 2 5 INDUCED SURGE FORCE VERSUS FREQUENCY SHALLOW WATER, B=4.00 193 FIGURE 1 0 . 2 6 HEAVE ADDED MASS COEFFICIENT VERSUS FREQUENCY DEEP WATER, B = 2 . 0 5 194 FIGURE 1 0 . 2 7 HEAVE ADDED MASS COEFFICIENT VERSUS FREQUENCY DEEP WATER, B = 2 . 4 8 195 FIGURE 1 0 . 2 8 HEAVE ADDED MASS COEFFICIENT VERSUS FREQUENCY DEEP WATER, B =3 .00 . . . 196 FIGURE 1 0 . 2 9 HEAVE ADDED MASS COEFFICIENT VERSUS FREQUENCY DEEP WATER, B - 3 .47 197 FIGURE 1 0 . 3 0 HEAVE ADDED MASS COEFFICIENT VERSUS FREQUENCY DEEP WATER, B - 4 . 0 0 198 FIGURE 1 0 . 3 1 HEAVE DAMPING COEFFICIENT VERSUS FREQUENCY DEEP WATER, B = 2 . 0 5 199 FIGURE 1 0 . 3 2 HEAVE DAMPING COEFFICIENT VERSUS FREQUENCY DEEP WATER, B - 2 . 4 8 200 FIGURE 1 0 . 3 3 HEAVE DAMPING COEFFICIENT VERSUS FREQUENCY DEEP WATER, B - 3 . 0 0 201 FIGURE 1 0 . 3 4 HEAVE DAMPING COEFFICIENT VERSUS FREQUENCY DEEP WATER, B - 3 .47 202 FIGURE 1 0 . 3 5 HEAVE DAMPING COEFFICIENT VERSUS FREQUENCY DEEP WATER, B - 4 . 0 0 203 x i i FIGURE 1 0 . 3 6 HEAVE ADDED MASS COEFFICIENT VERSUS FREQUENCY SHALLOW WATER, B = 2 . 0 8 204 FIGURE 1 0 . 3 7 HEAVE ADDED MASS COEFFICIENT VERSUS FREQUENCY SHALLOW WATER, B = 2 . 4 8 205 FIGURE 1 0 . 3 8 HEAVE ADDED MASS COEFFICIENT VERSUS FREQUENCY SHALLOW WATER, B = 3 . 0 0 206 FIGURE 1 0 . 3 9 HEAVE ADDED MASS COEFFICIENT VERSUS FREQUENCY SHALLOW WATER, B - 3 . 4 8 207 FIGURE 1 0 . 4 0 HEAVE ADDED MASS COEFFICIENT VERSUS FREQUENCY SHALLOW WATER, B = 4 . 0 0 208 FIGURE 1 0 . 4 1 HEAVE DAMPING COEFFICIENT VERSUS FREQUENCY SHALLOW WATER, B - 2 . 0 8 209 FIGURE 1 0 . 4 2 HEAVE DAMPING COEFFICIENT VERSUS FREQUENCY SHALLOW WATER, B - 2 . 4 8 210 FIGURE 1 0 . 4 3 HEAVE DAMPING COEFFICIENT VERSUS FREQUENCY SHALLOW WATER, B = 3 . 0 0 211 FIGURE 1 0 . 4 4 HEAVE DAMPING COEFFICIENT VERSUS FREQUENCY SHALLOW WATER, B - 3 . 4 8 212 FIGURE 1 0 . 4 5 HEAVE DAMPING COEFFICIENT VERSUS FREQUENCY SHALLOW WATER, B - 4 . 0 0 213 FIGURE 1 0 . 4 6 HEAVE DAMPING COEFFICIENT EQUIPMENT ACCURACY TEST 214 x i i i L I S T OF TABLES TABLE 3 . 2 - 1 RANGE OF NON-DIMENSIONAL QUANTITIES ENCOUNTERED 25 TABLE 5 . 3 . 1 . 1 - 1 VALUES OF PEAK INDUCED SURGE FORCE IN DEEP WATER 67 TABLE 5 . 3 . 1 . 2 - 1 VALUES OF PEAK INDUCED SURGE FORCE IN SHALLOW WATER 73 TABLE 5 . 4 . 1 . 1 - 1 VALUES OF MINIMUM INDUCED ADDED MASS COEFFICIENTS IN DEEP WATER 80 TABLE 5 . 4 . 2 . 1 - 1 VALUES OF MINIMUM INDUCED ADDED MASS COEFFICIENTS IN SHALLOW WATER 85 TABLE 5 . 5 . 2 . 1 - 1 ESTIMATE OF ERROR FOR PLOTTED PARAMETERS 100 TABLE 6 . 2 . 3 - 1 A M P L I F I E R CONFIGURATION 120 TABLE 6 . 2 . 5 . 1 - 1 DYNAMOMETER SPECIF ICATIONS 127 TABLE 6 . 2 . 5 . 2 - 1 UNIVERSAL SHEAR BEAMS SPECIF ICATIONS 129 TABLE 6 . 2 . 5 . 3 - 1 YO-YO POSITION TRANSDUCER SPECIF ICATIONS 131 TABLE 7 . 2 . 1 - 1 TRANSDUCER CHANNEL ASSIGNMENTS AND USER UNITS ASSUMED BY "QW" 138 TABLE 7 . 2 . 1 - 2 F I L E NAME EXTENSIONS USED BY "QW" 139 x i v ACKNOWLEDGEMENTS O v e r t h e c o u r s e o f c o n d u c t i n g t h i s r e s e a r c h , I h a v e d rawn upon t h e k n o w l e d g e and e x p e r t i s e o f many i n d i v i d u a l s , t o o numerous t o m e n t i o n h e r e . I w i s h t o t h a n k a l l o f them f o r t h e i r i n p u t and a s s i s t a n c e . I mus t s i n g l e o u t a few i n d i v i d u a l s whose c o n t r i b u t i o n s have b e e n s i g n i f i c a n t t o t h i s r e s e a r c h . I am i n d e b t e d t o D r . S a n d e r C a l i s a l , my s u p e r v i s o r , f o r h i s g u i d a n c e and s u p p o r t t h r o u g h o u t t h e c o u r s e o f t h i s r e s e a r c h . I a l s o must t h a n k t h e N a t i o n a l S c i e n c e s and E n g i n e e r i n g R e s e a r c h C o u n c i l (NSERC) o f Canada f o r f u n d i n g o f t h i s p r o j e c t . I w i s h t o t h a n k B . C . R e s e a r c h and t he s t a f f a t t h e Ocean E n g i n e e r i n g C e n t r e f o r t h e u s e o f t h e t o w i n g t a n k f a c i l i t i e s . Thanks t o , G e r r y S t e n s g a a r d f o r k i n d l y a l l o w i n g me t o wo rk i n t h e f a c i l i t y d u r i n g n o r m a l w o r k i n g h o u r s . Thanks t o Geo rge Roddan and G a r y N o v l e s k y f o r t h e i r v a l u a b l e a d v i c e on t h e i n s t r u m e n t a t i o n and f o r p r o v i d i n g s o l u t i o n s t o p r a c t i c a l p r o b l e m s . I a l s o w i s h t o t h a n k J o h n H o a r and Tony B e s i c o f t h e M e c h a n i c a l E n g i n e e r i n g M a c h i n e Shop f o r t h e i r q u a l i t y wo rkmansh ip p u t i n t o v a r i o u s components o f t h e e x p e r i m e n t a l a p p a r a t u s f a b r i c a t e d a t t h e i r s h o p . I am d e e p l y i n d e b t e d t o Doug G o o d r i d g e , f o r h i s h e l p and k n o w l e d g e i n t h i s s u b j e c t f i e l d . Much o f t h e e q u i p m e n t , s o f t w a r e and b a c k g r o u n d t h e o r y u s e d b y t h i s r e s e a r c h e r was u s e d o r m o d i f i e d xv f r o m M r . G o o d r i d g e ' s r e s e a r c h . I w i s h t o t h a n k a l l my c o l l e a g u e s i n t h e N a v a l A r c h i t e c t u r e g r o u p f o r a l l t h e h e l p t h e y have p r o v i d e d . Thanks t o Dan M c G r e e r , G e r r y R o h l i n g , J o h n s o n C h a n , F r a n k y C h u , and G i r e e s h S a d i s a v a n f o r h e l p on t h e e x p e r i m e n t s . F i n a l l y , I w i s h t o t h a n k T i n a G r a b e n h o r s t f o r h e r p a t i e n t h e l p i n e d i t i n g t h i s t h e s i s . x v i 1. INTRODUCTION S i n c e t h e b e g i n n i n g o f t i m e t h e o c e a n h a s b e e n a p r e o c c u p a t i o n o f man. The v a s t s u p p l y o f r e s o u r c e s h i d d e n b e n e a t h t h e d e p t h s have b e e n e l u s i v e t o h i m u n t i l j u s t r e c e n t l y . O n l y now does t e c h n o l o g y e n a b l e h i m t o e x p l o r e t h e o c e a n d e p t h s and a t t e m p t t o e x p l o i t i t s r i c h e s . W i t h t h e e v e r i n c r e a s i n g demand f o r o i l t h e q u e s t f o r new s u p p l i e s h a s gone o f f s h o r e and h a s p r o p e l l e d t h e d e v e l o p m e n t o f o f f s h o r e p l a t f o r m s f r o m w h i c h o i l e x p l o r a t i o n c a n t a k e p l a c e . These p a s t few d e c a d e s h a v e s e e n t h e d e v e l o p m e n t o f o f f s h o r e d r i l l i n g p l a t f o r m s , s e m i - s u b m e r s i b l e p l a t f o r m s as w e l l as g r a v i t y p l a t f o r m s , w h i c h c a n w i t h s t a n d some o f t h e r o u g h e s t o c e a n e n v i r o n m e n t s f o u n d on E a r t h . I t was a t t h e t u r n o f t h e c e n t u r y t h a t t h e f i r s t o f f s h o r e o i l e x p l o r a t i o n t o o k p l a c e . O f f t h e c o a s t o f C a l i f o r n i a , n e a r S a n t a B a r b a r a , l o n g wooden w h a r v e s r e a c h i n g up t o 360 m e t r e s o f f s h o r e were c o n s t r u c t e d t o e n a b l e o i l w e l l s t o be d r i l l e d . U p d a t e d v e r s i o n s o f t h e s e w h a r v e s c a n s t i l l be s e e n t o d a y . S h o r t l y a f t e r w a r d s t h e f o c u s o f o f f s h o r e o i l e x p l o r a t i o n s h i f t e d t o t h e G u l f o f M e x i c o . H e r e , i n l a n d l a k e t e c h n o l o g y was i m p l e m e n t e d f o r o f f s h o r e e x p l o r a t i o n . B a r g e s w o u l d be towed o f f s h o r e and l a i d t o r e s t upon wooden p i l e s d r i v e n i n t o t h e s e a b e d . L a t e r , b a r g e s were d e s i g n e d w h i c h c o u l d be towed o u t t o t h e d e s i r e d l o c a t i o n and f l o o d e d so t h a t t h e y w o u l d come t o r e s t on t h e b o t t o m . T h e s e s e m i - s u b m e r s i b l e b a r g e s h a d a l a r g e enough f r e e b o a r d t o p r o v i d e a d r y d e c k f o r t h e o i l d r i l l i n g e q u i p m e n t . T h e s e t y p e s o f f i x e d 1 g r a v i t y p l a t f o r m s a r e s t i l l i n u s e t o d a y a t t h e G u l f o f M e x i c o , and t h i s same t y p e i o f t e c h n o l o g y h a s b e e n i m p l e m e n t e d t o a l l o w e x p l o r a t i o n b e n e a t h t h e B e a u f o r t S e a . D u r i n g t h e 1 9 4 0 ' s t h e f i r s t f l o a t i n g b a r g e s w i t h an o i l d e r r i c k s u p e r s t r u c t u r e were d e s i g n e d and i m p l e m e n t e d . These b a r g e s c o u l d a l l o w d r i l l i n g i n up t o 12 m e t r e s o f w a t e r . The e a r l y 1 9 5 0 ' s saw t h e i n t r o d u c t i o n o f t h e f i r s t d r i l l s h i p s f o l l o w i n g a U n i t e d S t a t e s Navy e x p e r i m e n t a l p r o g r a m . U n f o r t u n a t e l y , t h e u n d e s i r a b l e m o t i o n c h a r a c t e r i s t i c s o f t h e s e d r i l l s h i p s d u r i n g a p p a r e n t l y mode ra te s e a s t a t e s c a u s e d d r i l l i n g o p e r a t i o n s t o f u n c t i o n i n f r e q u e n t l y . I t was n o t u n t i l t h e e a r l y 1 9 6 0 ' s t h a t t h e f i r s t s e m i - s u b m e r s i b l e p l a t f o r m was i n t r o d u c e d . A t y p i c a l s e m i - s u b m e r s i b l e c o n s i s t s o f a d e c k and s u p e r s t r u c t u r e s u p p o r t e d b y a number o f v e r t i c a l c o l u m n s , c r o s s b r a c e s , and p o n t o o n s , w h i c h h a v e s u f f i c i e n t b u o y a n c y t o f l o a t t h e e n t i r e s t r u c t u r e . F l o o d i n g chambers i n t h e p o n t o o n s a l l o w t h e d r i l l i n g p l a t f o r m t o f l o a t a t v a r i o u s d r a f t s . L a r g e t u g b o a t s a r e u s e d t o t r a n s p o r t t h e s e m i - s u b m e r s i b l e t o t h e s p e c i f i c wo rk s i t e where i t w i l l be a n c h o r e d t o t h e s e a b o t t o m w i t h l a r g e c a b l e s . T o d a y , a new t y p e o f s e m i - s u b m e r s i b l e h a s b e e n d e v e l o p e d . G u l f Canada h a s b e e n u s i n g a n e n t i r e l y a x i s y m m e t r i c f l o a t i n g d r i l l i n g u n i t i n t h e B e a u f o r t S e a . T h i s u n i t c a l l e d "The K u l l a c " i s shown i n F i g u r e 9 . 1 . T h e s e o f f s h o r e s t r u c t u r e s a r e v e r y l a r g e and c o m p l e x . Enormous c a p i t a l i n v e s t m e n t s a r e r e q u i r e d and o f t e n t h e s e 2 s t r u c t u r e s a r e d e s i g n e d t o h a v e an o p e r a t i n g l i f e o f up t o one h u n d r e d y e a r s . W h i l e one w o u l d e x p e c t t h a t d e s i g n e r s i n v o l v e c o m p l e x a n a l y t i c a l methods i n o r d e r t o p r e d i c t t h e h y d r o d y n a m i c s , t h i s i s o f t e n n o t t h e c a s e . A n a l y t i c a l methods w h i c h do e x i s t r e q u i r e numerous s i m p l i f i c a t i o n s t o mode l t h e f l o w . I n s t e a d , d e s i g n e r s emp loy s c a l e mode l t e s t r e s u l t s f r o m t o w i n g t a n k s as w e l l as s t a n d a r d d a t a f r o m e x i s t i n g o f f s h o r e p l a t f o r m s . I t mus t be n o t e d t h a t t h e f u n d a m e n t a l componen ts w h i c h c o n s t i t u t e t h e b e l o w d e c k p o r t i o n o f t h e s e m i - s u b m e r s i b l e a r e c y l i n d r i c a l i n n a t u r e . T h e r e f o r e , t h e s t u d y o f f l o w s a r o u n d a c y l i n d e r a r e o f c o n s i d e r a b l e i m p o r t a n c e and much r e s e a r c h h a s b e e n d e v o t e d t o t h i s t o p i c . I n o r d e r t o s t u d y t h e h y d r o d y n a m i c l o a d s a f l u i d i m p a r t s upon a f l o a t i n g body one must i n t r o d u c e t h e c o n c e p t o f h y d r o d y n a m i c c o e f f i c i e n t s . C o n s i d e r t h e g e n e r a l c o n f i g u r a t i o n o f a body m o v i n g i n t h e C a r t e s i a n c o - o r d i n a t e s y s t e m as shown i n F i g u r e 8 . 1 . T h r e e t r a n s l a t o r y m o t i o n s a l o n g t h e x , y , and z a x i s a r e p o s s i b l e : t h e s e a r e c l a s s i f i e d b y h y d r o d y n a m i c r e s e a r c h e r s as s u r g e , sway and h e a v e . A l s o t o be c o n s i d e r e d a r e t h r e e r o t a t i o n a l m o t i o n s a b o u t t h o s e a x i s known as r o l l , p i t c h , and yaw r e s p e c t i v e l y . The g e n e r a l dynam ic e q u a t i o n o f t h i s s i x d e g r e e o f f r e e d o m s y s t e m i s : X j 9 i j j j Where , 3 F = F o r c e v e c t o r (3x1 ) M = Moment v e c t o r (3x1 ) X , X , X = T r a n s l a t o r y m o t i o n v e c t o r s (3x1 ) 8 ,8 ,8 = R o t a r y m o t i o n v e c t o r s (3x1 ) m . . = Mass c o e f f i c i e n t (6x6 ) i j a . . = Added mass c o e f f i c i e n t (6x6 ) b „ = Damping c o e f f i c i e n t (6x6 ) c „ = R e s t o r i n g f o r c e c o e f f i c i e n t (6x6 ) The h y d r o d y n a m i c c o e f f i c i e n t s a r e t h e t e rms a , b , a n d c i j i j i j w h i c h a r e u s e d t o d e f i n e t h e h y d r o d y n a m i c l o a d i n g . The t e rms i n p h a s e w i t h t h e a c c e l e r a t i o n , v e l o c i t y , and d i s p l a c e m e n t a r e c l a s s i f i e d r e s p e c t i v e l y as t h e added m a s s , d a m p i n g , and r e s t o r i n g f o r c e . E a c h o f t h e c o e f f i c i e n t s c a n be e x p r e s s e d as a t e n s o r : a . . , b . . , and c . . . Where s u b s c r i p t " i " r e f e r s t o t h e d i r e c t i o n o f t h e body m o t i o n and s u b s c r i p t " j " r e f e r s t o t h e d i r e c t i o n o f t h e f o r c e on t h e b o d y . T h e r e e x i s t s 36 p o s s i b l e c o e f f i c i e n t s f o r e a c h o f t h e t h r e e m a t r i c e s . Howeve r , i t c a n be shown t h a t t h e m a t r i c e s a r e s y m m e t r i c , t h a t i s : a „ = a ^ ; t h u s t h e r e e x i s t a c t u a l l y o n l y 21 i n d e p e n d e n t t e n s o r s . I t c a n a l s o be shown a l s o t h a t f o r a body s y m m e t r i c a b o u t one o r more a x i s , t h e s e 21 v a r i a b l e s c a n be r e d u c e d f u r t h e r . F o r t h e c a s e o f a body s y m m e t r i c a b o u t t h e v e r t i c a l a x i s , t h e r e w i l l e x i s t 8 i n d e p e n d e n t v a r i a b l e s i n t h e added mass and damp ing f o r c e m a t r i c e s , and o n l y 4 v a r i a b l e s i n t h e r e s t o r i n g f o r c e m a t r i x w i t h a l l o t h e r v a r i a b l e s e q u a l t o z e r o . I n t h e s t u d y c a r r i e d o u t i n t h i s t h e s i s , t h e m o t i o n o f t h e c y l i n d e r 4 i s a l w a y s r e s t r i c t e d t o one d i r e c t i o n , t h e r e b y a l l o w i n g o n l y one t e r m t o be c o n s i d e r e d i n e a c h m a t r i x . They a r e a , b , and c J 22 2 2 2 2 i n t h e c a s e o f h e a v e and a , b , and c i n t h e c a s e o f s u r g e . i i i i n & I n g e n e r a l , any body m o v i n g i n a f l u i d w i l l e x p e r i e n c e h y d r o d y n a m i c f o r c e s . I f a body i s m o v i n g p e r i o d i c a l l y i n a f l u i d a t t h e f r e e s u r f a c e , t h e h y d r o d y n a m i c f o r c e s i t e x p e r i e n c e s a r e i n - p h a s e and o u t - o f - p h a s e w i t h t h e a c c e l e r a t i o n . T h i s i n - p h a s e component c o n t r i b u t e s t o t h e added mass c o e f f i c i e n t w h i l e t h e o u t - o f - p h a s e component c o n t r i b u t e s t o t h e damping c o e f f i c i e n t . A c l o s e e x a m i n a t i o n o f t h e s i t u a t i o n r e v e a l s t h a t t h e c a u s e o f h y d r o d y n a m i c c o e f f i c i e n t s a r e due t o f l u c t u a t i o n s o f t h e p r e s s u r e s u r r o u n d i n g t h e b o d y . F l u i d p a r t i c l e s i n t h e n e i g h b o u r h o o d o f t h e body w i l l be a c c e l e r a t e d b y t h e m o v i n g body a t v a r y i n g r a t e s d e p e n d i n g on t h e i r p r o x i m i t y t o t h e b o d y . The a d d e d mass c a n be t h o u g h t o f as a measure o f t h e q u a n t i t y o f f l u i d w h i c h i s a c c e l e r a t e d a l o n g w i t h t h e b o d y . I n p r i n c i p l e , e v e r y p a r t i c l e o f f l u i d i s a c c e l e r a t e d t o some f i n i t e d e g r e e , h o w e v e r , t h e t e r m added mass i s e x p r e s s e d as a n e q u i v a l e n t mass o f f l u i d a c c e l e r a t e d a t t h e same r a t e as t h a t o f t h e b o d y . The damp ing c o e f f i c i e n t t a k e s i n t o a c c o u n t t h e o u t w a r d f l u x o f e n e r g y d i s s i p a t e d b y t h e b o d y . M o s t t h e o r i e s u s e d t o p r e d i c t h y d r o d y n a m i c f o r c e s i n v o l v e a p o t e n t i a l f l o w t h e o r y , t h a t i s , t h e f l u i d i s assumed i n c o m p r e s s i b l e , and i n v i s c i d , and t h e f l o w i r r o t a t i o n a l , t h e r e b y n e g l e c t i n g a l l v i s c o u s and s t r u c t u r a l d a m p i n g . F o r t h i s i d e a l f l u i d t h e o u t w a r d f l u x o f e n e r g y m a n i f e s t s i t s e l f as wave e n e r g y , and so a d i r e c t r e l a t i o n s h i p 5 b e t w e e n wave h e i g h t and t h e damping c o e f f i c i e n t c a n be f o u n d (Wehausen, 1 9 7 1 ) . P r e d i c t i o n o f h y d r o d y n a m i c c o e f f i c i e n t s t h r o u g h t h e o r e t i c a l a n a l y s i s h a s p r e o c c u p i e d many r e s e a r c h e r s o v e r t h e p a s t 200 y e a r s . The f i r s t p e r s o n t o s t u d y t h i s p r o b l e m was C h e v a l i e r du B u a t . E v e n C h a r l e s D a r w i n s t u d i e d t h e p r o b l e m and was t h e f i r s t t o show t h a t a c y l i n d e r m o v i n g t h r o u g h a f l u i d d i s p l a c e s f l u i d p a r t i c l e s i n t h e d i r e c t i o n o f i t s m o t i o n . He went on t o show t h a t t h e d i s p l a c e d mass o f f l u i d e n c l o s e d b e t w e e n t h e i n i t i a l and f i n a l p o s i t i o n o f t h e f l u i d p a r t i c l e s i s t h e added mass i t s e l f . O t h e r w e l l known r e s e a r c h e r s t o work on t h i s p r o b l e m i n c l u d e B e s s e l , G r e e n , P l a m a , S t o k e s , and Lamb. I n o r d e r t o d e v e l o p a t r u e t h e o r e t i c a l mode l o f t h e i n t e r a c t i o n b e t w e e n a f l o a t i n g body and t h e f l u i d med ium, t h e e f f e c t s o f v i s c o s i t y and t h e t i m e h i s t o r y o f t h e m o t i o n mus t be c o n s i d e r e d . W i t h i n a v i s c o u s f l u i d , t h e r e c a n be s e p a r a t e d f l o w p a s t t h e o b j e c t . T h i s c a v i t y f u r t h e r i n d u c e s an added mass d e p e n d i n g on t h e shape o f t h e c a v i t y . The p r o b l e m c a n be s i m p l i f i e d i f one c o n s i d e r s h a r m o n i c m o t i o n o f an o b j e c t i n a p o t e n t i a l f i e l d . B a s e d on t h i s s i m p l i f i c a t i o n , a number o f t h e o r e t i c a l t e c h n i q u e s h a v e b e e n d e v i s e d t o t r y t o d e t e r m i n e t h e h y d r o d y n a m i c c o e f f i c i e n t s o f a w i d e r a n g e o f b o d i e s . i C o n f o r m a l mapp ing i s a w i d e l y u s e d t e c h n i q u e , h o w e v e r , i t c a n o n l y be u s e d i n two d i m e n s i o n a l a p p l i c a t i o n s . V a r i o u s s i n g u l a r i t y methods do n o t h a v e t h i s p r o b l e m and a r e u t i l i z e d i n t h e d e s i g n o f 6 o f f s h o r e s t r u c t u r e s . H a v e l o c k (1955) d e t e r m i n e d t h e h y d r o d y n a m i c c o e f f i c i e n t s o f a s p h e r e , Wang and Shen (1966) u s e d a s i m i l a r a p p r o a c h t o c a l c u l a t e t h e added mass and damping c o e f f i c i e n t o f a s p h e r e i n w a t e r o f f i n i t e d e p t h . K i m (1974) d e t e r m i n e d t h e h y d r o d y n a m i c c o e f f i c i e n t s o f e l l i p s o i d b o d i e s o s c i l l a t i n g n e a r t h e f r e e s u r f a c e . G a r r i s o n (1975) e m p l o y e d d i s t r i b u t e d s i n g u l a r i t i e s i n c a l c u l a t i o n s o f h y d r o d y n a m i c c o e f f i c i e n t s o f v e r t i c a l c i r c u l a r c y l i n d e r s i n w a t e r o f f i n i t e d e p t h . B a i and Y e u n g (1974) u s e d d i s t r i b u t e d s i n g u l a r i t i e s t o c a l c u l a t e h y d r o d y n a m i c c o e f f i c i e n t s o f a x i s y m m e t r i c o c e a n p l a t f o r m s . K r i t i s (1979) u s e d t h e h y b r i d i n t e g r a l me thod o f Yeung (1975) t o c a l c u l a t e t h e h y d r o d y n a m i c c o e f f i c i e n t s o f a c i r c u l a r c y l i n d e r . W h i l e t h e u s e o f s i n g u l a r i t y methods i s e x t e n s i v e t h e r e a r e d i s a d v a n t a g e s i n v o l v e d w i t h t h e i r a p p l i c a t i o n : c o n s i d e r a b l e c o m p u t a t i o n a l t i m e i s r e q u i r e d and c a r e must be t a k e n i n d i s c r e t i z i n g t h e body s u r f a c e . A l s o , t h e method some t imes g i v e s e r r o n e o u s v a l u e s a t c e r t a i n f r e q u e n c i e s r e f e r r e d t o as " i r r e g u l a r f r e q u e n c i e s . " F i n i t e e l e m e n t t e c h n i q u e s do n o t h a v e t h i s d i s a d v a n t a g e . Howeve r , i n o r d e r t o u s e f i n i t e e l e m e n t m e t h o d s , t h e e n t i r e f l o w f i e l d must be d i s c r e t i z e d w h i c h may i n v o l v e c o n s i d e r a b l e more c o m p u t a t i o n a l t i m e . The t h e o r e t i c a l mode l u s e d , i n t h i s t h e s i s i s a p p l i c a b l e m a i n l y f o r t h e c a s e o f an a x i s y m m e t r i c body i n a f l u i d . T h i s t h e o r e t i c a l mode l i s i n v e s t i g a t e d and compared w i t h e x p e r i m e n t a l r e s u l t s . The t h e o r e t i c a l mode l u s e d i s known as t h e M a t c h i n g T e c h n i q u e . 7 The M a t c h i n g T e c h n i q u e was f i r s t i n t r o d u c e d b y C . J . G a r r e t t (1971) i n o r d e r t o c a l c u l a t e wave f o r c e s on a c i r c u l a r d o c k . The method was a d a p t e d b y T . Sabuncu and S . M . C a l i s a l (1980) t o d e t e r m i n e h y d r o d y n a m i c c o e f f i c i e n t s o f s i n g l e and d o u b l e v e r t i c a l c y l i n d e r s . The t e c h n i q u e i s b a s e d on d i v i d i n g t h e s o l u t i o n doma in i n t o a x i s y m m e t r i c s u b - r e g i o n s where t h e v a l u e o f t h e p o t e n t i a l i s e x p r e s s e d i n a s e r i e s o f unknown c o e f f i c i e n t s . The unknown c o e f f i c i e n t s a r e t h e n d e t e r m i n e d b y m a t c h i n g n o r m a l v e l o c i t i e s as w e l l as p r e s s u r e s on t h e b o u n d a r i e s o f e a c h o f t h e s u b - r e g i o n s . The t e c h n i q u e h a s r e c e n t l y b e e n a p p l i e d b y S a b u n c u and C a l i s a l (1988) t o d e t e r m i n e t h e h y d r o d y n a m i c c o e f f i c i e n t s o f c y l i n d e r s i n n a r r o w t a n k s and a l s o t o d e t e r m i n e i n d u c e d s u r g e h y d r o d y n a m i c c o e f f i c i e n t s on c y l i n d e r s due t o a n a d j a c e n t , i d e n t i c a l c y l i n d e r i n h e a v e m o t i o n . 8 2. EXPERIMENTAL WORK A l l e x p e r i m e n t a l work f o r t h i s t h e s i s was c o n d u c t e d a t t h e Ocean E n g i n e e r i n g C e n t r e o f B . C . R e s e a r c h , l o c a t e d i n t h e D i s c o v e r y P a r k r e g i o n o f t h e U . B . C . Campus. V i e w s o f t h e e x t e r i o r and t h e i n t e r i o r o f t h e f a c i l i t y a r e f o u n d i n F i g u r e s 9 . 2 and 9 . 3 . T h r e e d i f f e r e n t t y p e s o f e x p e r i m e n t s were c o n d u c t e d a t t h e t o w i n g t a n k . The f i r s t was a d e t e r m i n a t i o n o f h y d r o d y n a m i c c o e f f i c i e n t s o f a compound c y l i n d e r i n h e a v e m o t i o n . These t e s t s we re done t o v e r i f y p r e v i o u s l y done e x p e r i m e n t s as w e l l as t o compare t h e e x p e r i m e n t a l r e s u l t s w i t h t h e p r e d i c t e d r e s u l t s f r o m t h e M a t c h i n g T e c h n i q u e . The s e c o n d s e t o f e x p e r i m e n t s were p e r f o r m e d t o d e t e r m i n e t h e i n d u c e d s i d e f o r c e s on a s i n g l e c y l i n d e r due t o an i d e n t i c a l , a d j a c e n t c y l i n d e r i n h e a v e m o t i o n . T h e s e i n d u c e d s i d e f o r c e s a r e compared t o v a l u e s p r e d i c t e d b y t h e M a t c h i n g T e c h n i q u e . A l s o , h y d r o d y n a m i c t e s t s were c o n d u c t e d t o e v a l u a t e t h e i n d u c e d h e a v e added mass and damping c o e f f i c i e n t s o f a s i n g l e c y l i n d e r , a g a i n due t o a n i d e n t i c a l , a d j a c e n t c y l i n d e r i n h e a v e m o t i o n . These t e s t s were a l s o compared t o p r e d i c t e d r e s u l t s compu ted u s i n g t h e M a t c h i n g T e c h n i q u e . 9 2 . 1 HYDRODYNAMIC COEFFICIENTS OF A COMPOUND CYLINDER IN HEAVE MOTION H y d r o d y n a m i c t e s t s were c o n d u c t e d on a t r i p l e c y l i n d e r m o d e l . F o r e a c h t e s t a m o t i o n g e n e r a t o r was u s e d t o i m p a r t s m a l l a m p l i t u d e s i n u s o i d a l h e a v e m o t i o n t o a compound c y l i n d e r a t a p a r t i c u l a r f r e q u e n c y . The h e a v e f o r c e was c o n t i n u o u s l y m e a s u r e d f r o m an i n s t r u m e n t e d dynamometer and r e c o r d e d b y a d a t a a c q u i s i t i o n c o m p u t e r . A y o - y o p o s i t i o n t r a n s d u c e r was u s e d t o measu re t h e d i s p l a c e m e n t o f t h e c y l i n d e r . A s i g n a l c o n d i t i o n e r p r o v i d e d t h e r e q u i r e d a m p l i f i c a t i o n and e x c i t a t i o n o f t h e s i g n a l b e f o r e i t was r e c o r d e d . A l l o f t h e r e c o r d e d d a t a was w r i t t e n i n a m u l t i p l e x e d f o r m on f l o p p y TM d i s k e t t e s b y a MINC 11 c o m p u t e r . P r o c e s s i n g o f t h e d a t a was c a r r i e d o u t w i t h a d a t a a n a l y s i s p r o g r a m c o m p i l e d on a VAX™ 1 1 / 7 5 0 c o m p u t e r . D e t a i l s on t h e d a t a a c q u i s i t i o n s y s t e m c a n be f o u n d i n A p p e n d i x A and d e t a i l s o f t h e d a t a a n a l y s i s p r o g r a m c a n be f o u n d i n A p p e n d i x B. A c a l c u l a t i o n o f t h e 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 s a r e l a t i v e l y s i m p l e p r o c e d u r e . From t h e d i s p l a c e m e n t t r a c e r e c o r d o f t h e y o - y o t r a n s d u c e r , one i s e a s i l y a b l e t o d e t e r m i n e t h e m a g n i t u d e o f t h e v e l o c i t y and t h e a c c e l e r a t i o n a t any t i m e . A l s o , b e c a u s e t h e m o t i o n g e n e r a t e d i s p u r e l y s i n u s o i d a l , i t i s e a s y t o d e t e r m i n e p o i n t s where t h e a c c e l e r a t i o n i s maximum and v e l o c i t y i s minimum as w e l l as t h e r e v e r s e c a s e o f minimum a c c e l e r a t i o n and maximum v e l o c i t y . A t t h e s e p o i n t s , i t i s p o s s i b l e t o i s o l a t e t h e 10 i n d i v i d u a l c o n t r i b u t i o n o f added mass and damp ing t e r m s . The g o v e r n i n g e q u a t i o n o f t h e dynamic s y s t e m i s : F = ( m + a ) x + ( b ) x + ( c ) x 2 . 1 - 1 H 2 2 22 2 2 Where F = Heave f o r c e H m = body mass a = added mass c o e f f i c i e n t 22 b 2 2 = damping c o e f f i c i e n t c = r e s t o r i n g f o r c e c o e f f i c i e n t x = a c c e l e r a t i o n i n h e a v e m o t i o n • x = v e l o c i t y i n heave m o t i o n x = d i s p l a c e m e n t i n h e a v e d i r e c t i o n From t h i s e q u a t i o n i t i s o b v i o u s t h a t once t h e r e s t o r i n g f o r c e component h a s b e e n s u b t r a c t e d f r o m t h e o v e r a l l m e a s u r e d f o r c e ; t h e added mass and damp ing c o e f f i c i e n t s c a n be d e t e r m i n e d b y i s o l a t i n g t h e r e s p e c t i v e c o e f f i c i e n t s c o r r e s p o n d i n g t o when e i t h e r t h e a c c e l e r a t i o n o r t h e v e l o c i t y t e rms a r e z e r o . D u r i n g t h e c o u r s e o f t h e s e t e s t s t h e c y l i n d e r was o s c i l l a t e d a t v a r i o u s f r e q u e n c i e s r a n g i n g f r o m 0 . 2 5 H e r t z t o 2 . 5 H e r t z . I t was f o u n d t h a t a t f r e q u e n c i e s g r e a t e r t h a n 2 . 0 H e r t z , g r e a t c a r e h a d t o be e x e r t e d i n o r d e r t o p r e v e n t e x c e s s i v e wear on t h e e q u i p m e n t . A t t i m e s t h e f o r c e s were so g r e a t t h a t t h e e n t i r e r i g was f l e x i n g n o t i c e a b l y ; t h e r e f o r e , o n l y a few t e s t s a t f r e q u e n c i e s 11 above 2 . 0 H e r t z were c o n d u c t e d . Two o t h e r p a r a m e t e r s were a l s o v a r i e d i n a d d i t i o n t o t h e f r e q u e n c y o f e x c i t a t i o n : a m p l i t u d e o f m o t i o n and d r a f t o f t h e c y l i n d e r m o d e l . The t h e o r i e s u s e d t o p r e d i c t t h e h y d r o d y n a m i c c o e f f i c i e n t s u s e l i n e a r i z a t i o n t h e o r y and assume s m a l l a m p l i t u d e s o f m o t i o n i n r e s p e c t t o t h e d i a m e t e r o f t h e c y l i n d e r . T e s t s on t h e l i m i t a t i o n s o f t h e s e a s s u m p t i o n s c a n be done b y v a r y i n g t h e a m p l i t u d e o f o s c i l l a t i o n . By v a r y i n g t h e d r a f t o f t h e c y l i n d e r i t i s p o s s i b l e t o i n v e s t i g a t e t h e e f f e c t s o f g r e a t e r s u r f a c e a r e a s u b j e c t e d t o h y d r o d y n a m i c l o a d s . L a s t l y , t h e r e s u l t s were compared w i t h p r e v i o u s l y g a t h e r e d e x p e r i m e n t a l d a t a and n u m e r i c a l r e s u l t s u s i n g t h e M a t c h i n g T e c h n i q u e . The r e p e a t o f t e s t s was deemed n e c e s s a r y due t o d o u b t s a b o u t t h e r e l i a b i l i t y o f t h e dynamometer u s e d i n t h e o r i g i n a l e x p e r i m e n t s . The dynamometer was s u b s e q u e n t l y f o u n d t o c o n t a i n s m a l l c r a c k s when a n o t h e r r e s e a r c h e r ( G o o d r i d g e 1986) was t o u s e i t i n a n o t h e r e x p e r i m e n t a l i n v e s t i g a t i o n . 2 . 2 INDUCED SIDE FORCES ON A CYLINDER FROM A SECOND CYLINDER  IN HEAVE MOTION F o r t h e h e a v e i n d u c e d s i d e f o r c e e x p e r i m e n t s , two g e o m e t r i c a l l y s i m i l a r s i n g u l a r c y l i n d e r s were r e q u i r e d . These t e s t s i n v o l v e d one s t a t i o n a r y c y l i n d e r as w e l l as a n a d j a c e n t c y l i n d e r n e a r b y s u b j e c t e d t o s i n u s o i d a l h e a v e m o t i o n b y t h e same m o t i o n g e n e r a t o r u s e d i n t h e H y d r o d y n a m i c Heave T e s t s o f a 12 Compound C y l i n d e r e x p e r i m e n t s . The s t a t i o n a r y c y l i n d e r h a d dynamometers l o c a t e d i n s i d e t h e c y l i n d e r w h i c h we re o r i e n t e d i n s u c h a way t h a t t h e s u r g e f o r c e s i n d u c e d b y t h e h e a v e m o t i o n o f t h e n e a r b y c y l i n d e r c o u l d be r e c o r d e d b y t h e d a t a a c q u i s i t i o n s y s t e m . A g a i n a y o - y o t r a n s d u c e r was u s e d t o measu re t h e d i s p l a c e m e n t o f t h e one c y l i n d e r i n heave m o t i o n . The same s i g n a l c o n d i t i o n e r , d a t a a c q u i s i t i o n c o m p u t e r , d a t a p r o c e s s i n g c o m p u t e r , and d a t a a n a l y s i s s o f t w a r e were u s e d as i n t h e p r e v i o u s e x p e r i m e n t s . The maximum i n d u c e d s u r g e f o r c e was o b t a i n e d f r o m t h e f o r c e t r a c e o f t h e dynamometer l o c a t e d i n s i d e t h e c y l i n d e r . The peak s u r g e f o r c e was t h e n n o n - d i m e n s i o n a l i s e d w i t h r e s p e c t t o t h e w e i g h t o f t h e c y l i n d e r and compared t o n u m e r i c a l r e s u l t s made u s i n g t h e M a t c h i n g T e c h n i q u e . These s e r i e s o f t e s t s were a g a i n c o n d u c t e d f o r v a r i o u s f r e q u e n c i e s r a n g i n g f r o m b e t w e e n 0 . 2 5 H e r t z t o 2 . 5 H e r t z . The d i s t a n c e b e t w e e n t h e c y l i n d e r c e n t e r s were v a r i e d b e t w e e n f i v e p r e s c r i b e d d i s t a n c e s . The l e a s t s e p a r a t i o n was a p p r o x i m a t e l y 2 . 0 r a d i i , a t w h i c h t h e c y l i n d e r s a r e v i r t u a l l y t o u c h i n g one a n o t h e r ; t h e g r e a t e s t was a d i s t a n c e o f 4 . 0 r a d i i s e p a r a t i n g t h e c e n t e r s o f t h e c y l i n d e r s . T h i s was done t o v e r i f y t h e p r e d i c t i o n o f d e c r e a s i n g i n d u c e d f o r c e f r o m t h e a d j a c e n t o s c i l l a t i n g c y l i n d e r as t h e d i s t a n c e b e t w e e n t h e c y l i n d e r s i s i n c r e a s e d . As w e l l as c y l i n d e r s e p a r a t i o n , t h e a m p l i t u d e o f o s c i l l a t i o n o f t h e one c y l i n d e r was v a r i e d t o i n v e s t i g a t e t h e l i m i t s o f l i n e a r i z a t i o n . F i n a l l y , t h e t e s t s were c o n d u c t e d i n a deep w a t e r as w e l l as 13 a s h a l l o w w a t e r e n v i r o n m e n t . The s h a l l o w w a t e r c a s e c o n s i s t e d o f a f a l s e b o t t o m s u s p e n d e d i n a t o w i n g t a n k t o r e d u c e t h e d e p t h o f t a n k a r o u n d t h e c y l i n d e r f r o m 2 . 3 3 m e t r e s t o 0 . 8 8 m e t r e s . 2 . 3 INDUCED HEAVE HYDRODYNAMIC COEFFICIENTS ON A SINGLE CYLINDER FROM A SECOND CYLINDER IN HEAVE MOTION The d e t e r m i n a t i o n o f i n d u c e d h e a v e h y d r o d y n a m i c c o e f f i c i e n t s was c o n d u c t e d i n t h e e x a c t same manner as t h e d e t e r m i n a t i o n o f t h e i n d u c e d s i d e f o r c e s . A s i n g l e c y l i n d e r was i n s t r u m e n t e d w i t h dynamometers o r i e n t e d t o measure f o r c e s i n t h e h e a v e d i r e c t i o n w h i l e a n o t h e r s i n g l e c y l i n d e r was s u b j e c t e d t o s i n u s o i d a l h e a v e m o t i o n . Deep w a t e r t e s t s as w e l l as s h a l l o w w a t e r t e s t s u s i n g t h e s u s p e n d e d b o t t o m were c o n d u c t e d f o r f i v e s e p a r a t e c y l i n d e r s e p a r a t i o n s . The M a t c h i n g T e c h n i q u e n u m e r i c a l mode l d e a l s w i t h t h e d e t e r m i n a t i o n o f i n d u c e d f o r c e s f o r a p a i r o f c y l i n d e r s i n s i m p l e h e a v e m o t i o n . The n u m e r i c a l mode l s i m p l i f i e s t h e p r o b l e m o f d e t e r m i n i n g h e a v e h y d r o d y n a m i c c o e f f i c i e n t s b y c a l c u l a t i n g t h e i n d u c e d h e a v e f o r c e f r o m t h e a d d i t i o n o f t h e d i f f r a c t i o n p o t e n t i a l and t h e r a d i a t i o n p o t e n t i a l . I n t e rms o f e x p e r i m e n t a l p r o c e d u r e h o w e v e r , t h e d e t e r m i n a t i o n o f i n d u c e d f o r c e s due t o d i f f r a c t i n g waves i s n o t an e a s y t a s k . When one c o n s i d e r s t h a t t h e r a d i a t i o n t e r m i s a t l e a s t a n o r d e r o f m a g n i t u d e h i g h e r t h a n t h e d i f f r a c t i o n t e r m , i t i s s i m p l e r t o t r y t o e l i m i n a t e t h e r a d i a t i o n t e r m and c o n c e n t r a t e on t h e d i f f r a c t i o n t e r m . 14 E x p e r i m e n t a l l y , t h e e l i m i n a t i o n o f t h e r a d i a t i o n t e r m i s c a r r i e d o u t b y h o l d i n g one c y l i n d e r s t a t i o n a r y w h i l e t h e o t h e r c y l i n d e r i s s u b j e c t e d t o s i m p l e s i n u s o i d a l h e a v e m o t i o n . By h o l d i n g t h e c y l i n d e r s t a t i o n a r y , t h e r a d i a t i o n t e r m i s e l i m i n a t e d , and one i s l e f t w i t h j u s t t h e d i f f r a c t i o n t e r m . The n u m e r i c a l mode l i s e a s i l y f i x e d t o s e t t h e r a d i a t i o n p o t e n t i a l t o z e r o and d e t e r m i n e t h e h e a v e h y d r o d y n a m i c c o e f f i c i e n t s b a s e d o n l y on t h e d i f f r a c t i o n p o t e n t i a l . Howeve r , t h i s s i m p l i f i c a t i o n i n t h e e x p e r i m e n t n e g l e c t s t h e a d d i t i o n o f r a d i a t i n g waves b e i n g r e f l e c t e d f r o m t h e o t h e r c y l i n d e r b a c k t o t h e s o u r c e c y l i n d e r . Y e t on a p r a c t i c a l p o i n t , s i n c e t h e f o r c e s b e i n g m e a s u r e d a r e v e r y s m a l l , i t was d e c i d e d t h a t i n o r d e r t o o b t a i n a r e l i a b l e e s t i m a t e o f t h e i n d u c e d h e a v e 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 t i s b e s t t o assume t h e e f f e c t o f t h e s e waves t o be s m a l l and c o n c e n t r a t e on t h e i n c o m i n g r a d i a t i n g waves o f t h e o t h e r c y l i n d e r . The d e t e r m i n a t i o n o f t h e 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 s a c c o m p l i s h e d i n a s i m i l a r as i n t h e f i r s t e x p e r i m e n t a l i n v e s t i g a t i o n . From t h e d i s p l a c e m e n t t r a c e on t h e o s c i l l a t i n g c y l i n d e r , t h e h e a v e added mass and damp ing c o e f f i c i e n t s o f t h e s t a t i o n a r y c y l i n d e r were c a l c u l a t e d f r o m t h e f o r c e s e x e r t e d on i t . The d y n a m i c e q u a t i o n i s t h e same as e q u a t i o n ( 2 . 1 - 1 ) and t h e d e t e r m i n a t i o n o f t h e 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 s c a r r i e d o u t i n t h e same manner as b e f o r e . 15 3. BASIC THEORY 3 . 1 . GENERAL CASE OF A BODY IN A FLUID DOMAIN B e f o r e any a n a l y s i s o f t h e p r o b l e m c a n b e g i n , t h e c o o r d i n a t e s y s t e m must be e s t a b l i s h e d . F i g u r e 8 .1 shows a g e n e r a l a r r a n g e m e n t o f a f r e e f l o a t i n g b o d y . The o r i g i n o f t h i s s t a n d a r d C a r t e s i a n c o o r d i n a t e s y s t e m i s a t t h e b o t t o m w i t h t h e z a x i s b e i n g p o s i t i v e i n t h e upward d i r e c t i o n . The u n d i s t u r b e d f r e e s u r f a c e i s d e f i n e d a t z = d . 3 . 1 . 1 GOVERNING EQUATION I f t h e f l u i d s u r r o u n d i n g t h e body i s assumed t o be i n v i s c i d , i r r o t a t i o n a l , and i n c o m p r e s s i b l e , t h e f l u i d may be d e f i n e d w i t h a v e l o c i t y p o t e n t i a l , $ , w h i c h s a t i s f i e s t h e L a p l a c e E q u a t i o n : V 2 $ = 0 3 . 1 . 1 - 1 S i n c e o n l y a x i s y m m e t r i c b o d i e s a r e b e i n g c o n s i d e r e d , t h e L a p l a c e E q u a t i o n i s b e s t e x p r e s s e d i n c y l i n d r i c a l c o o r d i n a t e s : dZv r dr r 2 d2B 32z 16 I n o r d e r t o s o l v e t h e L a p l a c e E q u a t i o n , a s e p a r a t i o n o f v a r i a b l e s t e c h n i q u e i s a p p l i e d a l o n g w i t h t h e a p p r o p r i a t e b o u n d a r y c o n d i t i o n . I t c a n be assumed t h a t t h e p o t e n t i a l f u n c t i o n c a n be d e f i n e d as a p r o d u c t o f f o u r i n d e p e n d e n t f u n c t i o n s , e a c h c o n t a i n i n g one v a r i a b l e so t h a t $ c a n be d e f i n e d a s : $ ( r , 0 , z , t ) = R ( r ) 9 ( 0 ) Z ( z ) T ( t ) 3 . 1 . 1 - 3 Howeve r , s i n c e t h i s p r o b l e m d e a l s o n l y w i t h p e r i o d i c m o t i o n , one c a n s u b s t i t u t e f o r T ( t ) : T ( t ) = e " i W t 3 . 1 . 1 - 4 H e n c e , t h e p o t e n t i a l c a n be w r i t t e n : * ( r , 0 , z , t ) = R ( r ) 0 ( 0 ) Z ( z ) e " i w t 3 . 1 . 1 - 5 T h i s p a r t i a l d i f f e r e n t i a l e q u a t i o n c a n be b r o k e n down i n t o t h r e e o r d i n a r y d i f f e r e n t i a l e q u a t i o n s o f t h e f o r m : d 2 R + — r ~ 2 - 2 d r 2 ^ + [ J _ A | R = 0 3 . 1 . 1 - 6 r d r ^ r d2e 2 — + A e - 0 3 . 1 . 1 - 7 d2* d 2 Z 2 — + A Z - 0 3 . 1 . 1 - 8 d 2 z 17 3 . 1 . 2 BOUNDARY CONDITIONS F o r t h e c a s e o f a f l o a t i n g b o d y a t t h e f r e e s u r f a c e as shown i n F i g u r e 8 . 1 , t h e f o l l o w i n g f i v e b o u n d a r y c o n d i t i o n s a p p l y : i ) On t h e b o t t o m s u r f a c e , z = 0 , t h e r e e x i s t s a n i m p e r m e a b l e b o t t o m b o u n d a r y c o n d i t i o n whe reby t h e f l u i d v e l o c i t y i n a d i r e c t i o n n o r m a l t o t h e b o t t o m must be z e r o . - 0 a t z = 0 3 . 1 . 2 - 1 dz i i ) On t h e submerged s u r f a c e o f t h e b o d y , S , t h e r e i s a l s o b a n i m p e r m e a b l e b o u n d a r y c o n d i t i o n whe reby t h e f l u i d v e l o c i t y i n a n o r m a l d i r e c t i o n t o t h e body s u r f a c e must be e q u a l t o t h e n o r m a l v e l o c i t y component o f t h e body i t s e l f . T h i s c a n be e x p r e s s e d a s : = V • n + fi • ( r x n ) 3 . 1 . 2 - 2 b where V i s t h e v e l o c i t y v e c t o r , Q i s t h e a n g u l a r v e l o c i t y o f t h e f l o a t i n g b o d y , and n i s a u n i t v e c t o r t o t h e body s u r f a c e . i i i ) F o r a wave i n c i d e n t on t h e submerged b o d y , S , t h e n o r m a l d e r i v a t i v e o f t h e d i f f r a c t e d wave p o t e n t i a l , $ , i s e q u a l t o t h e n e g a t i v e o f t h e n o r m a l d e r i v a t i v e o f t h e i n c i d e n t wave p o t e n t i a l , $ , t h a t i s , t h e r e i s no wave e n e r g y a b s o r p t i o n i n t o an 18 t h e f l o a t i n g b o d y . T h i s b o u n d a r y c o n d i t i o n i s e x p r e s s e d a s ; 3$ 3$ dn 5n The i n c i d e n t wave p o t e n t i a l $ , c a n be d e f i n e d u s i n g numerous a n a l y t i c a l m e t h o d s . The most common method as p r o p o s e d b y I s a a c s o n e t a l . (1981) i s t h e l i n e a r t h e o r y wave p o t e n t i a l f o r an i n c i d e n t wave t r a v e l l i n g o v e r a f i n i t e d e p t h f l u i d . T h i s i n c i d e n t p o t e n t i a l i s e x p r e s s e d a s ; $ = iMS cosh k(z+d) gi(kx-wt) 3 2 4 i 2co cosh (kd) where H i s t h e wave h e i g h t , d i s t h e w a t e r d e p t h , w i s t h e a n g u l a r wave f r e q u e n c y , and k i s t h e wave number f o u n d f r o m t h e l i n e a r d i s p e r s i o n r e l a t i o n s h i p w h i c h i s a l s o f o r m u l a t e d f r o m t h e l i n e a r wave t h e o r y ; 2 — = k t a n h (kh ) 3 . 1 . 2 - 5 g i v ) The r a d i a t i o n b o u n d a r y c o n d i t i o n e x i s t s a t an i m a g i n a r y b o u n d a r y d e f i n e d b y a c o n t r o l s u r f a c e , S , w h i c h i s " s u f f i c i e n t l y " f a r away. T h i s r a d i a t i o n b o u n d a r y c o n d i t i o n e n s u r e s t h a t t h e d i s s i p a t i n g e n e r g y r a d i a t e away f r o m t h e b o d y . T h e r e a r e numerous f o r m s o f t h e r a d i a t i o n c o n d i t i o n w h i c h a r e n o r m a l l y u s e d , t h e e s s e n t i a l component o f t h e c o n d i t i o n i s a n e g a t i v e e x p o n e n t i a l i n t e rms o f d i s t a n c e away f r o m t h e b o d y . F o r t h i s a x i s y m m e t r i c 19 b o d y , t h e r a d i a t i o n b o u n d a r y c o n d i t i o n h a s b e e n d e t e r m i n e d b y B a i (1972) t o be d e f i n e d a s : [ ^ + i k ] « i S - - I  k I • 3 . 1 . 2 - 6 r where k i s t h e wave number , and R i s t h e d i s t a n c e f r o m t h e body c e n t e r t o t h e c o n t r o l s u r f a c e , S . r v ) The f i n a l b o u n d a r y c o n d i t i o n i s known as t h e " f r e e s u r f a c e b o u n d a r y c o n d i t i o n . " I t i s made up o f two b o u n d a r y c o n d i t i o n s : t h e k i n e m a t i c b o u n d a r y c o n d i t i o n and t h e dynam ic b o u n d a r y c o n d i t i o n . The k i n e m a t i c b o u n d a r y c o n d i t i o n s t a t e s t h a t f l u i d p a r t i c l e s a t t h e f r e e s u r f a c e , rj, must r e m a i n a t t h e f r e e s u r f a c e . T h i s c a n be d e f i n e d m a t h e m a t i c a l l y a s : + IT + T~ a = 3"" a t z = r? 3 . 1 . 2 - 7 dt dx 3x ay ay az The dynam ic b o u n d a r y c o n d i t i o n i s t h e B e r n o u l l i e n e r g y e q u a t i o n w h i c h s t a t e s t h a t t h e p r e s s u r e , p , i s u n i f o r m o v e r t h e f r e e s u r f a c e . T h i s i s g i v e n m a t h e m a t i c a l l y a s : 4. Z Z a$ i f fa$l ra$l ra$l 1 P N a t z = i| 3 . 1 . 2 - 8 B o t h o f t h e s e b o u n d a r y c o n d i t i o n s a r e n o n - l i n e a r and c o n s i d e r a b l y d i f f i c u l t t o u t i l i z e i n t h e i r c u r r e n t f o r m . Howeve r , 20 i f one l i n e a r i z e s t h e two e q u a t i o n s and n e g l e c t s a l l o f t e rms o f o r d e r g r e a t e r t h a n u n i t y , t h e n a c o m b i n a t i o n o f t h e two b o u n d a r y c o n d i t i o n s y i e l d s a s i m p l e l i n e a r i z e d f r e e s u r f a c e b o u n d a r y c o n d i t i o n f o r a p u r e l y p e r i o d i c p o t e n t i a l o f t h e f o r m : flfc 2 p = - $ a t z = d 3 . 1 . 2 - 9 dz g 3 . 1 . 3 DETERMINATION OF HYDRODYNAMIC COEFFICIENTS Once t h e p o t e n t i a l , h a s b e e n d e t e r m i n e d b y a n u m e r i c a l scheme o r a n a n a l y t i c a l method f o r a l l p o i n t s i n t h e f l u i d d o m a i n , i t i s t h e n p o s s i b l e t o d e t e r m i n e t h e h y d r o d y n a m i c c o e f f i c i e n t s o f t h e f l o a t i n g b o d y . The f o r c e s and moments a r e d e f i n e d b y t a k i n g t h e t i m e d e r i v a t i v e s o f t h e p r e s s u r e i n t e g r a l s t a k e n o v e r t h e e n t i r e b o d y s u r f a c e , S . M a t h e m a t i c a l l y , t h e s e e q u a t i o n s i n t e rms b o f l i n e a r f o r c e o r t r a n s l a t o r y moments a r e g i v e n a s : 3 . 1 . 3 - 1 3 . 1 . 3 - 2 b T h e s e e q u a t i o n s c a n be r e w r i t t e n i n a g e n e r a l i z e d n o n - d i m e n s i o n a l f o r m i n t e rms o f " i " and " j " w h i c h c o v e r a l l s i x d e g r e e s o f f r e e d o m . The f o r c e t e r m i s s p l i t i n t o i t s r e a l and i m a g i n a r y 21 p a r t s w h i c h c o r r e s p o n d t o t he added mass and damp ing t e r m s . The s o l u t i o n i s g i v e n i n a a c c e p t e d f o r m p r o p o s e d b y Wehausen (1971) a s : F a J i J + i - s II 4 ( r J , z ) L * , J- dS 3 . 1 . 3 - 3 u p W pV wpV s b w h e r e , F = T o t a l h y d r o d y n a m i c f o r c e . a = Added mass c o e f f i c i e n t , b^ = Damping c o e f f i c i e n t . p = D e n s i t y o f f l u i d m o t i o n . V = D i s p l a c e d vo lume o f b o d y , w = F r e q u e n c y o f m o t i o n . V = V e l o c i t y i n t h e i d i r e c t i o n . i J n = U n i t n o r m a l i n t h e i t r a n s l a t o r y d i r e c t i o n , r x n = U n i t n o r m a l i n t h e i r o t a t i o n a l d i r e c t i o n . i JJ* . ds = I n t e g r a l o v e r t h e body s u r f a c e . S b $ ( r , 0 , z ) = P o t e n t i a l o f f l u i d medium. I n o r d e r t o o b t a i n t h e h y d r o d y n a m i c c o e f f i c i e n t s , one must c a l c u l a t e t h e p o t e n t i a l , a t a l l p o i n t s s u r r o u n d i n g t h e b o d y . The n u m e r i c a l scheme i n v e s t i g a t e d i n t h i s s t u d y c a l c u l a t e s v a l u e s f o r $ f r o m w h i c h t h e h y d r o d y n a m i c c o e f f i c i e n t s a r e c o m p u t e d . 22 3 . 2 DIMENSIONAL ANALYSIS OF PROBLEM A d i m e n s i o n a l a n a l y s i s o f t h e p e r f o r m e d e x p e r i m e n t s p r i o r t o any t h e o r e t i c a l a p p l i c a t i o n w i l l p r o v e b e n e f i c i a l i n e s t a b l i s h i n g t h e r e l a t i v e i m p o r t a n c e o f t h e f l o w s e p a r a t i o n and d i f f r a c t i o n e f f e c t s . A t i m e i n v a r i a n t maximum f o r c e on a f i x e d s t r u c t u r e due t o a n i n c i d e n t wave c a n be e x p r e s s e d a s : w h e r e , D L VD VT D 2 : f\— — — —— — 1 3 2 - 1 H 2 D 2 L L ' L ' L ' i/ ' D J * U d i s p e r s i o n t e r m d L H — = wave s t e e p n e s s t e r m J_j d i f f r a c t i o n t e r m ^ R e y n o l d s number , R K e u l e g a n - C a r p e n t e r number , K 0 d = w a t e r d e p t h D = r e p r e s e n t a t i v e d i a m e t e r o f body g = g r a v i t a t i o n a l c o n s t a n t H = wave h e i g h t L = w a v e l e n g t h T = p e r i o d o f m o t i o n V = r e p r e s e n t a t i v e r e l a t i v e v e l o c i t y o f f l u i d w i t h r e s p e c t t o t h e body t h e body p = d e n s i t y o f w a t e r v = k i n e m a t i c v i s c o s i t y o f w a t e r 23 F o r t h e t e s t s i n o r d e r t o d e t e r m i n e t h e i n d u c e d s i d e f o r c e s o r t h e i n d u c e d h e a v e h y d r o d y n a m i c c o e f f i c i e n t s o f a s i n g l e c y l i n d e r , one c a n t r e a t t h e n o n - d i m e n s i o n a l a n a l y s i s i n t h e same manner as i f one assumes an i n c i d e n t f l o w on t o a c y l i n d e r . A l t h o u g h t h i s i s a c r u d e a s s u m p t i o n , i t g i v e s an i n s i g h t t o t h e v a l i d i t y o f some o f t h e a s s u m p t i o n s made i n t h e s e t - u p o f t h e n u m e r i c a l m o d e l . K n o w i n g t h a t t h e f r e q u e n c y o f o s c i l l a t i o n s f o r a l l o f t h e e x p e r i m e n t s i s a l w a y s b e t w e e n 0 . 2 5 and 2 . 5 H e r t z , i t i s p o s s i b l e t o c a l c u l a t e some o f t h e s e n o n - d i m e n s i o n a l c o n s t a n t s . T a b l e 3 . 2 - 1 shows t h e v a l u e s o f t h e t e r m s f o r a r a n g e o f f r e q u e n c i e s t e s t e d as w e l l as t h e maximum and minimum o s c i l l a t i o n a m p l i t u d e t h a t t h e c y l i n d e r was s u b j e c t e d t o . The r e p r e s e n t a t i v e d i a m e t e r o f t h e b o d y , D, i s t a k e n t o be t h e maximum d i a m e t e r o f t h e c y l i n d e r ( 3 8 . 6 c m . ) ; t h e k i n e m a t i c v i s c o s i t y v, i s t a k e n t o be 1 .15 x 10 6 2 . m / s . 24 T a b l e 3 . 2 - 1 RANGE OF NON-DIMENSIONAL QUANTITIES ENCOUNTERED F r e q . A m p l i t u d e K e u l e g a n - C a r p e n t e r R e y n o l d s D i f f r a c t i o n ( H z . ) o f M o t i o n Number Number Term (cm. ) 0 . 2 5 1 .5 0 . 2 4 7 . 9 x l 0 3 0 . 2 0 0 . 2 5 4 . 5 0 . 7 3 2 . 4 x 1 0 * 0 . 2 0 1 .00 1 .5 0 . 2 4 3 . 1 x 1 0 * 0 . 2 6 1 .00 4 . 5 0 . 7 3 9 . 5 x 1 0 * 0 . 2 6 2 . 5 0 1 .5 0 . 2 4 7 . 9 x 1 0 * 1 .61 2 . 5 0 4 . 5 0 . 7 3 2 . 4 x l 0 5 1 .61 A c c o r d i n g t o t h e s e f i r s t o r d e r a p p r o x i m a t e c a l c u l a t i o n s , one c a n n o t i c e t h a t t h e R e y n o l d s Number i s u s u a l l y l e s s t h a n 2 . 0 x l 0 5 , t h e u p p e r l i m i t o f t h e l a m i n a r f l o w r e g i o n . T h e r e f o r e , most o f t h e t e s t i n g i s done i n t h e u p p e r r e g i o n o f t h e l a m i n a r f l o w r a n g e a n d , h e n c e , a p o t e n t i a l f l o w s o l u t i o n may be a p p l i e d t o t h e o u t e r f l o w r e g i o n . I s a a c s o n e t . a l . (1981) showed t h a t f l o w s e p a r a t i o n e f f e c t s a r e c o n s i d e r e d i m p o r t a n t f o r K v a l u e s g r e a t e r t h a n 2 . 0 . He a l s o c showed t h a t f l o w d i f f r a c t i o n e f f e c t s a r e c o n s i d e r e d i m p o r t a n t f o r d i s p e r s i o n r a t i o s : ^ g r e a t e r t h a n 0 . 2 . S i n c e i n t h e s e e x p e r i m e n t s t h e K u p p e r v a l u e 0 . 7 3 f a l l s f a r b e l o w 2 . 0 , one c a n c o n c l u d e t h a t c f l o w s e p a r a t i o n s h o u l d n o t be a m a j o r f a c t o r . Howeve r , t h e l a t t e r c r i t e r i a i s s a t i s f i e d o n l y f o r f r e q u e n c i e s g r e a t e r t h a n 0 . 9 H e r t z . T e s t s c o n d u c t e d above 0 . 9 H e r t z a r e w i t h i n t h e d i f f r a c t i o n r e g i m e and c a n be s o l v e d n e g l e c t i n g v i s c o u s e f f e c t s w i t h a p o t e n t i a l f l o w 25 s o l u t i o n t h r o u g h o u t t h e f l u i d d o m a i n . F o r f r e q u e n c i e s f a l l i n g b e l o w 0 . 9 H e r t z , n e i t h e r v i s c o u s e f f e c t s n o r d i f f r a c t i o n may be c o n s i d e r e d d o m i n a n t . I s a a c s o n e t . a l . (1981) s u g g e s t s t h a t t he f o r c e s w h i c h may a r i s e a r e m a i n l y i n e r t i a l and may be s o l v e d u s i n g t h e M o r r i s o n E q u a t i o n , w h i c h i s b a s e d on e m p i r i c a l d a t a . Howeve r , a p o t e n t i a l f l o w mode l w i l l be a p p l i e d a t t h e s e l o w f r e q u e n c i e s t o i n v e s t i g a t e t h e v a l i d i t y o f t h e mode l f o r t h i s c o m b i n a t i o n o f K c a n d l o w ^ r a t i o s . I n t h e s u b s e q u e n t c h a p t e r , a p o t e n t i a l f l o w f i e l d w i l l be assumed a r o u n d t h e c y l i n d e r d o m a i n . A n u m e r i c a l scheme w i l l be i n t r o d u c e d and a p p l i e d t o t h r e e v a r i o u s t e s t c a s e s i n o r d e r t o t r y t o p r e d i c t t h e h y d r o d y n a m i c c o e f f i c i e n t s . 26 4. THEORETICAL MODELS A n u m e r i c a l mode l scheme i s d e a l t w i t h i n t h i s t h e s i s i n o r d e r t o d e t e r m i n e t h e v e l o c i t y p o t e n t i a l , <f>, w h i c h w i l l e n a b l e one t o p r e d i c t t h e h y d r o d y n a m i c c o e f f i c i e n t s . The me thod i s known as t h e M a t c h i n g T e c h n i q u e . T h i s method u t i l i z e s t h e g o v e r n i n g e q u a t i o n and t h e b o u n d a r y c o n d i t i o n s i n t r o d u c e d i n C h a p t e r 3 . The M a t c h i n g T e c h n i q u e w i l l be a p p l i e d t o e a c h o f t h e t h r e e c a s e s o f c y l i n d e r e x p e r i m e n t s . The M a t c h i n g T e c h n i q u e Me thod was o r i g i n a l l y f o r m u l a t e d b y G a r r e t t (1971) i n o r d e r t o s t u d y t h e s c a t t e r i n g o f waves i n c i d e n t on a c i r c u l a r d o c k . Sabuncu and C a l i s a l (1981) a p p l i e d t h e t e c h n i q u e t o d e t e r m i n e t h e h y d r o d y n a m i c c o e f f i c i e n t s o f compound c y l i n d e r s i n v a r i o u s o r i e n t a t i o n s and s i z e s . The t e c h n i q u e i s a p p l i e d b y d i v i d i n g t h e s o l u t i o n doma in i n t o s u b - r e g i o n s and t h e n r e p r e s e n t i n g t h e a n a l y t i c a l s o l u t i o n f o r e a c h o f t h e s e s u b - r e g i o n s i n s e r i e s f o r m w i t h unknown c o e f f i c i e n t s . T h e s e unknown c o e f f i c i e n t s a r e t h e n o b t a i n e d b y m a t c h i n g t h e n o r m a l v e l o c i t i e s and p r e s s u r e s a t t h e b o u n d a r i e s d i v i d i n g t h e r e g i o n s . 27 4 . 1 MATCHING TECHNIQUE FOR PREDICTION OF HYDRODYNAMIC COEFFICIENTS  OF A T R I P L E CYLINDER 4 . 1 . 1 DEFIN IT ION OF FLOWFIELD I n o r d e r t o u s e t h e M a t c h i n g T e c h n i q u e , t h e f l o w f i e l d r e g i o n must be d i v i d e d i n t o s u b - r e g i o n s . T h i s s u b d i v i s i o n o f t h e f l o w f i e l d c o n t a i n i n g t h e t r i p l e c y l i n d e r i s shown i n F i g u r e 8 . 2 . I t must be remembered t h a t t h e e l e m e n t s a r e c y l i n d r i c a l t o b e n e f i t f r o m t h e a x i a l symmetry o f t h e p r o b l e m . The f o u r r e g i o n s a r e d e f i n e d as t h e vo lume c o n t a i n e d w i t h i n t h e f o l l o w i n g r e g i o n s : R e g i o n P o t e n t i a l 1 0 < r < a 0 < z < d $ l i i 2 a < r < a 0 < z < d $ 1 2 2 2 3 a < r < a d < z < d $ 3 2 3 3 4 a < r 0 < z < d $ 2 E The p o t e n t i a l f u n c t i o n s a r e d e f i n e d u s i n g t h e same method as G a r r e t t . 4 . 1 . 2 DEFIN IT ION OF POTENTIALS The v e l o c i t y p o t e n t i a l must s o l v e t h e L a p l a c e e q u a t i o n and a l l t h e b o u n d a r y c o n d i t i o n s d e f i n e d i n C h a p t e r 3 . R e c a l l t h a t t h e p o t e n t i a l c a n be d e f i n e d a s : $ ( r , 0 , z , t ) = He. { R ( r ) 9 ( 0 ) Z ( z ) e 1 W t ) 4 . 1 . 2 - 1 w h i c h s a t i s f i e s : 28 v2$ = 0 S i n c e t h e f l o w i s a x i s y m m e t r i c t h e a x i a l f u n c t i o n , 6 ( 0 ) must be an e v e n f u n c t i o n o f t h e f o r m : 9 ( 0 ) = cos (m0) 4 . 1 . 2 - 2 F o r t h e c a s e o f p u r e h e a v e m o t i o n , t h i s c o e f f i c i e n t , m, i s t a k e n as z e r o a s s u m i n g t h a t t h e most s i g n i f i c a n t c o n t r i b u t i o n comes f r o m t h e f i r s t t e r m o f t h e c o s i n e s e r i e s f o r 9. A l s o t h e g e n e r a l s o l u t i o n o f t h e L a p l a c e e q u a t i o n w i l l c o n t a i n a p a r t i c u l a r s o l u t i o n as w e l l as a homogeneous s o l u t i o n . H e n c e , t h e p o t e n t i a l c a n be d e f i n e d a s : $ ( r , 0 , z , t ) = { <t> ( r , z ) + <f> ( r , z ) } e " l w t 4 . 1 . 2 - 3 P h The p o t e n t i a l s $ , $ 2 > $ 3 > and $ £ a r e now d e f i n e d a c c o r d i n g t o t h e a n a l y s i s o f Sabuncu and C a l i s a l ( 1 9 8 4 ) . I n t h e f o l l o w i n g f o r m u l a t i o n t h e n o t a t i o n i s d e f i n e d as f o l l o w s (See A p p e n d i x F ) : I ( x ) M o d i f i e d B e s s e l f u n c t i o n o f t h e f i r s t k i n d n o f o r d e r n . J ( x ) B e s s e l f u n c t i o n o f t h e f i r s t k i n d o f o r d e r n . n K (x ) M o d i f i e d B e s s e l f u n c t i o n o f t h e s e c o n d k i n d n o f o r d e r n . H ( x ) H a n k e l f u n c t i o n o f t h e f i r s t k i n d o f o r d e r n . n 29 4 . 1 . 2 . 1 REGION 1 W i t h V d e f i n e d as t h e h e a v e v e l o c i t y , t h e v e l o c i t y p o t e n t i a l H h a s t h e f o r m : $ = <j> V d e " i W t 4 . 1 . 2 . 1 . - 1 1 1 H w i t h t h e f o l l o w i n g b o u n d a r y c o n d i t i o n s : 3$ = V a t z - d 4 . 1 . 2 . 1 - 2 3z H l 3$ 1 = 0 a t z = 0 4 . 1 . 2 . 1 - 3 3z The p a r t i c u l a r s o l u t i o n i s g i v e n b y : K " [ ( z 2 ' —2 ) 2^ ] and t h e homogeneous s o l u t i o n b y : ( n?rr ] o l ~d~ J v o ' [ n?ra \ n=l j I 1 I 1 4 . 1 . 2 . 1 - 4 I A °° c , , JL 0 • V A 0 n 7 r Z / 1 1 1 C <p = — + X A cos —7- 4 . 1 . 2 . 1 - 5 IH 2 ^ n • n?ra \ d I i s a m o d i f i e d B e s s e l F u n c t i o n o f t h e f i r s t k i n d ; A and A a r e 0 O n unknown c o e f f i c i e n t s . 30 4 . 1 . 2 . 2 REGION 2 The p o t e n t i a l , . , . - iwt $ = d> V d e 2 2 H must s a t i s f y t h e two b o u n d a r y c o n d i t i o n s : a t z = d 2 a t z = 0 The p a r t i c u l a r s o l u t i o n i s once a g a i n g i v e n b y : K - [[* - —2) m ] and t h e homogeneous s o l u t i o n b y : — 2 = V 3z H 2 = 0 dz where ^ 2 H = B ~ + ^ { B - V - < r > + C_W_(r) } c o s n 7 r z n = l 1 I " 2 ? -o I d t njra x l " l d V 9 I V ( r ) = n [ n7ra •> ((1171 c t \ d I f — - v — I + ^ K d J r nvra * o ( K 31 and W ( r ) = i + 2—L K f-2JL. o l d f xma. •> o I d I ' 1 [ 2 | v 2 ' ° l \ J K i 4 . 1 . 2 . 2 - 7 i ra-a x I 1 • K f - ^ I 1 11 d I /• n^ra % 11 d I B and B a r e unknown c o e f f i c i e n t s w h i c h a r e d e t e r m i n e d b y t h e O n J r a d i a l b o u n d a r y e q u a t i o n . 4 . 1 . 2 . 3 REGION 3 F o r r e g i o n 3 t h e p o t e n t i a l i s g i v e n b y : $ = <j> V d e " i W t 4 . 1 . 2 . 3 - 1 3 3 H and must s a t i s f y t h e two b o u n d a r y c o n d i t i o n s : - r - 3 = V a t z = d 4 . 1 . 2 . 3 - 2 dz H 3 3$ -w 2 $ + g ^ = 0 a t z = d 4 . 1 . 2 . 2 - 3 3 ° 3z The p a r t i c u l a r s o l u t i o n i s g i v e n b y : z S A = ± + i 4 . 1 . 2 . 3 - 4 3p d 2 , to d 32 and t h e homogeneous s o l u t i o n b y : <f> = D X Y + V D X Y 2H 0 0 0 u n n n n = l where X = 0 X = n J (m r ) -o o J (m a 1 0 3 H (m a 1 0 3 J (m a ) o o 2 I (m r ) 0 n J (m a 1 0 3 H (m a 1 0 3 I (m a 1 n 3 K (m a 1 n 3 I (m a ) 0 n Z' I (m a 1 n 3 K (m a 1 n 3 H (m r ) o o H (m a ) o o 2 K (m r ) O n ' K (m a ) 0 n 2 -1/2 Y Q = M q c o s h { m Q ( z - d 3 ) } s i n h { 2m ( d - d ) } 1 + °-2m ( d - d ) 0 3 "1/2 Y = M c o s h { m ( z - d 0 ) } o o o 3 s i n { 2m ( d - d ) } M = £ 1 1 + n 3 _ 1 2 n [ 2m ( d - d ) n 3 and m and m a r e t h e r o o t s o f t h e f o l l o w i n g equa 33 w - gnt t a n h [ m ( d - d ) ] = 0 4 . 1 . 2 . 3 - 1 2 & o 1 o 3 J w 2 + gm t a n [ m ( d - d ) ] = 0 4 . 1 . 2 . 3 - 1 3 n n 3 4 . 1 . 2 . 4 REGION 4 F o r r e g i o n 4 t h e p o t e n t i a l i s g i v e n b y : $ = <j> V d e " i W t 4 . 1 . 2 . 4 - 1 E E H and must s a t i s f y t h e two b o u n d a r y c o n d i t i o n s : a$ ~ = V a t z = 0 4 . 1 . 2 . 4 - 2 3z H a$ -w 2 $ + g - r - £ = 0 a t z = d 4 . 1 . 2 . 4 - 3 E dz. The v e l o c i t y p o t e n t i a l , 4> , w h i c h d e s c r i b e s t h e o u t e r r e g i o n as E w e l l as t h e waves d i s p e r s e d f r o m t h e c y l i n d e r , i s g i v e n b y : r H ( k r ) =o K ( k r ) $ = V d e " i W t i E ° ° , Z ( z ) - I E ° n , Z ( z ) i r E H 1 o H ( k a ) o ^ n K ( k a ) n J v 1 0 2 n=l 1 n 2 ' 4 . 1 . 2 . 4 - 4 where c o s h ( k z ) Z = - 4 . 1 . 2 . 4 - 5 o 34 s i n h ( 2 k d) -> N o = - f - | 1 + 2k d ° 4 . 1 . 2 . 4 - 6 o J "H 1 c o s h (k z ) Z = 4 . 1 . 2 . 4 - 7 r s i n h ( 2 k d) •, N n = — 1 + 2 k d " 4 . 1 . 2 . 4 - 8 E and E a r e unknown c o e f f i c i e n t s t o be d e t e r m i n e d ; k and k a r e O n O n t h e r o o t s o f : w 2 - gk t a n h (k d) = 0 4 . 1 . 2 . 4 - 9 & o o </ + gk t a n (k d) = 0 4 . 1 . 2 . 4 - 1 0 n n 4 . 1 . 3 SOLVING FOR THE UNKNOWN COEFFICIENTS The unknown c o e f f i c i e n t s o f t h e s e r i e s f o r t h e d e f i n i t i o n o f t h e p o t e n t i a l s A , B , C , D and E a r e now d e t e r m i n e d b y n n n n n m a t c h i n g t h e n o r m a l v e l o c i t i e s and p r e s s u r e s a t e a c h o f t h e b o u n d a r i e s s e p a r a t i n g t h e r e g i o n s . T h i s w i l l c r e a t e f i v e s y s t e m s o f s i m u l t a n e o u s e q u a t i o n s t o s o l v e . D e t a i l s o f t h e s o l u t i o n c a n be f o u n d i n V e n u g o p a l ( 1 9 8 4 ) . 35 4 . 1 . 4 CALCULATION OF HYDRODYNAMIC COEFFICIENTS Once a l l o f t h e p o t e n t i a l s w i t h i n t h e r e g i o n h a v e b e e n d e t e r m i n e d , t h e added mass and damping c o e f f i c i e n t s o f t h e compound c y l i n d e r c a n be f o u n d f r o m t h e f o l l o w i n g e q u a t i o n ; a b i rr + i = ± \\ <f»n dS 4 . 1 . 4 - 1 pV pVto V J J ^ 2 s where S d e n o t e s a s u r f a c e i n t e g r a l t a k e n o v e r t h e s u r f a c e o f t h e i n t e g r a l . U s i n g t h e f a c t t h a t t h e c y l i n d e r i s a x i a l s y m m e t r i c and t h a t t h e p o t e n t i a l i s d e f i n e d as d i f f e r e n t p o t e n t i a l s i n d i f f e r e n t r e g i o n s , t h e e q u a t i o n c a n be r e - w r i t t e n as ; a a a o a a 1 2 4 . 1 . 4 - 2 2 2 7v + i 2 2 pVw S a b u n c u and C a l i s a l (1981) e v a l u a t e d t h e t h r e e i n t e g r a l s i n d e p e n d e n t l y t o o b t a i n t h e v a l u e s o f t h e added mass and damping c o e f f i c i e n t s . Numerous c u r v e s o f added mass and damping c o e f f i c i e n t s v e r s u s i n d u c e d f r e q u e n c y were compu ted u s i n g t h i s method and p u b l i s h e d i n a p a p e r [ C a l i s a l and S a b u n c u 1 9 8 1 ] . F u r t h e r d i s c u s s i o n on t h e f o r m u l a t i o n and t h e n u m e r i c a l c a l c u l a t i o n s c a n be f o u n d i n t h e p u b l i s h e d p a p e r . R e s u l t s u s i n g t h e M a t c h i n g T e c h n i q u e have b e e n t r a n s f e r r e d t o F i g u r e s 1 0 . 8 t o 36 1 0 . 1 5 i n o r d e r f o r c o m p a r i s o n w i t h t h e e x p e r i m e n t a l r e s u l t s p e r f o r m e d b y t h i s r e s e a r c h e r as w e l l as p r e v i o u s p e r f o r m e d e x p e r i m e n t s b y V e n u g o p a l ( 1 9 8 4 ) . 37 4 . 2 MATCHING TECHNIQUE FOR PREDICTION OF HEAVE MOTION INDUCED SIDE FORCES ON A SECOND VERTICAL CYLINDER I n t h i s s t u d y t h e M a t c h i n g T e c h n i q u e i s u s e d t o d e t e r m i n e t h e m a g n i t u d e o f t h e i n d u c e d s i d e f o r c e s on one c y l i n d e r due t o a n e a r b y , i d e n t i c a l c y l i n d e r o s c i l l a t i n g i n t h e h e a v e d i r e c t i o n . I n t h e f o r m u l a t i o n o f t h i s p r o b l e m , two i d e n t i c a l v e r t i c a l c y l i n d e r s a r e o s c i l l a t i n g i n heave m o t i o n . A n i l l u s t r a t i o n o f t h e b a s i c s y s t e m as w e l l as t h e c y l i n d e r g e o m e t r i e s a r e shown i n F i g u r e s 8 . 3 and 8 . 4 . The f i r s t c y l i n d e r i s l o c a t e d a t t h e o r i g i n and t h e s e c o n d c y l i n d e r i s l o c a t e d a t a p r e s c r i b e d d i s t a n c e , B f r o m t h e o r i g i n . T h e r e a r e two r e g i o n s a s s o c i a t e d w i t h t h i s p r o b l e m , an e x t e r n a l r e g i o n and an i n t e r n a l r e g i o n , as shown on F i g u r e 8 . 3 . The two r e g i o n s f o r e a c h o f t h e c y l i n d e r s a r e d e f i n e d a s ; R e g i o n P o t e n t i a l E x t e r i o r a < r , 0 < z < d $ E I n t e r i o r 0 < r < a , 0 < z < d $ l i The b o u n d a r y c o n d i t i o n s f o r t h e p r o b l e m a r e ; i ) The i m p e r m e a b l e b o t t o m c o n d i t i o n , ^ = 0 a t z = 0 3z i i ) The i m p e r m e a b l e body c o n d i t i o n , on t h e b o t t o m o f t h e c y l i n d e r , 38 a V e dz - i W t a t z= d ; 1 0 < r < a and t h e s i d e o f t h e c y l i n d e r a t r = a ; d < z < d i i i ) The l i n e a r i z e d f r e e s u r f a c e c o n d i t i o n , 3J> dz 2 a t z = d 4 . 2 . 1 DEFIN IT ION OF THE POTENTIALS I n t h i s f o r m u l a t i o n , a l l v a r i a b l e s r e l a t e d t o t h e c y l i n d e r l o c a t e d a t t h e o r i g i n w i l l be d e f i n e d u s i n g t h e s u b s c r i p t , 1 w h i l e a l l v a r i a b l e s r e l a t e d t o t h e s e c o n d c y l i n d e r w i l l be d e f i n e d u s i n g t h e s u b s c r i p t , 2 . The v e l o c i t y p o t e n t i a l , $ must s a t i s f y t h e L a p l a c e e q u a t i o n as w e l l as t h e l i s t e d b o u n d a r y c o n d i t i o n s . T h e r e a r e two p o t e n t i a l s a s s o c i a t e d w i t h t h e e x t e r i o r r e g i o n o f e a c h o f t h e c y l i n d e r s . The r a d i a t i o n p o t e n t i a l , d e f i n e d as $ , f o r i = 1, 2 , and a d i f f r a c t i o n p o t e n t i a l w h i c h i s a s s o c i a t e d w i t h t h e waves g e n e r a t e d b y one o f t h e c y l i n d e r s due t o i n c o m i n g waves o f t h e s e c o n d c y l i n d e r , d e f i n e d as f o r i = 1, 2 . The i n t e r i o r r e g i o n i s d e f i n e d f o r b o t h c y l i n d e r s as $ . S i n c e t h e i n t e r i o r r e g i o n does n o t e x t e n d t o t h e s u r f a c e , i t o n l y c o n t r i b u t e s i n d i r e c t l y t o t h e f o r m a t i o n o f w a v e s . Two m a j o r s e t s o f t e rms a r e a s s o c i a t e d w i t h t h e i n t e r i o r r e g i o n , t h e f i r s t one c o r r e s p o n d s t o t h e p o t e n t i a l a s s o c i a t e d w i t h t h e t h e h e a v e m o t i o n o f t h e c y l i n d e r , and t h e s e c o n d o n e , r e p r e s e n t s a d d i t i o n a l t e rms n e e d e d ERi 3 9 t o b a l a n c e t h e e f f e c t s o f v e l o c i t y and p r e s s u r e i n d u c e d b y t h e e x t e r n a l p o t e n t i a l o f t h e s e c o n d c y l i n d e r . The f o r m u l a t i o n i s e x p r e s s e d i n t h e same manner as p r o p o s e d b y S a b u n c u and C a l i s a l ( 1 9 8 6 ) . The f o l l o w i n g n o t a t i o n i s u s e d i n t h e f o r m u l a t i o n (See A p p e n d i x F ) : I ( x ) M o d i f i e d B e s s e l f u n c t i o n o f t h e f i r s t k i n d o f k o r d e r k. J ( x ) B e s s e l f u n c t i o n o f t h e f i r s t k i n d o f o r d e r k . k K ( x ) M o d i f i e d B e s s e l f u n c t i o n o f t h e s e c o n d k i n d o f k o r d e r k. H ( x ) H a n k e l f u n c t i o n o f t h e f i r s t k i n d o f o r d e r k. k Y ( x ) B e s s e l f u n c t i o n o f t h e s e c o n d k i n d o f o r d e r k. k k Q and k a r e s o l u t i o n s o f t h e d i s p e r s i o n r e l a t i o n s h i p , w 2 - gk t a n h (k d) = 0 ° o o w 2 - gk t a n h (k d) = 0 n n Z ( z ) and Z ( z ) a r e o r t h o n o r m a l f u n c t i o n s f o r 0 < z < d 0 n d e f i n e d a s , Z ( z ) = c o s h (k z ) 0 0 / I T 0 Z ( z ) = c o s (k z ) / I F n where N and N a r e . 0 n , r s i n h (2k d) -, L 0 J 40 i and n r s i n (2k d) n n J L and L a r e d e f i n e d a s , nO nq L = nO 2 ( -1 ) (k d ) s i n h (k d ) 0 1 0 1 [ ( k d ) + (n»r) ] / N o l J o L = nq 2 ( -1 ) (k d ) s i n (k d ) q _ i q i K k d ) 2 + ( n 7 r ) 2 ] v n r q 1 n 4 . 2 . 1 . 1 EXTERIOR REGION A c c o r d i n g t o Sabuncu and C a l i s a l (1986) t h e r a d i a t i o n p o t e n t i a l s f o r t h e two c y l i n d e r s c a n be e x p r e s s e d a s ; i ) f o r t h e f i r s t c y l i n d e r : CO $ ( r ,8 , z , t ) = V d [ E H (k r ) Z + Y E K (k r ) Z ] e E R I 1 1 0 0 0 0 U q 0 q q J q = l - i W t 4 . 2 . 1 . 1 - 1 i i ) f o r image c y l i n d e r 1, CO $ ( r ,8 , z , t ) = V d [ E H (k r ) Z + Y E K ( k r ) Z ] e " i W t E R 2 2 2 ' ' ' 0 0 0 0 U q 0 q q J q = l 4 . 2 . 1 . 1 - 2 Where V i s t h e maximum h e a v e v e l o c i t y and E and E a r e c o m p l e x o Q c o n s t a n t s . S i n c e t h e s y s t e m i s s y m m e t r i c , $ c a n be e x p r e s s e d i n te rms E R 2 o f r and 6 . U s i n g G r a f ' s summat ion t h e o r e m f o r B e s s e l f u n c t i o n s 1 1 b 41 w h i c h d e f i n e s t h e f o l l o w i n g i d e n t i t y ; CO e 1 ^ Zv (mr 2 ) = £ Z , + £ ( m b ) J ^ m r ^ £ = - c o 4 . 2 . 1 . 1 - 3 f o r an a r b i t r a r y c o m p l e x number m, an i n t e g e r v, and a B e s s e l f u n c t i o n Z , and u s i n g g e o m e t r i c r e l a t i o n s h i p s w h i c h d e f i n e t h e f o l l o w i n g ; a = 7; - 6 ,8=6 - ^ 5 , and m=kq. 2 1 ^ 2 2 n One c a n o b t a i n t h e f o l l o w i n g e q u a t i o n s f o r t h e B e s s e l f u n c t i o n s H and K : im6 0 0 iie e 2 H (k r ) = I ( i ) k " m H , (k b ) J . ( k r ) e 1 m 0 2 i) m + C 0 I 0 1 •t = - o o 4 . 2 . 1 . 1 - 4 im0 «> i i 0 e 2 ( k q r z ) = £ ( i ) k " m ( k £ b ) J £ ( k e 1 4 . 2 . 1 . 1 - 5 So now t h e e x t e r i o r r a d i a t i o n p o t e n t i a l f o r t h e s e c o n d c y l i n d e r c a n be e x p r e s s e d i n te rms o f r and 8^ a s ; CO . $ - V d e " i W t [ I ( i ) [E H, ( k b ) J , ( k r ) Z E R 2 ' - „ ' - ' L 0 £ N 0 £ 0 1 0 + lE K. ( k b ) I - (k r ) Z.] e 1 £ = 1 q q q 4 . 2 . 1 . 1 - 6 H e n c e , t h e e x t e r i o r p o t e n t i a l a r o u n d t h e f i r s t c y l i n d e r c a n be 42 e x p r e s s e d a s ; $ ( r J , z , t ) = <S> ( r ,6 , z , t ) + $ ( r ,0 , z , t ) + $ E l l ERI 1 1 E R 2 1 1 Dl 4 . 2 . 1 . 1 - 7 where $ i s t h e d i f f r a c t i o n p o t e n t i a l . Upon i n s p e c t i o n o f t h e f o r m o f $ , one c a n c o n c l u d e t h a t t h e f o r m o f $ s h o u l d b e ; E R 2 Dl 4 . 2 . 1 . 1 - 8 The c o l l e c t i o n o f l i k e t e rms c a n now be c a r r i e d o u t f o r t h e e x t e r i o r p o t e n t i a l , * . Due t o symmetry i n t h e geome t r y o f the* p r o b l e m , t h e c o e f f i c i e n t s o f t h e t e rms c o n t a i n i n g c o s ( 0 ) must v a n i s h . S i m i l a r l y , t h e c o e f f i c i e n t s o f t h e t e r m s c o n t a i n i n g s i n ( £ 0 ) must v a n i s h wheneve r I i s an e v e n i n t e g e r . T h e s e c o n d i t i o n s r e s u l t i n t h e f o l l o w i n g r e l a t i o n s f o r D „ : D . = ( -1 ) D „ and D „ = D'„ 0 ( - t ) 01 q ( - £ ) q£ U s i n g t h e s e r e s u l t s t h e e x t e r i o r p o t e n t i a l c a n now be e x p r e s s e d a s ; $ ( r , 0 , z , t ) - V d e " i W t U ( r , z ) + Y li> cos (2£0 ) + rp s i n ( 2 £ 0 + 0 ) }1 E 0 « even odd J £ = 1 4 . 2 . 1 . 1 - 9 43 w h e r e ; 1=1 4 . 2 . 1 . 1 - 1 0 V> ( r , z ) - ( - 1 / 2 [ r E H (k b ) J (k r ) + D H , ( k r ) 1 Z even ^ L 0 p 0 P 0 Op p £ v 0 J 0 + Y f E K (k b ) I (k r ) + D K (k r ) 1 Z L q p q P q q p p q i j q = l f o r I = 0 , 1, 2, . . . and p = 2£ 4 . 2 . 1 . 1 - 1 1 a n d ; V> ( r , z ) = ( - 1 / 2 I [E H (k b ) J (k r ) + D H (k r ) 1 Z odd ^ L 0 p 0 P 0 Op p 0 J 0 + V r E K (k b ) I (k r ) + D K (k r ) I Z i - q p q p q q p p q J q I q = l f o r I = 0 , 1, 2 , . . . and p = 21 + 1 4 . 2 . 1 . 1 - 1 2 E and D a r e t h e unknown c o e f f i c i e n t s w h i c h must be c o m p u t e d , q qp 4 . 2 . 1 . 2 INTERIOR REGION The i n t e r i o r p o t e n t i a l , $^ c a n a l s o be e x p r e s s e d i n a f o r m v e r y s i m i l a r t o t h a t o f t h e p o t e n t i a l w i t h i n t h e e x t e r i o r r e g i o n . The r a d i a t i o n p o t e n t i a l c o n s i s t s o f two p a r t s ; a homogeneous p a r t s a t i s f y i n g t h e n u l l b o u n d a r y c o n d i t i o n s , and a p a r t i c u l a r p a r t s a t i s f y i n g t h e h e a v e v e l o c i t y c o n d i t i o n s a t t h e b o t t o m o f t h e c y l i n d e r , z = d . C a l i s a l and Sabuncu (1986) d e r i v e a f o r m o f t h e p o t e n t i a l i n t h e i n t e r i o r r e g i o n a s : $ i = V d e " i W t [ ^ ^ r , z ) + I { ^ e v e n i c o s ( 2 ^ ) + ^^^(219+9)}] -iWt 1 ',= 1 4 . 2 . 1 . 2 - 1 44 w h e r e , + I A. c o s [ W l "I 1 , 2 r \ 4 . 2 . 1 . 2 - 2 [ B f -v p «> f o r £ = 1, 2 , . . . and p =2£ c o s 4 . 2 . 1 . 2 - 3 and w h e r e , £?rr ] c o s f o r £ = 1 , 2 , . . . and p = 2£ + 1 4 . 2 . 1 . 2 - 4 A and B a r e t h e unknown c o e f f i c i e n t s w h i c h must be c o m p u t e d . n n 4 . 2 . 2 MATCHING CONDITIONS B a l a n c i n g t h e p r e s s u r e and t h e r a d i a l v e l o c i t i e s a t t h e b o u n d a r y b e t w e e n t h e i n t e r i o r and e x t e r i o r r e g i o n s w i l l y i e l d t h e s o l u t i o n s t o t h e unknown c o e f f i c i e n t s . F o r t h e c a s e o f e q u a l p r e s s u r e a t t h e b o u n d a r y one o b t a i n s t h e f o l l o w i n g : 45 d l d l f <E> (a ,z) cos dz = f $ (a ,z) cos dz J Interior I d I J Exterior I d ] 4 . 2 . 2 - 1 for n = 0 , 1 , . . . «> a t boundary r = a and 0 < z < d . Where $ i s t h e a p p r o p r i a t e d i f f r a c t i o n o r r a d i a t i o n p o t e n t i a l f o r t h e r e g i o n . E a c h o f t h e te rms r e p r e s e n t i n g t h e d i f f r a c t e d p o t e n t i a l s and t h e te rms r e p r e s e n t i n g t h e r a d i a t e d p o t e n t i a l s must be b a l a n c e d s e p a r a t e l y . S i m i l a r l y f o r t h e c a s e o f c o n t i n u i t y o f t h e r a d i a l v e l o c i t y , 3$ a t t h e b o u n d a r y s e p a r a t i n g t h e i n t e r i o r and e x t e r i o r r e g i o n s one o b t a i n s ; d 1 a * ( a , z ) I Inter ior , -x— Z dz J 3 r n 0 f o r n = 0 , 1 , . . . °o a t b o u n d a r y r = a and 0 < z < d . - I d 3$ ( a , z ) Exterior _ , 7T— Z dz o r n 4 . 2 . 2 - 2 A t t h e s i d e o f t h e c y l i n d e r t h e i m p e r m e a b l e b o u n d a r y c o n d i t i o n must a l s o be s a t i s f i e d ; 3$ ( a , z ) -g^- = 0 a t b o u n d a r y d i < z < d and r = a . 4 . 2 . 2 - 3 The s o l u t i o n o f t h e unknown c o e f f i c i e n t s c a n now be s o l v e d b y p e r f o r m i n g t h e a p p r o p r i a t e i n t e g r a t i o n s and a p p l y i n g t h e m a t c h i n g c o n d i t i o n s w h i c h w i l l g e n e r a t e a s e r i e s o f l i n e a r e q u a t i o n s t o be 46 s o l v e d . A l s o , t h e odd and t he e v e n components o f t h e p o t e n t i a l s i n t h e i n t e r i o r and e x t e r i o r r e g i o n s a r e computed a n d m a t c h e d s e p a r a t e l y so t h a t one o b t a i n s p a i r s o f e q u a t i o n s t o s a t i s f y t h e m a t c h i n g c o n d i t i o n s o f e q u a l p r e s s u r e and r a d i a l v e l o c i t y a t t h e b o u n d a r y . T h a t i s ; rj> ( a , z ) = V ( a , z ) 4 . 2 . 2 - 4 T o 10 a ^ Q ( a , z ) d t f> i o (a,z) a r~ = 3 r~ 4 . 2 . 2 - 5 i> ( a , z ) = i> ( a , z ) 4 . 2 . 2 - 6 e v e n I e v e n di> ( a , z ) 8rl)T ( a , z ) e v en I e v en 3 r 3 r 4 . 2 . 2 - 7 V- „ , ( a , z ) = „„(a ,z) 4 . 2 . 2 - 8 odd l o a d 9V ^ ( a , z ) 3V»T ^ ( a . z ) o dd Io dd a r ~ ~ a r - 4 . 2 . 2 - 9 E a c h component o f t h e e x t e r n a l p o t e n t i a l s must h a v e no r a d i a l v e l o c i t y a t t h e b o u n d a r y b e t w e e n t h e c y l i n d e r and t h e f l u i d , t h a t i s a t d < z < d and r = a . l U s i n g t h e f i r s t e q u a l i t y , 4 . 2 . 2 - 4 , t h e e q u a l i t y o f p r e s s u r e w h i c h i n t h i s c a s e t r a n s l a t e s t o a e q u a l i t y i n p o t e n t i a l v a l u e s , a l i n e a r s e t o f e q u a t i o n s o f unknown c o e f f i c i e n t s A^ and E c a n be o b t a i n e d o f t h e f o r m : =0 , d 2 \ A = E H (k a) L + Y E„ K ( k „ a ) L „ - \ — - \ o o o oo ^ I I ol \ 3 d 2 ( J d / l 4 . 2 . 2 - 1 0 and 47 A - E H (k a) L + I E- K ( k „ a ) L . - 2 (— 1 n 0 0 0 nO £ 0 £ n£ 1 , , . 2 £= i v d (n7r) ' 4 . 2 . 2 - 1 1 w i t h t h e f u n c t i o n L „ d e f i n e d e a r l i e r . n£ The s e c o n d e q u a l i t y , 4 . 2 . 2 - 5 , w h i c h i s t h e e q u a l i t y o f r a d i a l v e l o c i t i e s g i v e s t h e f o l l o w i n g two e q u a t i o n s : x- r -%=-i co o ^  d J E H (k a) L + * Y A„ £ 1 4k d o £= i _ o I 0 4 . 2 . 2 - 1 2 The o t h e r s e t s o f e q u a t i o n s c o r r e s p o n d i n g t o t h e b a l a n c e o f p o t e n t i a l s and r a d i a l v e l o c i t i e s a r e as f o l l o w s : F o r 4 . 2 . 2 - 6 : B = [ E H (k b ) J (k a ) + D H (k a) ] L + nO 0 0 0 0 0 00 0 0 nO co I [B£ K o ( k £ b ) I o(k £a) + D f o K ( k £ a ) ] J. f o r m=0 t = i 4 . 2 . 2 - 1 4 B = ( - l ) m 2 { [ E H (k b ) J (k a ) + D H (k a) ] L + np 0 p 0 p 0 Op p 0 nO CO I [E, K (k.b) I (k,a)+ D. K (k-a)] L , ) f o r 2m=p ^ C p c p c c p p c nc 4 . 2 . 2 - 1 5 48 f o r 4 . 2 . 2 - 7 : f o r m = 0 CO E H ( k b ) j ' (k a ) + D H ' (k a ) = „ . * , Y B n o o v o y o v o ' oo o v o ' 2 k d Z j n O 0 n= 1 nO 4 . 2 . 2 - 1 6 and E K (k b ) I ( k a) + D K (k a) 0 1 * Y B n - L ~ « v " ' « N _ ' q 0 0 q 2k d ^ nO q 0 0 0 q q n= 1 n?ra ] l ^ T J nq 4 . 2 . 2 - 1 7 and f o r p = 2m, m = 1 , 2 , md ( -1 ) 2 [ E H (k b ) J (k a) + D H (k a) ] = B * — L L q P o p o op p o 1 Op 2k d a oo o £ = i n?ra ^ nO 4 . 2 . 2 - 1 8 and md ( -1) ' " 2 [ E K ( k b ) I (k a) + D K (k a ) ] = B * L q p q P <I I P q Op Z K o a Oq 2k d £ B £ P 1 q £ = 1 una ( una ^ nq 4 . 2 . 2 - 1 9 The e q u a t i o n s c o r r e s p o n d i n g t o 4 . 2 . 2 - 8 and 4 . 2 . 2 - 9 a r e i d e n t i c a l t o t h e l a s t two e q u a t i o n s e x c e p t t h a t p = 2m + 1 i n s t e a d o f p = 2m. These e q u a t i o n s a r e e x p r e s s e d as 4 . 2 . 2 - 2 0 and 4 . 2 . 2 - 2 0 . C a l c u l a t i o n s o f t h e p o t e n t i a l s c a n now be c o n d u c t e d b y s o l v i n g 49 t h i s s e t l i n e a r e q u a t i o n s . 4 . 2 . 3 CALCULATION OF HEAVE INDUCED SIDE FORCE Once a l l t h e c o e f f i c i e n t s h a v e b e e n c a l c u l a t e d t h e i n d u c e d s i d e f o r c e c a n now be d e t e r m i n e d b y u s i n g a l i n e a r i z e d e x p r e s s i o n f o r t h e p r e s s u r e i n t e g r a t e d o v e r t h e s u r f a c e o f t h e c y l i n d e r . F o r known p o t e n t i a l s , $ , t h e f o r c e , F a c t i n g o v e r t h e s u r f a c e S w i t h a n o r m a l v e c t o r , n c a n be e x p r e s s e d a s : F = - p | ^ J $ n ds 4 . 2 . 3 - 1 I n t h e c a s e o f d e t e r m i n i n g t h e f o r c e , F , a l o n g t h e l i n e c o n n e c t i n g t h e c e n t e r s o f t h e c y l i n d e r s , t h i s e x p r e s s i o n c a n be d e r i v e d a s : 27T d F j - - P ft J J $ ( a , z , 0 , t ) n • j a d$ ds 0 d 1 4 . 2 . 3 - 2 S a b u n c u and C a l i s a l (1986) d e r i v e d t h e f o l l o w i n g e x p r e s s i o n f r o m t h i s t h e o r y i n o r d e r t o p r e d i c t t h e i n d u c e d s u r g e f o r c e : F = p ( i w ) V d e " i W t Y 4 . 2 . 3 - 3 j where V i s t h e vo lume o f t h e c y l i n d e r a n d , 50 Y = 2TT ad\ \E H (k b ) J (k a) + D H (k a)l j [_ o 1 o 1 o 0 1 l o J s i n h ( k a) - s i n h ( k d ) o 0 1 k d V N o o a> r -, s in(k^a) - sinCk^d^) k^d 4 . 2 . 3 - 4 The maximum v a l u e o f F d e t e r m i n e d f r o m t h e modu lus o f Y , j | Y | , i s o f i m p o r t a n c e i n t h i s s t u d y and i s n o n - d i m e n s i o n a l i s e d w i t h r e s p e c t t o t h e w e i g h t o f t h e c y l i n d e r a s : | F I d w V I Y I j m a x - — 4 . 2 . 3 - 5 V p g T r a 2 ( d - d i ) g The M a t c h i n g T e c h n i q u e p r o g r a m w i l l s o l v e t h e l i n e a r s e t o f e q u a t i o n s 4 . 2 . 2 - 1 0 t o 4 . 2 . 2 - 2 1 t o o b t a i n t h e i n t e r i o r and e x t e r i o r p o t e n t i a l s f o r a g i v e n i n p u t o f f r e q u e n c y . Then e q u a t i o n 4 . 2 . 3 - 5 i s u s e d t o d e t e r m i n e t h e i n d u c e d s i d e f o r c e s . D o u b l e p r e c i s i o n c o m p l e x numbers a r e u s e d and a l l t h e s e r i e s a r e e x p r e s s e d b y t w e n t y t e r m s . Sabuncu and C a l i s a l f o u n d t h a t t w e n t y t e rms t o r e p r e s e n t t h e s e r i e s w i l l e n s u r e a n u m e r i c a l c o n v e r g e n c e o f t h e s e r i e s . I n t h e f o r m u l a t i o n o n l y t h e f i r s t d i f f r a c t i o n p o t e n t i a l o f an i n f i n i t e s e t o f s u c h p o t e n t i a l s i s c o n s i d e r e d . T h i s may e f f e c t t h e s o l u t i o n f o r v e r y s m a l l c y l i n d e r s e p a r a t i o n s . F u r t h e r d i s c u s s i o n on t h e t h i s f o r m u l a t i o n c a n be f o u n d i n a s o o n t o p u b l i s h e d p a p e r on t h i s s u b j e c t b y Sabuncu and C a l i s a l . F o r c o m p a r i s o n w i t h e x p e r i m e n t a l d a t a c o l l e c t e d b y t h e a u t h o r t h e p r o g r a m was r u n w i t h t h e v a r i o u s i n p u t v a r i a b l e s s e t t o ma tch t h e e x p e r i m e n t a l c o n d i t i o n s i n t h e t o w i n g t a n k . C u r v e s were 51 t a b u l a t e d u s i n g t h i s t e c h n i q u e f o r b o t h a deep w a t e r c a s e and a s h a l l o w w a t e r c a s e i n o r d e r f o r a c o m p a r i s o n w i t h e x p e r i m e n t a l r e s u l t s . These r e s u l t s c a n be f o u n d on F i g u r e s 1 0 . 1 6 t o 1 0 . 2 5 . 52 4 . 3 MATCHING TECHNIQUE FOR PREDICTION OF HEAVE MOTION INDUCED HEAVE HYDRODYNAMIC COEFFICIENTS ON A SECOND VERTICAL CYLINDER The p r e d i c t i o n o f h e a v e i n d u c e d h e a v e 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 s v i r t u a l l y i d e n t i c a l t o t h e p r e d i c t i o n h e a v e i n d u c e d s i d e f o r c e s d e s c r i b e d i n S e c t i o n 4 . 2 . The same g e o m e t r i c c o n f i g u r a t i o n , as w e l l as t h e same g o v e r n i n g e q u a t i o n and b o u n d a r y c o n d i t i o n s , a r e u t i l i z e d t o d e v e l o p a scheme t o d e s c r i b e t h e i n t e r n a l and e x t e r n a l v e l o c i t y p o t e n t i a l s . These p o t e n t i a l s a r e t h e n u t i l i z e d a t t h e b o u n d a r y b e t w e e n t h e two r e g i o n s f o r t h e m a t c h i n g o f t h e r a d i a l v e l o c i t i e s and p r e s s u r e s c o r r e s p o n d i n g t o e q u a t i o n s 4 . 2 . 2 - 4 t o 4 . 2 . 2 - 9 . The c o r r e s p o n d i n g e q u a t i o n s d e s c r i b i n g t h e s e m a t c h i n g c o n d i t i o n s a r e f o u n d i n e q u a t i o n s 4 . 2 . 2 - 1 0 t o 4 . 2 . 2 - 2 1 . Once a l l t h e s e r i e s c o e f f i c i e n t s a r e d e t e r m i n e d , t h e n b o t h t h e i n t e r n a l and e x t e r n a l v e l o c i t y p o t e n t i a l s c a n be d e s c r i b e d . 4 . 3 . 1 CALCULATION OF HEAVE INDUCED HYDRODYNAMIC COEFFICIENTS Once t h e v e l o c i t y p o t e n t i a l i s d e f i n e d , t h e i n d u c e d h e a v e h y d r o d y n a m i c c o e f f i c i e n t s c a n now be d e t e r m i n e d b y , once a g a i n , u s i n g a l i n e a r i z e d e x p r e s s i o n f o r t h e p r e s s u r e i n t e g r a t e d o v e r t h e b o t t o m s u r f a c e o f t h e c y l i n d e r . The f o r c e , F , i s w r i t t e n i n t h e same manner as i n t h e d e t e r m i n a t i o n o f t h e i n d u c e d s i d e f o r c e s . 53 F = a t J $ n ds 4 . 2 . 3 - 1 I n t h i s c a s e , t h e d i r e c t i o n o f i n t e r e s t i s t h e v e r t i c a l d i r e c t i o n w h i l e f o r t h e i n d u c e d s i d e f o r c e s i t was t h e h o r i z o n t a l d i r e c t i o n . M u l t i p l y i n g b o t h s i d e s o f 4 . 2 . 3 - 1 b y a u n i t v e c t o r , k, i n t h e z d i r e c t i o n y i e l d s t h e f o l l o w i n g : F z = -p ( - i w ) e " i W t J $ n • k ds 4 . 3 . 1 - 1 I n t h i s c a s e , t h e i n t e g r a t i o n t a k e s p l a c e o v e r t h e b o t t o m d i s k a r e a o f t h e c y l i n d e r d e f i n e d b y z = d i and 0 < r < a . The c o r r e s p o n d i n g p o t e n t i a l i n t h i s r e g i o n i s t h e i n n e r p o t e n t i a l , and i n t e g r a t i o n o v e r t h e b o t t o m d i s k a r e a f o r t h e f o r c e i n t h e z d i r e c t i o n y i e l d s t h e f o l l o w i n g r e l a t i o n s h i p : 2 F = -p ( - i w ) e " V d 2TT 4- T 4 . 3 . 1 - 2 z 4 w h e r e , d a 2 -i 4dd l 4d » + (A + B ) + Y (A. + B„ ) o oo ? r a ^ I to I 0 4 . 3 . 1 - 3 S e p a r a t i n g t h e t e rms i n p h a s e w i t h t h e a c c e l e r a t i o n and v e l o c i t y w i l l y i e l d t h e heave added mass c o e f f i c i e n t , a , and t h e damping c o e f f i c i e n t , b . These a r e f u r t h e r n o n - d i m e n s i o n a l i s e d t o o b t a i n t h e f o l l o w i n g r e s u l t s : 54 a b T 1 1 + i 1 1 4 . 3 . 1 - 4 pV pVw 7" d 2 1 -l d where V i s t h e vo lume o f t h e c y l i n d e r . A s i n t h e p r e v i o u s s e c t i o n , d o u b l e p r e c i s i o n c o m p l e x numbers a r e u s e d and a l l t h e s e r i e s a r e r e p r e s e n t e d b y t w e n t y t e rms i n o r d e r t o c a l c u l a t e t h e h e a v e i n d u c e 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 . A s i n t h e p r e v i o u s f o r m u l a t i o n , o n l y one d i f f r a c t i o n p o t e n t i a l o f a n i n f i n i t e number o f s u c h p o t e n t i a l s i s u s e d i n t h e f o r m u l a t i o n , t h i s may c a u s e e r r o r i f t h e c y l i n d e r s e p a r a t i o n i s v e r y s m a l l . Once a g a i n , t h e M a t c h i n g T e c h n i q u e p r o g r a m was r u n w i t h t h e v a r i o u s i n p u t p a r a m e t e r s s e t t o mode l t h e e x p e r i m e n t a l c o n d i t i o n s i n t h e t o w i n g t a n k . Added mass and damping c o e f f i c i e n t c u r v e s were t a b u l a t e d f o r b o t h a deep w a t e r e n v i r o n m e n t and a s h a l l o w w a t e r e n v i r o n m e n t and compared w i t h e x p e r i m e n t a l r e s u l t s . The deep w a t e r r e s u l t s c a n be f o u n d on F i g u r e s 1 0 . 2 6 t o 1 0 . 3 6 , w h i l e t h e s h a l l o w w a t e r r e s u l t s c a n be f o u n d on F i g u r e s 1 0 . 3 6 t o 1 0 . 4 5 . 55 5. PRESENTATION AND ANALYSIS OF RESULTS A l l o f t h e e x p e r i m e n t a l r e s u l t s a r e g r a p h i c a l l y i l l u s t r a t e d i n A p p e n d i x E . T h r e e s e t s o f e x p e r i m e n t a l r e s u l t s a r e d i v i d e d i n t o s i x c a t e g o r i e s as f o l l o w s : a ) samp le d a t a p l o t s , b ) h e a v e added mass c o e f f i c i e n t s o f compound c y l i n d e r h e a v e e x p e r i m e n t s , c ) h e a v e damp ing c o e f f i c i e n t s o f compound c y l i n d e r h e a v e e x p e r i m e n t s , d) i n d u c e d s i d e f o r c e s on a s i n g l e c y l i n d e r due t o an a d j a c e n t , i d e n t i c a l c y l i n d e r i n h e a v e m o t i o n , e ) i n d u c e d h e a v e added mass c o e f f i c i e n t s o f a c y l i n d e r due t o an a d j a c e n t , i d e n t i c a l c y l i n d e r i n h e a v e m o t i o n , f ) i n d u c e d heave damping c o e f f i c i e n t s o f a c y l i n d e r due t o a n a d j a c e n t , i d e n t i c a l c y l i n d e r i n h e a v e m o t i o n . I n a l l f i v e c a s e s , t h e e x p e r i m e n t a l v a l u e s a r e compared w i t h t h e r e s u l t s o f t h e M a t c h i n g T e c h n i q u e t h e o r y . E x p e r i m e n t a l d a t a w h i c h was o b t a i n e d f r o m t e s t s w i t h known e q u i p m e n t f a i l u r e s o r p r o c e d u r a l e r r o r s was n o t i n c l u d e d i n t h e a n a l y s i s . F o r e x a m p l e , a number o f t e s t s were b e i n g c o n d u c t e d t o d e t e r m i n e t h e h y d r o d y n a m i c c o e f f i c i e n t s o f a compound c y l i n d e r i n h e a v e m o t i o n when i t was d i s c o v e r e d t h a t t h e d i s p l a c e m e n t y o - y o t r a n s d u c e r was n o t f u n c t i o n i n g p r o p e r l y , t h e s e t e s t r e s u l t s a r e 56 n o t i n c l u d e d i n any o f t h e f i g u r e s i n A p p e n d i x E . A l l o f t h e r e s u l t s a r e p r e s e n t e d i n a n o n - d i m e n s i o n a l f o r m a t so t h a t t h e y may be u n i v e r s a l l y a p p l i e d t o s h a p e s o f e q u a l p r o p o r t i o n s . I n many a n a l y s i s w h i c h r e q u i r e t h e f r e q u e n c y t o be n o n - d i m e n s i o n a l i s e d , t h e f r e q u e n c y i s l e f t i n a l i n e a r o r d e r and n o t i n a s q u a r e d o r d e r . Howeve r , i n t h i s g e n r e o f i n v e s t i g a t i o n t h e a c c e p t e d f o r m o f t h e n o n - d i m e n s i o n a l i s e d f r e q u e n c y i s w i t h r e s p e c t t o t h e f r e q u e n c y s q u a r e d . R e a d e r s s h o u l d keep t h i s i n m i n d when i n s p e c t i n g t h e g r a p h s i n A p p e n d i x E . The f r e q u e n c y , w, i s a l s o n o n - d i m e n s i o n a l i s e d w i t h r e s p e c t t o t h e maximum r a d i u s , a , and t h e g r a v i t a t i o n a l c o n s t a n t , g , a s : 2 w a w = . (nd ) g The added mass c o e f f i c i e n t , a , i s n o n - d i m e n s i o n a l i s e d w i t h 2 2 r e s p e c t t o t h e d i s p l a c e d v o l u m e , V , and t h e d e n s i t y o f t h e f l u i d medium, p, a s : a 2 2 a — . 22(nd) pV The damp ing c o e f f i c i e n t , b , i s a l s o n o n - d i m e n s i o n a l i s e d w i t h r e s p e c t t o t h e d i s p l a c e d v o l u m e , V , and t h e d e n s i t y , p, as w e l l as w i t h t h e f r e q u e n c y o f o s c i l l a t i o n , w, a s : b b = 2 2 22 (nd) pVw 57 The n u m e r i c a l l y and e x p e r i m e n t a l l y d e t e r m i n e d i n d u c e d s i d e f o r c e s due t o a n e a r b y , i d e n t i c a l c y l i n d e r o s c i l l a t i n g i n s i n u s o i d a l h e a v e m o t i o n i s a l s o n o n - d i m e n s i o n a l i s e d i n o r d e r t o be u n i v e r s a l l y a p p l i e d . The maximum s i d e f o r c e i s n o n - d i m e n s i o n a l i s e d w i t h r e s p e c t t o t h e d i s p l a c e m e n t o f t h e c y l i n d e r V , t h e d e n s i t y o f t h e f l u i d p, and t h e g r a v i t a t i o n a l c o n s t a n t , g , a s , F F . Knd) pVg 5 . 1 SAMPLE DATA PLOTS F i g u r e s 1 0 . 1 t h r o u g h 1 0 . 7 show samp le p l o t s f r o m a t y p i c a l t e s t t o d e t e r m i n e t h e h y d r o d y n a m i c c o e f f i c i e n t s o f a s i n g l e c y l i n d e r . The t e s t SB7D15 c o n s i s t e d o f a s i n g l e c y l i n d e r i n s i m p l e s i n u s o i d a l h e a v e m o t i o n . The c y l i n d e r h a d a mass o f 26 k i l o g r a m s , a d r a f t o f 0 . 1 8 m e t r e , and was o s c i l l a t i n g w i t h an a m p l i t u d e o f 3 cm. The t e s t r e s u l t s i s t y p i c a l o f any o f t h e e x p e r i m e n t s c o n d u c t e d i n t h i s t h e s i s . A n u n f i l t e r e d t r a c e o f t h e raw y o - y o p o s i t i o n t r a n s d u c e r s i g n a l i s shown i n F i g u r e 1 0 . 1 . S a m p l i n g was done a t a r a t e o f once e v e r y 15 m i l l i s e c o n d s and t h e e n t i r e s a m p l i n g d u r a t i o n was a p p r o x i m a t e l y 12 s e c o n d s ( o n l y t h e f i r s t 3 . 2 5 s e c o n d s a r e shown i n t h e F i g u r e ) . F i g u r e 1 0 . 2 shows t h e same s i g n a l a f t e r i t h a s gone t h r o u g h t h e DC o f f s e t r e m o v a l p e r f o r m e d b y t h e s u b r o u t i n e TREND and t h e f i l t e r i n g p r o c e d u r e s p e r f o r m e d b y t h e s u b r o u t i n e F I L T E R . 58 I t i s o b v i o u s t h a t t h e y o - y o p o s i t i o n t r a n s d u c e r p r o d u c e s a smooth w e l l d e f i n e d s i g n a l f r o m w h i c h t h e o p e r a t i n g f r e q u e n c y c a n e a s i l y be d e t e r m i n e d . The f i l t e r i n g s u b r o u t i n e h a s t h e e f f e c t o f s l i g h t l y l o w e r i n g t h e a m p l i t u d e , as w e l l as a l t e r i n g t h e b e g i n n i n g and t h e end o f t h e t r a c e d e p e n d i n g upon where i n t h e s i n u s o i d a l c u r v e t h e s i g n a l t r a c e b e g i n s o r e n d s . T h i s 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 as w e l l as end e f f e c t s h a v e no e f f e c t on t h e d a t a a n a l y s i s , b e c a u s e t h e o p e r a t i n g a m p l i t u d e o f m o t i o n i s i n p u t i n t o t h e d a t a a n a l y s i s p r o c e d u r e a n d n o t d e t e r m i n e d f r o m t h e d i s p l a c e m e n t t r a c e . F i g u r e 1 0 . 3 shows a p l o t o f t h e f r e q u e n c y s p e c t r u m c a l c u l a t e d u s i n g a F a s t F o u r i e r T r a n s f o r m b y t h e s u b r o u t i n e F F T . I t i s o b v i o u s t h a t t h e s p e c t r u m c o n t a i n s v i r t u a l l y no e n e r g y o u t s i d e t h e d e s i r e d n a r r o w b a n d . The p l o t i n d i c a t e s a n a m p l i t u d e o f 3 c e n t i m e t r e s a t a f r e q u e n c y o f 1 .89 H e r t z . F i g u r e 1 0 . 4 shows t h e u n f i l t e r e d s i g n a l o f t h e raw d a t a f r o m t h e s u r g e c h a n n e l o f t h e f o r c e dynamometer f o r t h e same t e s t c a s e (The dynamometer i s o r i e n t e d so t h a t t h e s u r g e c h a n n e l i s r e a d i n g a h e a v e f o r c e ) . The u n f i l t e r e d t r a c e shows some n o i s e e x i s t i n g as w e l l as t h e s i g n a l b e i n g c l i p p e d a t t h e p e a k s . T h e s e i n a c c u r a c i e s may be c a u s e d b y t h e f a c t t h a t a 2200N. dynamometer i s b e i n g u s e d t o measu re f o r c e s o f l e s s t h a n 200N. The amount o f n o i s e i n t h e s i g n a l was r e d u c e d as t h e dynamometer was r e a d i n g h i g h e r f o r c e s , and c o n v e r s e l y , t h e n o i s e i n c r e a s e d i n t h e t e s t s where t h e f o r c e s were q u i t e s m a l l . F i g u r e 1 0 . 5 shows t h e f i l t e r e d t r a c e o f t h e same s i g n a l . The 59 f i l t e r i n g r o u t i n e was a b l e t o smooth t h e t r a c e q u i t e w e l l by-n a r r o w i n g t h e s i g n a l n e a r t he p e a k s . T h i s i s o b v i o u s i n F i g u r e 1 0 . 6 where a s u p e r p o s i t i o n o f t h e f i l t e r e d and u n f i l t e r e d s i g n a l i s shown. F i g u r e 1 0 . 7 shows t h e f r e q u e n c y s p e c t r u m o f t h e u n f i l t e r e d s u r g e d a t a . The g r a p h shows t h a t t h e m a j o r f r e q u e n c y c o r r e s p o n d s t o t h e f r e q u e n c y o f o s c i l l a t i o n . However , o t h e r m i n o r p e a k s do o c c u r i n t h e s p e c t r u m i n d i c a t i n g o t h e r p o s s i b l e h i g h e r f r e q u e n c y n o i s e p r o d u c e d b y r e f l e c t i o n o f waves i n t h e t a n k o r p e r h a p s a r e s o n a n c e phenomenon o c c u r r i n g w i t h i n t h e e x p e r i m e n t a l a p p a r a t u s o r some o t h e r phenomena. V i r t u a l l y a l l t h e e n e r g y i s c o n c e n t r a t e d a t t h e f r e q u e n c y , 1 .89 H e r t z , w h i c h c o r r e s p o n d s t o t h e d r i v i n g f r e q u e n c y o f t h e c y l i n d e r . I n s p e c t i o n o f t h e samp le d a t a p l o t s s u g g e s t r e a s o n a b l e a c c u r a c y i n t h e r e s u l t s . A l l t h e t r a c e s show c l e a r i n d i c a t i o n o f b e h a v i o u r and c a n be u s e d i n t h e d e t e r m i n a t i o n o f t h e h y d r o d y n a m i c c o e f f i c i e n t s o f c y l i n d e r s . The d a t a f i l t e r i n g r o u t i n e s do n o t a l t e r t h e d a t a t o a s u c h an e x t e n t t h a t e r r o n e o u s v a l u e s o f t h e h y d r o d y n a m i c c o e f f i c i e n t s w i l l r e s u l t . 60 5 . 2 HYDRODYNAMIC COEFFICIENTS OF A COMPOUND CYLINDER IN HEAVE  MOTION Two e x p e r i m e n t s o r i g i n a l l y c a r r i e d o u t b y Madan V e n u g o p a l (1984) were r e p e a t e d i n o r d e r t o d e t e r m i n e t h e h y d r o d y n a m i c c o e f f i c i e n t s o f a compound c y l i n d e r i n h e a v e m o t i o n . The a n a l y s i s was r e p e a t e d due t o a s u s p i c i o n t h a t t h e r e s u l t s o b t a i n e d p r e v i o u s l y may be e r r o n e o u s due t o a f a u l t y f o r c e dynamometer b e i n g u s e d . Doug G o o d r i d g e (1986) i n s p e c t e d t h e dynamometer i n h o p e s o f u s i n g t h e i n s t r u m e n t h i m s e l f i n t h e d e t e r m i n a t i o n o f h y d r o d y n a m i c c o e f f i c i e n t s o f compound c y l i n d e r s i n s u r g e m o t i o n . He f o u n d t i n y c r a c k s i n t h e i n s t r u m e n t and t h e r e f o r e deemed i t i n o p e r a b l e . A new f o r c e dynamometer was c o n s t r u c t e d and u s e d f o r h i s r e s e a r c h . I t was d e c i d e d t h a t t h i s p a r t i c u l a r dynamometer be u s e d t o r e p e a t some e x p e r i m e n t s t h a t V e n u g o p a l h a d p e r f o r m e d t o r e - e v a l u a t e some o f h i s r e s u l t s . The same compound c y l i n d e r was s e t a t t h e same d r a f t as t h a t o f t h e p r e v i o u s l y c o n d u c t e d e x p e r i m e n t s . The r e s u l t s a r e compared w i t h t h e o r e t i c a l p r e d i c t i o n s o f t h e h y d r o d y n a m i c c o e f f i c i e n t s done u s i n g t h e M a t c h i n g T e c h n i q u e . 5 . 2 . 1 ADDED MASS COEFFICIENT OF A COMPOUND CYLINDER IN HEAVE  MOTION The added mass c o e f f i c i e n t s o f two t e s t s c o n d u c t e d b y t h i s r e s e a r c h e r a r e shown i n F i g u r e s 1 0 . 8 and 1 0 . 1 0 . The p r i o r r e s u l t s 61 o b t a i n e d b y V e n u g o p a l (1984) a r e shown on F i g u r e s 1 0 . 9 and 1 0 . 1 1 . B o t h s e t s o f e x p e r i m e n t a l r e s u l t s a r e compared w i t h t h e v a l u e s p r e d i c t e d b y t h e M a t c h i n g T e c h n i q u e . L o o k i n g a t t h e r e s u l t s one s e e s t h a t t h e r e p e a t e d e x p e r i m e n t s g i v e v a l u e s g e n e r a l l y l o w e r t h a n t h o s e o b t a i n e d b y V e n u g o p a l . H o w e v e r , t h e v a l u e s a r e g e n e r a l l y s l i g h t l y h i g h e r t h a n t h o s e p r e d i c t e d b y t h e M a t c h i n g T e c h n i q u e o f Sabuncu and C a l i s a l (1981) The t r e n d o f t h e d a t a f i t s n i c e l y w i t h t h e t r e n d o f t h e d a t a p r e d i c t e d b y t h e M a t c h i n g T e c h n i q u e , e s p e c i a l l y f o r h i g h e r f r e q u e n c i e s above a n o n - d i m e n s i o n a l i s e d v a l u e o f w = 1 . 0 . The maximum d e v i a t i o n i s a b o u t 20% f o r v a l u e s above w = 1 .0 f o r t h e r e p e a t e d e x p e r i m e n t s w h i l e t h e r e s u l t s o f t h e p r e v i o u s l y p e r f o r m e d e x p e r i m e n t s r a n g e as h i g h as 110%. B e l o w a v a l u e o f w = 1 . 0 , t h e r e s u l t s a r e n o t as g o o d . T h e o r y p r e d i c t s v a l u e s b e g i n n i n g a t a b o u t a = 0 . 2 f o r t h e F i g u r e 1 0 . 8 and a = 0 . 2 3 f o r F i g u r e 1 0 . 9 , and r i s e e v e r so s l i g h t l y b e f o r e f a l l i n g t o t h e s t e a d y s t a t e v a l u e s o f a = 0 . 2 3 f o r F i g u r e 1 0 . 8 and a ^ = 0 . 1 8 f o r F i g u r e 1 0 . 1 0 . I n t h e e x p e r i m e n t s h o w e v e r , t h e a d d e d mass c o e f f i c i e n t s b e g i n a t v e r y l o w v a l u e s and q u i c k l y r i s e t o t h e s t e a d y s t a t e v a l u e s o f a ^ = 0 . 2 5 f o r F i g u r e 1 0 . 8 and a ^ = 0 . 2 3 f o r F i g u r e 1 0 . 1 0 a t a b o u t a v a l u e o f w = 1 . 0 . T h i s seems t o c o n t r a d i c t t h e e x p e c t a t i o n o f h i g h added mass v a l u e s a t l o w f r e q u e n c i e s . The d i s c r e p a n c y may be due t o t o t h e f a c t t h a t a t l o w w v a l u e s , t h e f o r c e s a r e v e r y s m a l l and may n o t be a c c u r a t e l y m e a s u r e d w i t h t h e dynamometer . The s i g n a l t o n o i s e r a t i o may be so l o w as t o w a r r a n t s u s p i c i o n o f t h e r e s u l t s 62 o b t a i n e d a t t h e s e l o w f r e q u e n c i e s . V e n u g o p a l (1984) h a d a l s o t h e same p r o b l e m w i t h o b t a i n i n g r e s u l t s a t l o w co v a l u e s and i n h i s c a s e , d i d n o t b o t h e r t o r e p o r t some o f t h e s e p o i n t s . The new e x p e r i m e n t a l r e s u l t s show t h a t a s t e a d y s t a t e added mass c o e f f i c i e n t i s r e a c h e d as t h e f r e q u e n c y i s i n c r e a s e d , i n s t e a d o f a n i n c r e a s i n g added mass w i t h i n c r e a s i n g f r e q u e n c y as f o u n d b y V e n u g o p a l . The d e v i a t i o n i n t h e two e x p e r i m e n t a l r e s u l t s i s most l i k e l y due t o a f a u l t y dynamometer u s e d b y V e n u g o p a l . I f c r a c k s were p r e s e n t i n h i s f o r c e t r a n s d u c e r , t h e y p r o b a b l y c o n t r i b u t e d a g r e a t e r e r r o r as t h e f r e q u e n c y i n c r e a s e d and t h e t r a n s m i t t e d f o r c e s were g r e a t e r . The new e x p e r i m e n t a l r e s u l t s a l s o show t h a t v a r y i n g t h e a m p l i t u d e o f o s c i l l a t i o n does n o t s i g n i f i c a n t l y a l t e r t h e r e s u l t s a p p r e c i a b l y . A t u < 1 .5 t h e r e s u l t s v a r y s l i g h t l y and c o u l d be nd a t t r i b u t e d t o d i f f i c u l t y i n m e a s u r i n g l o w f o r c e s . A t co > 1 .5 t h e r e s u l t s f o l l o w t h e same t r e n d l i n e s w i t h r e l a t i v e l y l i t t l e d i f f e r e n c e b e t w e e n c u r v e s o f v a r y i n g a m p l i t u d e . 5 . 2 . 2 DAMPING COEFFICIENTS OF A COMPOUND CYLINDER IN HEAVE MOTION The damp ing c o e f f i c i e n t s o b t a i n e d b y t h i s r e s e a r c h e r a r e shown on F i g u r e s 1 0 . 1 2 and 1 0 . 1 4 w h i l e t h o s e o b t a i n e d b y V e n u g o p a l (1984) a r e shown on F i g u r e s 1 0 . 1 3 and 1 0 . 1 5 . The p l o t s o f t h e r e c e n t l y o b t a i n e d r e s u l t s seem t o g i v e r e a s o n a b l e c o r r e l a t i o n w i t h t h e t h e o r e t i c a l r e s u l t s p r e d i c t e d b y t h e M a t c h i n g T e c h n i q u e as does t h e e x p e r i m e n t a l r e s u l t s o b t a i n e d b y V e n u g o p a l (1984) f o r a 63 s t e p d r a f t o f 90 c e n t i m e t r e s . As i s t h e c a s e i n t h e added mass p l o t s , t h e n e w l y o b t a i n e d v a l u e s f a l l b e l o w t h o s e o b t a i n e d b y V e n u g o p a l and a r e s l i g h t l y above t h e r e s u l t s p r e d i c t e d b y t h e M a t c h i n g T e c h n i q u e . The t r e n d o f t h e new e x p e r i m e n t a l d a t a f i t s q u i t e w e l l w i t h t h e t r e n d p r e d i c t e d b y t h e M a t c h i n g T e c h n i q u e . The damp ing c o e f f i c i e n t v a l u e s a r e a t a maximum a t a n o n - d i m e n s i o n a l f r e q u e n c y , w, o f a p p r o x i m a t e l y 0 . 5 and s t e a d i l y d e c r e a s e t o w a r d s b = 0 as t h e f r e q u e n c y i n c r e a s e s . The e x p e r i m e n t a l v a l u e s o b t a i n e d b y V e n u g o p a l (1984) do n o t c o n c l u s i v e l y f o l l o w t h i s t r e n d . I n F i g u r e 1 0 . 1 4 , t h e d a t a f o l l o w s t h e p r e d i c t e d t r e n d v e r y n i c e l y w i t h t h e maximum damping c o e f f i c i e n t c o r r e s p o n d i n g t o a p p r o x i m a t e l y t h e same w v a l u e as t h e M a t c h i n g T e c h n i q u e p r e d i c t i o n and t h e n d e c r e a s i n g t o w a r d s z e r o as t h e f r e q u e n c y i n c r e a s e s . I n F i g u r e 1 0 . 1 5 , h o w e v e r , t h e d a t a i s much h i g h e r t h a n t h e p r e d i c t e d v a l u e s . The t r e n d o f t h e d a t a shows a s l i g h t d e c r e a s e o f b w i t h i n c r e a s i n g f r e q u e n c y u n t i l t h e l a s t two p o i n t s when t h e damp ing c o e f f i c i e n t s i n c r e a s e once a g a i n . T h e s e l a r g e d i s c r e p a n c i e s may h a v e b e e n due once a g a i n t o a f a u l t y dynamometer . The d a t a seems t o s u g g e s t t h a t t h e dynamometer was w o r k i n g p r o p e r l y f o r t h e F i g u r e 1 0 . 1 4 t e s t s b u t n o t f o r t h e F i g u r e 1 0 . 1 5 t e s t s . A t e n f o l d i n c r e a s e i n t h e damp ing c o e f f i c i e n t due t o a s l i g h t i n c r e a s e i n d r a f t does n o t seem p l a u s i b l e i n l i g h t o f t h e r e s u l t s o f t h e new e x p e r i m e n t s on t h e same c y l i n d e r . A g a i n t h e new e x p e r i m e n t a l r e s u l t s c o r r e s p o n d i n g t o t h e e x t r e m e l o w end o f t h e f r e q u e n c y s c a l e may be u n r e l i a b l e b e c a u s e 64 o f t h e l i m i t a t i o n s o f t h e dynamometer i n r e s o l v i n g s u c h l o w f o r c e s . T h i s may e x p l a i n a d i s c r e p a n c y b e t w e e n t h e t h e o r y and t h e e x p e r i m e n t a l l y o b t a i n e d p o i n t s i n F i g u r e s 1 0 . 1 4 and 1 0 . 1 5 . A t t h e h i g h e r f r e q u e n c i e s , a t r e n d o f i n c r e a s i n g damp ing c o e f f i c i e n t w i t h i n c r e a s i n g f r e q u e n c y i s n o t i c e a b l e . Howeve r , t h i s i n c r e a s i n g t r e n d i s n o t v e r y s t e e p , and i s p r o b a b l y due t o i n c r e a s i n g e r r o r i n t h e r e a d i n g s due t o f l e x i n g o f t h e s t r u c t u r e . S i n c e t h e t r i p l e c y l i n d e r mode l was q u i t e h e a v y , ( s 80 k g . ) , a c o n s i d e r a b l e amount o f f l e x i n g was n o t i c e a b l e as t h e d r i v i n g f r e q u e n c y was i n c r e a s e d w h i c h c a u s e s e r r o r i n t h e r e c o r d e d f o r c e s . 65 5 . 3 INDUCED SURGE FORCES ON A CYLINDER DUE TO A SECOND CYLINDER IN HEAVE MOTION T e s t s were c o n d u c t e d d u r i n g J a n u a r y , 1988 i n o r d e r t o d e t e r m i n e t h e i n d u c e d s u r g e and h e a v e f o r c e s upon a s i n g l e c y l i n d e r due t o a n a d j a c e n t c y l i n d e r o f t h e same s i z e and d r a f t s u b j e c t e d t o h e a v e m o t i o n . A deep w a t e r as w e l l as a s h a l l o w w a t e r c a s e i s i n v e s t i g a t e d . A f a l s e b o t t o m c o n s t r u c t e d f r o m s h e e t s o f p l y w o o d was s u s p e n d e d i n t h e deep w a t e r t a n k t o s i m u l a t e t h e s h a l l o w w a t e r s c e n a r i o . T h i s f a l s e b o t t o m was s u s p e n d e d t o a d e p t h o f a p p r o x i m a t e l y 68 c e n t i m e t r e s (= 1 .75 c y l i n d e r d i a m e t e r s ) . The d i s t a n c e b e t w e e n t h e two c y l i n d e r s , B, was m e a s u r e d f r o m c e n t e r t o c e n t e r o f t h e two c y l i n d e r s . F i v e d i f f e r e n t c y l i n d e r s p a c i n g s were n o r m a l l y u s e d , t h e y w e r e : B = 2 . 0 , 2 . 5 , 3 . 0 , 3 . 5 , and 4 . 0 c y l i n d e r r a d i i . The a m p l i t u d e o f o s c i l l a t o r y m o t i o n was s e l e c t e d as e i t h e r 1 . 0 , 1 . 5 , 2 . 5 , 3 . 5 , o r 4 . 5 c e n t i m e t r e s . The d r a f t o f t h e two c y l i n d e r s was s e t a t 38 c e n t i m e t r e s ( a p p r o x i m a t e l y 2 . 0 c y l i n d e r r a d i i ) . A c o m p a r i s o n w i t h t h e n u m e r i c a l r e s u l t s b a s e d on t h e M a t c h i n g T e c h n i q u e t h e o r y i s c a r r i e d o u t i n o r d e r t o i n v e s t i g a t e t h e v a l i d i t y o f t h e t h e o r y i n p r e d i c t i n g i n d u c e d s i d e f o r c e s . 66 5 . 3 . 1 INDUCED SURGE FORCES 5 . 3 . 1 . 1 DEEP WATER The r e s u l t s f r o m t h e e x p e r i m e n t s t o d e t e r m i n e t h e i n d u c e d s u r g e f o r c e s , e x p e r i m e n t a l l y and n u m e r i c a l l y , f o r t h e deep w a t e r s c e n a r i o a r e shown i n F i g u r e s 1 0 . 1 6 t o 1 0 . 2 0 , and a summary o f t h e r e s u l t s i s shown i n T a b l e 5 . 3 . 1 . 1 - 1 . TABLE 5 . 3 . 1 . 1 - 1 VALUES OF PEAK INDUCED SURGE FORCE IN DEEP WATER C y l . S e p . M a t c h i n g T e c h n i q u e E x p e r i m e n t a l R a t i o R a d i i F r e q . Peak F r c . F r e q . P e a k F r c . E x p . / T h e o . 2 . 0 5 0 . 5 0 0 . 0 1 2 9 0 . 6 1 0 . 0 1 0 0 0 . 7 6 2 . 4 8 0 . 5 0 0 . 0 1 3 1 0 . 7 5 0 . 0 1 0 5 0 . 8 0 3 . 0 0 0 . 6 0 0 .0127 * 0 . 8 1 0 . 0 0 5 0 * * 0 . 3 9 3 . 4 8 0 . 6 0 0 . 0 1 2 3 0 . 7 4 0 . 0 1 0 3 0 . 8 1 4 . 0 0 0 . 6 0 0 . 0 1 1 7 0 . 7 5 0 . 0 0 9 0 0 . 7 7 * A n i n s u f f i c i e n t number o f d a t a p o i n t s we re g a t h e r e d a t l o w f r e q u e n c i e s t o o b t a i n peak i n d u c e d f o r c e v a l u e s . F i g u r e 1 0 . 1 6 shows a p l o t o f t h e n o n - d i m e n s i o n a l i n d u c e d s u r g e f o r c e v e r s u s f r e q u e n c y f o r a c y l i n d e r v e r y n e a r l y t o u c h i n g t h e o t h e r c y l i n d e r (B = 2 . 0 c y l i n d e r r a d i i ) . The r e s u l t s seem t o s u g g e s t a r e a s o n a b l e ag reemen t b e t w e e n t h e e x p e r i m e n t a l r e s u l t s 67 and t h e p r e d i c t e d t h e o r y . The M a t c h i n g T e c h n i q u e p r e d i c t s t h a t as t h e f r e q u e n c y i s i n c r e a s e d f r o m co = 0, t h e i n d u c e d s u r g e f o r c e s q u i c k l y i n c r e a s e t o a maximum and t h e n q u i c k l y d e c r e a s e t o z e r o as t h e f r e q u e n c y i s i n c r e a s e d . The maximum v a l u e o f t h e i n d u c e d f o r c e i n t h i s c a s e i s Y = 0 . 0 1 2 9 a t co = 0 . 5 . The t h e o r y p r e d i c t s t h a t t h e i n d u c e d m a x J s u r g e f o r c e w i l l be n o n - e x i s t e n t a t f r e q u e n c i e s g r e a t e r t h a n w = 2 . 0 . The e x p e r i m e n t a l r e s u l t s f o l l o w s t o a d e g r e e t h e t r e n d p r e d i c t e d b y t h e M a t c h i n g T e c h n i q u e . The v a l u e s a r e n e a r z e r o f o r v e r y l o w f r e q u e n c i e s (co < 0 . 3 ) and q u i c k l y r i s e t o t h e i r maximum v a l u e s a t f r e q u e n c i e s n e a r t h e n u m e r i c a l l y p r e d i c t e d v a l u e . H o w e v e r , t h e v a l u e o f t h i s maximum i s l e s s t h a n t h e p r e d i c t e d v a l u e ; a l s o , t h e maximum i n d u c e d s u r g e f o r c e i s d e p e n d a n t on a m p l i t u d e o f o s c i l l a t i o n . I n t h i s c a s e , t h e maximum i n d u c e d s u r g e f o r c e f o u n d e x p e r i m e n t a l l y i s Y = 0 . 0 1 0 f o r a a m p l i t u d e o f 3 . 5 m a x c e n t i m e t r e s , w h i c h i s a p p r o x i m a t e l y 24% l e s s t h a n t h e p r e d i c t e d v a l u e , w h i l e t h e v a l u e i s o n l y Y = 0 . 0 0 6 5 f o r a m p l i t u d e s o f 1 .0 m a x and 2 . 5 c e n t i m e t r e s , a b o u t 49% l e s s t h a n t h e p r e d i c t e d v a l u e . The e x p e r i m e n t a l v a l u e s r a p i d l y d e c r e a s e as t h e f r e q u e n c y i s i n c r e a s e d b e y o n d co = 0 . 5 , h o w e v e r , t h e v a l u e s do n o t d e c r e a s e a l l t h e way t o z e r o b u t i n s t e a d seem t o s e t t l e a t v a l u e s n e a r Y = 0 . 0 0 2 . Hence a c l e a r i n d i c a t i o n o f a q u i c k l y i n c r e a s i n g and d e c r e a s i n g phenomenon o f t h e i n d u c e d s u r g e f o r c e i s v i s i b l e . F i g u r e 1 0 . 1 7 i l l u s t r a t e s t h e c a s e where t h e s e p a r a t i o n b e t w e e n t h e c y l i n d e r s i s i n c r e a s e d t o B = 2 . 5 . A g a i n a r e a s o n a b l e 68 amount o f ag reemen t b e t w e e n t h e p r e d i c t e d v a l u e s and t he e x p e r i m e n t a l r e s u l t s seems t o e x i s t a t t h e r a n g e o f f r e q u e n c i e s t e s t e d . The e x p e r i m e n t a l v a l u e s a r e i n g e n e r a l l o w e r t h a n t h e p r e d i c t e d v a l u e s The M a t c h i n g T e c h n i q u e once a g a i n p r e d i c t s a q u i c k r i s e i n t h e i n d u c e d f o r c e as t h e f r e q u e n c y i s i n c r e a s e d f r o m w = 0 . The p e a k v a l u e o f Y = 0 . 0 1 3 1 i s r e a c h e d a t u> = 0 . 5 ; t h i s peak v a l u e max i s s l i g h t l y h i g h e r (= 1.5%) t h a n t h e c a s e o f t h e c y l i n d e r s b e i n g c l o s e t o g e t h e r (B = 2) . T h i s c o u l d be a t t r i b u t e d t o t h e f a c t t h a t n o t enough f r e q u e n c y p o i n t s were c h o s e n t o g i v e a n a c c u r a t e p r e d i c t i o n n e a r t h e peak v a l u e s o f t h e i n d u c e d s u r g e f o r c e . The Y f o r c e q u i c k l y d e c r e a s e s t o z e r o as t h e f r e q u e n c y i s i n c r e a s e d b e y o n d w = 0 . 5 , r e a c h i n g v a l u e s c l o s e t o z e r o a t u> = 2 . 0 . The e x p e r i m e n t a l r e s u l t s show e v i d e n c e o f a q u i c k r i s e and f a l l o f t h e i n d u c e d s u r g e f o r c e v a l u e s n e a r w = 0 . 5 . T h i s i s mos t e v i d e n t f o r t h e c a s e o f t h e d r i v i n g a m p l i t u d e e q u a l t o 3 . 5 c e n t i m e t r e s . Howeve r , due t o t h e d i f f i c u l t n a t u r e o f o b t a i n i n g d a t a v e r y n e a r z e r o , c l e a r e v i d e n c e o f a r i s e i n t h e i n d u c e d s u r g e f o r c e as t h e f r e q u e n c y r i s e s f r o m u = 0 i s n o t shown. The v a l u e s o f t h e e x p e r i m e n t a l r e s u l t s a r e l o w e r t h a n t h e p r e d i c t e d v a l u e s w i t h i n t h e bounds u> = 0 and w = 1 . 7 5 . T h i s i s e s p e c i a l l y t r u e f o r t h e c a s e o f l o w d r i v i n g a m p l i t u d e . The peak v a l u e o b t a i n e d e x p e r i m e n t a l l y , Y = 0 . 0 1 0 5 , i s a p p r o x i m a t e l y 20% l o w e r t h a n t h a t max p r e d i c t e d n u m e r i c a l l y . F i g u r e 1 0 . 1 8 shows t h e r e s u l t s when t h e s e p a r a t i o n h a s b e e n i n c r e a s e d t o B = 3 . 0 r a d i i . T h i s g r a p h v e r y much r e s e m b l e s F i g u r e 69 1 0 . 1 7 i n t h a t an i n s u f f i c i e n t number o f d a t a p o i n t s were t a k e n a t t h e l o w f r e q u e n c i e s t h e r e b y n o t g i v i n g an i n d i c a t i o n o f t h e i n d u c e d s u r g e f o r c e b e h a v i o u r a t t h e s e l o w f r e q u e n c i e s . The M a t c h i n g T e c h n i q u e p r e d i c t s a s t e e p r i s e i n t h e i n d u c e d s u r g e f o r c e t o a maximum o f Y = 0 . 0 1 2 7 a t a f r e q u e n c y o f 0 . 6 0 . m a x T h i s f o r c e i s l o w e r t h a n t h e p r e v i o u s two c a s e s and o c c u r s a t a h i g h e r f r e q u e n c y . The Y F o r c e q u i c k l y d e c r e a s e s t o Y = 0 as t h e f r e q u e n c y i n c r e a s e s b e y o n d co = 0 . 0 6 . The e x p e r i m e n t a l r e s u l t s show a r a p i d d e c r e a s e o f i n d u c e d s u r g e as t h e f r e q u e n c y i s i n c r e a s i n g w i t h i n t h e r a n g e w = 0 . 5 t o w = 1 . 7 , e s p e c i a l l y f o r t h e c a s e s o f h i g h e r d r i v i n g a m p l i t u d e . T h i s seems t o a g r e e w i t h p r e d i c t e d r e s u l t s w i t h i n t h i s r a n g e o f f r e q u e n c i e s . W h i l e no c o n c l u s i o n c a n made as t o t h e b e h a v i o u r o f t h e Y F o r c e a t l o w f r e q u e n c i e s , one c a n assume t h a t t h e v a l u e s w i l l r i s e t o a maximum a t a f r e q u e n c y v a l u e n e a r co = 0 . 5 and t h e n b e g i n t o d e c r e a s e . T h i s c a n be c o n c l u d e d b y l o o k i n g a t t h e r e s u l t s o f t e s t s a t g r e a t e r s e p a r a t i o n b e t w e e n t h e c y l i n d e r s . The e x p e r i m e n t a l r e s u l t s a r e l o w e r t h a n t h e p r e d i c t e d v a l u e s f o r f r e q u e n c i e s b e l o w co = 1 . 5 . A t f r e q u e n c i e s above co = 1 . 5 , t h e e x p e r i m e n t a l v a l u e s seem t o c o n v e r g e t o Y = 0 . 0 0 1 4 w i t h t h e Y f o r c e v a l u e s o f t h e g r e a t e r d r i v i n g a m p l i t u d e b e i n g h i g h e r . F i g u r e 1 0 . 1 9 , s h o w i n g t h e c a s e o f B = 3 . 5 , shows a b e t t e r ag reemen t b e t w e e n t h e p r e d i c t e d v a l u e s and t h e e x p e r i m e n t a l r e s u l t s m a i n l y due t o f a c t t h a t a s u f f i c i e n t number o f l o w f r e q u e n c y d a t a p o i n t s were o b t a i n e d t o g i v e c l e a r u n d e r s t a n d i n g t o t h e b e h a v i o u r a l r e s p o n s e . 70 The M a t c h i n g T e c h n i q u e once a g a i n p r e d i c t s a r a p i d i n c r e a s e i n t h e i n d u c e d s u r g e f o r c e s t o a maximum o f Y = 0 . 0 1 2 3 a t a m a x f r e q u e n c y o f w = 0 . 0 6 . T h i s maximum f o r c e i s l e s s t h a n t h e o t h e r c a s e s w h i c h i s what one e x p e c t s as t h e c y l i n d e r s e p a r a t i o n i s i n c r e a s e d . The i n d u c e d s u r g e f o r c e r a p i d l y d e c r e a s e s t o w a r d s Y = 0 as t h e f r e q u e n c y i n c r e a s e s b e y o n d w = 0 . 0 6 . The e x p e r i m e n t a l r e s u l t s show good c o r r e l a t i o n w i t h t h e p r e d i c t e d r e s u l t s i n t h i s c a s e . A c l e a r i n c r e a s e i n t h e i n d u c e d s u r g e f o r c e as t h e f r e q u e n c y i s i n c r e a s e d f r o m w = 0 i s v i s i b l e . T h i s f o r c e r e a c h e s a maximum v a l u e o f a p p r o x i m a t e l y Y = 0 . 0 1 0 a t m a x a r o u n d w = 0 . 0 8 , w h i c h i s a b o u t 19% l o w e r t h a n t h e n u m e r i c a l l y p r e d i c t e d maximum. E v e n t h o u g h t h e r e s u l t s a r e l o w e r t h a n t h e p r e d i c t e d v a l u e s , a c l e a r t r e n d i n t h e d a t a shows t h a t t h e p r e d i c t e d phenomenon i s i n f a c t h a p p e n i n g . The e x p e r i m e n t a l r e s u l t s do r a p i d l y d e c r e a s e as t h e f r e q u e n c y i s i n c r e a s e d b e y o n d 0 . 0 8 , b u t do n o t d e c r e a s e t o Y = 0 as p r e d i c t e d , b u t r a t h e r t o Y = 0 . 0 0 1 4 . T h i s i n d i c a t e s t h a t i n d u c e d s u r g e f o r c e s a r e p r e s e n t a t t h e h i g h e r f r e q u e n c i e s . F i g u r e 1 0 . 2 0 shows t h e c a s e o f t h e c y l i n d e r s e p a r a t i o n b e i n g i n c r e a s e d t o B = 4 . 0 . The r e s u l t s , once a g a i n , seem t o i n d i c a t e a b e h a v i o u r w h i c h i s p r e d i c t e d b y t h e M a t c h i n g T e c h n i q u e . The p l o t does h o w e v e r , show a g r e a t e r s p r e a d i n t h e d a t a b e t w e e n t h e v a r i o u s d r i v i n g a m p l i t u d e compared t o t h e p r e v i o u s p l o t . A s i n t h e p r e v i o u s p l o t s t h e M a t c h i n g T e c h n i q u e p r e d i c t s a s t e e p r i s e and f a l l i n t h e v a l u e s o f t h e i n d u c e d s u r g e f o r c e as t h e f r e q u e n c y i s i n c r e a s e d f r o m u> = 0 . The maximum v a l u e o f t h i s 71 s u r g e f o r c e , Y = 0 . 0 1 1 7 , o c c u r s a t co = 0 . 6 and i s t h e l o w e s t m a x v a l u e o f Y p r e d i c t e d i n t h e f i v e d i f f e r e n t c y l i n d e r m a x s e p a r a t i o n s . T h i s t r e n d p r e d i c t i n g l o w e r v a l u e s o f i n d u c e d s u r g e f o r c e as t h e s e p a r a t i o n b e t w e e n t h e c y l i n d e r s i s i n c r e a s e d i s e x p e c t e d s i n c e r a d i a l wave f o r c e s a r e r e d u c e d w i t h i n c r e a s e d s e p a r a t i o n . The e x p e r i m e n t a l r e s u l t s , once a g a i n , seem t o s u g g e s t a n i n c r e a s e i n t h e i n d u c e d s u r g e f o r c e as t h e f r e q u e n c y i s i n c r e a s e d f r o m co = 0 , t h e v a l u e s r e a c h a maximum n e a r co = 0 . 8 b e f o r e d e c r e a s i n g t o a s t e a d y s t a t e v a l u e . The maximum v a l u e o f t h e i n d u c e d s u r g e f o r c e depends g r e a t l y on t h e a m p l i t u d e o f o s c i l l a t i o n , as t h e a m p l i t u d e i s i n c r e a s e d , t h e g r e a t e r i s t h e maximum i n d u c e d s u r g e f o r c e . I n t h i s c a s e , f r o m Y = 0 . 0 0 6 f o r m a x t h e 2 / 5 c e n t i m e t r e a m p l i t u d e , ( a b o u t 52% t h e o r e t i c a l maximum) t o Y = 0 . 0 0 9 f o r t h e 4 . 5 c e n t i m e t r e a m p l i t u d e c a s e ( a b o u t 77% m a x t h e o r e t i c a l maximum). Once a g a i n , most o f t h e v a l u e s o f t h e i n d u c e d f o r c e a r e l o w e r t h a n t h e p r e d i c t e d v a l u e s w i t h i n t h e r a n g e o f co = 0 t o co = 1 . 2 5 , w h i l e a t f r e q u e n c i e s above t h i s v a l u e t h e e x p e r i m e n t a l r e s u l t s c o n v e r g e t o a v a l u e o f Y = 0 . 0 0 1 5 i n s t e a d o f Y = 0 . 0 as t h e M a t c h i n g T e c h n i q u e p r e d i c t s . Howeve r , r e a s o n a b l e ag reemen t w i t h t h e p r e d i c t e d t r e n d i n d i c a t e s an a c c u r a t e p r e d i c t i o n i n t h e b e h a v i o u r t h e i n d u c e d s u r g e f o r c e s o f o s c i l l a t i n g c y l i n d e r s . 72 5 . 3 . 1 . 2 SHALLOW WATER P l o t s o f t h e n u m e r i c a l c a l c u l a t i o n s compu ted f r o m t h e M a t c h i n g T e c h n i q u e and t h e e x p e r i m e n t a l r e s u l t s a r e f o u n d i n F i g u r e s 1 0 . 2 1 t o 1 0 . 2 5 . The r e s u l t s a r e s u m m a r i z e d on T a b l e 5 . 3 . 1 . 2 - 1 . TABLE 5 . 3 . 1 . 2 - 1 VALUES OF PEAK INDUCED SURGE FORCE IN SHALLOW WATER C y l . S e p . M a t c h i n g T e c h n i q u e E x p e r i m e n t a l R a t i o R a d i i F r e q . Peak F r c . F r e q . Peak F r c . E x p . / T h e o . 2 . 0 8 0 . 5 0 0 . 0 1 4 8 0 . 7 1 0 . 0 1 2 1 0 . 9 3 2 . 4 7 0 . 5 0 0 . 0 1 5 2 0 . 4 5 0 . 0 1 1 8 0 . 7 8 3 . 0 0 0 . 5 0 0 . 0 1 4 6 0 . 7 9 0 . 0 1 1 2 0 . 7 7 3 . 4 8 0 . 6 0 0 . 0 1 4 4 0 . 7 7 0 . 0 1 1 8 0 . 8 2 4 . 0 0 0 . 6 0 0 . 0 1 3 8 0 . 7 6 0 . 0 1 0 3 0 . 7 5 F i g u r e 1 0 . 2 1 shows t h e r e s u l t s o f t h e s h a l l o w w a t e r c a s e where t h e c y l i n d e r s a r e v e r y n e a r l y t o u c h i n g one a n o t h e r , B = 2 . 0 . The r e s u l t s show a, r e a s o n a b l e ag reemen t w i t h t h e t r e n d p r e d i c t e d b y t h e M a t c h i n g T e c h n i q u e . The M a t c h i n g T e c h n i q u e p r e d i c t s a t r e n d i n t h e i n d u c e d s u r g e f o r c e t o be v e r y much l i k e t h e t r e n d i n t h e deep w a t e r c a s e . As t h e f r e q u e n c y i s i n c r e a s e d f r o m w = 0 , t h e i n d u c e d s u r g e f o r c e q u i c k l y r i s e s t o a peak v a l u e b e f o r e r a p i d l y d e c r e a s i n g t o w a r d s z e r o . I n t h i s c a s e t h e p e a k v a l u e o f t h e i n d u c e d s u r g e f o r c e i s Y = 0 . 0 1 4 8 a t a f r e q u e n c y o f w = 0 . 5 . T h i s p e a k v a l u e i s a b o u t max 73 15% h i g h e r t h a n t h e v a l u e p r e d i c t e d f o r t h e deep w a t e r c a s e (Y = max 0 . 0 1 2 9 ) f o r t h e same s e p a r a t i o n . A s i n t h e deep w a t e r c a s e , t h e p r e d i c t e d v a l u e s o f t h e i n d u c e d s u r g e f o r c e q u i c k l y d e c r e a s e t o z e r o as t h e f r e q u e n c y i s i n c r e a s e d b e y o n d co = 0 . 5 . The e x p e r i m e n t a l r e s u l t s seem t o a g r e e w i t h t h e p r e d i c t e d t r e n d e s p e c i a l l y f o r t h e h i g h e r o s c i l l a t i n g a m p l i t u d e ( 4 . 5 c m . ) . The e x p e r i m e n t a l r e s u l t s i n c r e a s e as t h e f r e q u e n c y i n c r e a s e s f r o m co = 0 , r e a c h a peak v a l u e n e a r co = 0 . 7 b e f o r e d e c r e a s i n g t o w a r d s a s t e a d y s t a t e v a l u e o f Y 2 0 . 0 0 2 a t t h e h i g h e r f r e q u e n c i e s . T h i s p e a k v a l u e o f t h e i n d u c e d s u r g e f o r c e i s h i g h l y d e p e n d e n t on t h e a m p l i t u d e o f o s c i l l a t i o n , t h e v a l u e i s o n l y Y = 0 . 0 0 5 f o r an max a m p l i t u d e o f 1 .5 c e n t i m e t r e s ( a p p r o x i m a t e l y 39% t h e o r e t i c a l maximum), w h i l e i t r i s e s t o Y = 0 . 0 1 2 f o r an a m p l i t u d e o f 4 . 5 max c e n t i m e t r e s ( a p p r o x i m a t e l y 93% t h e o r e t i c a l maximum). E a c h o f t h e s e v a l u e s a r e h o w e v e r , s t i l l b e l o w t h e maximum v a l u e f o r t h e i n d u c e d s u r g e f o r c e p r e d i c t e d b y t h e M a t c h i n g T e c h n i q u e . C o m p a r i n g t h i s c a s e w i t h t h e deep w a t e r c a s e shows t h a t t h e peak v a l u e s o b t a i n e d i n t h e s h a l l o w w a t e r c a s e a r e a b o u t 20% h i g h e r t h a n t h e deep w a t e r c a s e (Y = 0 . 0 1 2 f o r t h e s h a l l o w w a t e r c a s e max and Y = 0 . 0 1 0 f o r t h e deep w a t e r c a s e ) , y e t t h e s t e a d y s t a t e max v a l u e s f o r h i g h e r f r e q u e n c i e s a r e r o u g h l y t h e same i n b o t h c a s e s . F i g u r e 1 0 . 2 2 shows t h e r e s u l t s o f i n c r e a s i n g t h e s e p a r a t i o n t o a p p r o x i m a t e l y two and one h a l f c y l i n d e r r a d i i . The r e s u l t s seem t o m a t c h t h e t r e n d p r e d i c t e d b y t h e M a t c h i n g T e c h n i q u e e s p e c i a l l y a t t h e h i g h a m p l i t u d e o f o s c i l l a t i o n . The M a t c h i n g T e c h n i q u e p r e d i c t s a s t e e p r i s e i n t h e i n d u c e d 74 s u r g e f o r c e as t h e f r e q u e n c y i s i n c r e a s e d f r o m w = 0 . The peak v a l u e o f Y = 0 . 0 1 5 2 i s r e a c h e d a t a f r e q u e n c y o f w = 0 . 5 . T h i s max v a l u e o f Y i s h i g h e r t h a n t h e p r e v i o u s c a s e o f B = 2 , j u s t as max i t was i n t h e deep w a t e r c a s e . P e r h a p s t h i s i s once a g a i n due t o t h e f a c t t h a t an i n s u f f i c i e n t number o f d a t a p o i n t s were c o l l e c t e d i n t h e a l l i m p o r t a n t peak a r e a . Once t h e f r e q u e n c y i s i n c r e a s e d b e y o n d w = 0 . 5 t h e v a l u e s o f t h e i n d u c e d s u r g e q u i c k l y d e c r e a s e t o w a r d s z e r o . The d e g r e e t o w h i c h e x p e r i m e n t a l r e s u l t s f o l l o w t h e p r e d i c t e d t r e n d i s v e r y much dependen t upon t h e a m p l i t u d e o f o s c i l l a t i o n . The h i g h e r t h e a m p l i t u d e o f o s c i l l a t i o n t h e b e t t e r t h e a g r e e m e n t . The e x p e r i m e n t a l r e s u l t s i n c r e a s e as t h e f r e q u e n c y i s i n c r e a s e d f r o m z e r o , r e a c h a p e a k v a l u e , and t h e n d e c r e a s e t o a s t e a d y s t a t e v a l u e f o r h i g h f r e q u e n c i e s . T h i s peak v a l u e i s o n l y Y = 0 . 0 0 6 7 max ( a b o u t 44% t h e o r e t i c a l maximum) f o r an a m p l i t u d e o f 1 .5 c e n t i m e t r e s w h i l e i t r i s e s t o Y = 0 . 0 1 1 8 ( a b o u t 78% t h e o r e t i c a l max maximum) f o r a n a m p l i t u d e o f 4 . 5 c e n t i m e t r e s , y e t t h e y a r e s t i l l b e l o w t h e peak p r e d i c t e d v a l u e . The p e a k v a l u e s o b t a i n e d e x p e r i m e n t a l l y were a t a f r e q u e n c y o f w = 0 . 4 , a g r e e i n g w e l l w i t h t h e p r e d i c t e d f r e q u e n c y o f p e a k i n d u c e d s u r g e f o r c e . The e x p e r i m e n t a l r e s u l t s c o n v e r g e t o a v a l u e o f Y = 0 . 0 0 2 5 a t t h e h i g h e r f r e q u e n c i e s . T h i s i s r o u g h l y t h e same as t h e p r e v i o u s c a s e and i s s l i g h t l y h i g h e r t h a n t h e c o r r e s p o n d i n g c a s e f o r deep w a t e r (Y = 0 . 0 0 2 ) . F i g u r e 1 0 . 2 3 shows t h e r e s u l t s o f t h e c y l i n d e r s e p a r a t i o n b e i n g s e t a t B = 3 . 0 c y l i n d e r r a d i i . The r e s u l t s a g r e e w i t h t h e 75 p r e d i c t e d t r e n d , b u t n o t t o as good a d e g r e e as t h e p r e v i o u s g r a p h . The M a t c h i n g T e c h n i q u e p r e d i c t s a t r e n d i n t h e e x a c t same manner as a l l t h e p r e v i o u s g r a p h s . The maximum v a l u e o f t h e i n d u c e d s u r g e f o r c e i s i n t h i s c a s e Y = 0 . 0 1 4 6 w h i c h o c c u r s a t a max f r e q u e n c y o f w = 0 . 5 . T h i s r e s u l t i s l o w e r t h a n t h e p r e v i o u s two c y l i n d e r s e p a r a t i o n , w h i c h i s what one e x p e c t s t o h a p p e n as t h e c y l i n d e r s e p a r a t i o n i s i n c r e a s e d . A t f r e q u e n c i e s above w = 0 . 5 t h e i n d u c e d s u r g e f o r c e v a l u e s q u i c k l y d e c r e a s e t o w a r d s z e r o . The e x p e r i m e n t a l r e s u l t s o b t a i n e d i n t h i s e x p e r i m e n t show a d i s t i n c t i v e s t e e p r i s e and f a l l i n t h e i n d u c e d s u r g e f o r c e as t h e f r e q u e n c y i s i n c r e a s e d f r o m z e r o . The peak v a l u e i s a g a i n d e p e n d e n t upon t h e a m p l i t u d e o f o s c i l l a t i o n , r a n g i n g f r o m Y = max 0 . 0 0 4 5 ( a b o u t 31% t h e o r e t i c a l maximum) f o r t h e 1 .5 c e n t i m e t r e a m p l i t u d e t o Y = 0 . 0 1 1 2 ( a b o u t 77% t h e o r e t i c a l maximum) f o r t h e max 4 . 5 c e n t i m e t r e a m p l i t u d e . Howeve r , t h e f r e q u e n c y a t w h i c h t h e s e p e a k s o c c u r i s a l s o dependen t upon t h e a m p l i t u d e o f o s c i l l a t i o n and i n some c a s e s does n o t ma tch t h e f r e q u e n c y p r e d i c t e d b y t h e M a t c h i n g T e c h n i q u e ; t h e s e v a l u e s r a n g e f r o m w = 0 . 4 3 f o r 1 .5 c e n t i m e t r e a m p l i t u d e t o w = 0 . 8 1 f o r 4 . 5 c e n t i m e t r e a m p l i t u d e . The r e s u l t s o f maximum i n d u c e d s u r g e a r e a g a i n h i g h e r i n t h e s h a l l o w w a t e r c a s e t h a n t h o s e o b t a i n e d i n t h e deep w a t e r c a s e f o r t h e same c y l i n d e r s e p a r a t i o n , i n t h i s c a s e t h e d i f f e r e n c e i s a b o u t 16%. As t h e f r e q u e n c y i s i n c r e a s e d , t h e e x p e r i m e n t a l v a l u e s o f t h e i n d u c e d s u r g e f o r c e c o n v e r g e t o a v a l u e o f Y = 0 . 0 0 1 8 , w h i c h i s l o w e r t h a n t h e p r e v i o u s two c a s e s . 76 F i g u r e 1 0 . 2 4 shows t h e r e s u l t s when t h e c y l i n d e r s e p a r a t i o n i s i n c r e a s e d t o B = 3 . 5 c y l i n d e r r a d i i . The r e s u l t s show good ag reemen t w i t h t h e p r e d i c t e d v a l u e s computed b y t h e M a t c h i n g T e c h n i q u e . The M a t c h i n g T e c h n i q u e p r e d i c t s t h e same t r e n d as i n t h e p r e v i o u s g r a p h s . The peak v a l u e o f t h e i n d u c e d s u r g e f o r c e i s p r e d i c t e d as Y = 0 . 0 1 4 4 a t a f r e q u e n c y o f co = 0 . 6 . The v a l u e i s max s l i g h t l y l o w e r t h a n t h e c a s e o f B = 3 . 0 , b u t t h e p e a k v a l u e o c c u r s a t a h i g h e r f r e q u e n c y . The i n d u c e d s u r g e f o r c e q u i c k l y d e c r e a s e s t o z e r o as t h e f r e q u e n c y i s i n c r e a s e d above w = 0 . 6 . The e x p e r i m e n t a l r e s u l t s show good ag reemen t w i t h t h e p r e d i c t e d r e s u l t s . The e x p e r i m e n t a l v a l u e s q u i c k l y r i s e as t h e f r e q u e n c y i n c r e a s e s f r o m z e r o . Once a g a i n t h e peak v a l u e r e a c h e d depends on t h e a m p l i t u d e o f o s c i l l a t i o n . I n t h e c a s e o f t h e 1 .5 c e n t i m e t r e a m p l i t u d e t h e maximum i n d u c e d s u r g e f o r c e i s Y = max 0 . 0 0 8 ( a b o u t 56% t h e o r e t i c a l maximum) w h i l e i t i s Y = 0 . 0 1 1 8 max ( a b o u t 82% t h e o r e t i c a l maximum) f o r t h e 4 . 5 c e n t i m e t r e c a s e . The p e a k v a l u e s a l l seem t o o c c u r v e r y n e a r t h e p e a k f r e q u e n c y p r e d i c t e d b y t h e M a t c h i n g T e c h n i q u e . The v a l u e s a l l l i e b e l o w t h e p r e d i c t e d t r e n d l i n e f o r t h e l o w f r e q u e n c i e s (co < 1 .0 ) and l i e above i t f o r t h e h i g h e r f r e q u e n c i e s . The v a l u e s a r e a g a i n h i g h e r t h a n t h e v a l u e s o b t a i n e d i n t h e deep w a t e r c a s e f o r t h e same c y l i n d e r s e p a r a t i o n . The e x p e r i m e n t a l v a l u e s r e a c h a s t e a d y s t a t e v a l u e o f Y = 0 . 0 0 2 a t t h e h i g h e r v a l u e s o f f r e q u e n c i e s . F i g u r e 1 0 . 2 5 shows t h e r e s u l t s f o r t h e c a s e o f t h e c y l i n d e r s e p a r a t i o n b e i n g i n c r e a s e d t o B = 4 c y l i n d e r r a d i i . The 77 e x p e r i m e n t a l r e s u l t s , a l t h o u g h m o s t l y l o w e r t h a n t h e p r e d i c t e d v a l u e s , does f o l l o w t h e p r e d i c t e d t r e n d f a i r l y w e l l . As i n a l l t h e p r e v i o u s c a s e s , t h e M a t c h i n g T e c h n i q u e p r e d i c t s a s t e e p r i s e and f a l l i n t h e i n d u c e d s u r g e f o r c e p e a k i n g i n t h i s c a s e t o a v a l u e o f Y = 0 . 0 1 3 8 a t a f r e q u e n c y o f w = 0 . 6 . T h i s max v a l u e i s l o w e r t h a n t h e p r e v i o u s c a s e s as e x p e c t e d , f o r t h e c a s e o f maximum c y l i n d e r s e p a r a t i o n . As t h e f r e q u e n c y i s i n c r e a s e d above w = 0 . 6 t h e i n d u c e d f o r c e q u i c k l y d e c r e a s e s t o z e r o . The e x p e r i m e n t a l r e s u l t s shows t h e f a m i l i a r t r a i t o f i n c r e a s i n g and d e c r e a s i n g i n d u c e d s u r g e f o r c e as t h e f r e q u e n c y i s i n c r e a s e f r o m z e r o . I n t h i s c a s e t h e maximum p e a k v a l u e s a r e Y = 0 . 0 0 7 6 ( a b o u t 55% t h e o r e t i c a l maximum) f o r t h e 2 . 5 max c e n t i m e t r e a m p l i t u d e c a s e and Y = 0 . 0 1 0 3 ( a b o u t 75% t h e o r e t i c a l max maximum) f o r t h e 4 . 5 c e n t i m e t r e c a s e . The p e a k v a l u e s o c c u r a t a s l i g h t l y h i g h e r f r e q u e n c y (w = 0 . 8 ) t h a n t h a t p r e d i c t e d b y t h e M a t c h i n g T e c h n i q u e . The v a l u e s q u i c k l y d e c r e a s e t o a s t e a d y s t a t e v a l u e o f Y = 0 . 0 0 1 8 a t t h e h i g h e r f r e q u e n c i e s . C o m p a r i n g t h i s c a s e t o t h e deep w a t e r c a s e , one o b s e r v e s t h a t t h e s h a l l o w w a t e r c a s e v a l u e s a r e h i g h e r , a p p r o x i m a t e l y 14% h i g h e r , t h a n t h o s e o b t a i n e d i n t h e deep w a t e r c a s e . As i n t h e deep w a t e r c a s e , a r e a s o n a b l e ag reemen t w i t h t h e p r e d i c t e d t r e n d i n d i c a t e s a f a i r l y a c c u r a t e p r e d i c t i o n i n t h e b e h a v i o u r o f t h e i n d u c e d s u r g e f o r c e s o f o s c i l l a t i n g c y l i n d e r s i n a s h a l l o w w a t e r s c e n a r i o . 78 5 . 4 INDUCED HEAVE HYDRODYNAMIC COEFFICIENTS ON A CYLINDER DUE TO A SECOND CYLINDER IN HEAVE MOTION D u r i n g t h e same t e s t s t o g a t h e r d a t a f o r t h e i n d u c e d s u r g e f o r c e s on a c y l i n d e r due t o an a d j a c e n t , i d e n t i c a l c y l i n d e r i n h e a v e m o t i o n , t h e i n d u c e d h e a v e f o r c e s on t h e c y l i n d e r was m e a s u r e d as w e l l . A deep w a t e r as w e l l as a s h a l l o w w a t e r s c e n a r i o u s i n g t h e s u s p e n d e d f a l s e b o t t o m was i n v e s t i g a t e d . The same p a r a m e t e r s w h i c h were v a r i e d i n t h e i n d u c e d s u r g e f o r c e e x p e r i m e n t s were v a r i e d i n t h e same manner i n t h i s e x p e r i m e n t . A l s o , as i n t h e p r e v i o u s s e c t i o n , a c o m p a r i s o n w i t h v a l u e s o b t a i n e d f r o m t h e M a t c h i n g T e c h n i q u e i s c a r r i e d o u t . 5 . 4 . 1 DEEP WATER 5 . 4 . 1 . 1 ADDED MASS COEFFICIENT The r e s u l t s f r o m t h e e x p e r i m e n t s t o d e t e r m i n e t h e i n d u c e d h e a v e added mass c o e f f i c i e n t s , e x p e r i m e n t a l l y and n u m e r i c a l l y , f o r t h e deep w a t e r s c e n a r i o a r e shown i n F i g u r e s 1 0 . 2 6 t o 1 0 . 3 0 , and a summary o f t h e r e s u l t s i s shown i n T a b l e 5 . 4 . 1 . 1 - 1 . 79 TABLE 5 . 4 . 1 . 1 - 1 VALUES OF MINIMUM INDUCED ADDED MASS COEFFICIENTS IN DEEP WATER C y l . S e p . M a t c h i n g T e c h n i q u e E x p e r i m e n t a l R a t i o R a d i i F r e q . M i n . F r c . F r e q . M i n . F r c . E x p . / T h e o . 2 . 0 5 0 . 5 0 0 . 0 4 0 8 1 .05 0 . 0 0 4 7 0 . 1 2 2 . 4 8 0 . 5 0 0 . 0 1 8 6 0 . 7 5 0 . 0 0 2 7 0 . 1 4 3 . 0 0 0 . 5 0 0 . 0 0 6 7 0 . 8 0 0 . 0 0 1 5 0 . 2 2 3 . 4 7 0 . 5 0 - 0 . 0 0 0 3 0 . 7 6 0 . 0 0 1 3 - 4 . 3 3 4 . 0 0 0 . 5 0 - 0 . 0 0 5 2 0 . 7 4 0 . 0 0 0 5 - 0 . 1 0 C o n s i d e r i n g t h a t t h e i n d u c e d h e a v e f o r c e s a r e v e r y l o w i n m a g n i t u d e , i t i s q u i t e s u r p r i s i n g t h a t a d e f i n i t e t r e n d i n t h e i n d u c e d h e a v e d a t a c a n be o b s e r v e d . A t t i m e s t h e m a g n i t u d e o f t h e s e f o r c e s was l e s s t h a n one n e w t o n . The M a t c h i n g T e c h n i q u e p r e d i c t i o n s , a l t h o u g h s i m i l a r i n t r e n d f o r e a c h c y l i n d e r s e p a r a t i o n , v a r y g r e a t l y i n r e l a t i v e m a g n i t u d e . A s t h e c y l i n d e r s e p a r a t i o n i n c r e a s e s , t h e o v e r a l l m a g n i t u d e o f t h e a d d e d mass c o e f f i c i e n t s d e c r e a s e . Howeve r , f o r e a c h o f t h e f i v e c y l i n d e r s e p a r a t i o n s , t h e M a t c h i n g T e c h n i q u e p r e d i c t s t h a t as t h e f r e q u e n c y i n c r e a s e s f r o m w = 0 t h e added mass c o e f f i c i e n t d e c r e a s e s f r o m h i g h v a l u e s v e r y r a p i d l y . These d e c r e a s i n g v a l u e s r e a c h a minimum a t a p p r o x i m a t e l y w = 0 . 5 , w i t h t h e m a g n i t u d e o f t h i s minimum d e p e n d i n g on t h e c y l i n d e r s e p a r a t i o n . I n t h e c a s e o f t h e two c y l i n d e r s v e r y n e a r l y t o u c h i n g one a n o t h e r , B = 2 . 0 5 ( F i g u r e 1 0 . 2 6 ) , t h i s minimum v a l u e i s a ^ = 0 . 0 4 d e c r e a s i n g t o a ^ = - 0 . 0 0 7 f o r a c y l i n d e r s e p a r a t i o n o f B = 4 . 0 ( F i g u r e 1 0 . 3 0 ) . 80 A s t h e f r e q u e n c y i n c r e a s e s f r o m w = 0 . 5 , t h e M a t c h i n g T e c h n i q u e p r e d i c t s t h a t t h e added mass c o e f f i c i e n t s w i l l r i s e f r o m t h e s e minimum v a l u e s and r e a c h an a p p r o x i m a t e s t e a d y s t a t e v a l u e a t a f r e q u e n c y o f w = 2 . 4 . The v a l u e o f t h i s s t e a d y s t a t e v a l u e i s a g a i n , d e p e n d a n t upon t h e c y l i n d e r s e p a r a t i o n ; f o r t h e c a s e o f t h e c y l i n d e r s v e r y n e a r l y t o u c h i n g one a n o t h e r , ( F i g u r e 1 0 . 2 6 ) , t h e v a l u e i s a^= 0 . 0 5 4 , d e c r e a s i n g t o a ^ = 0 . 0 1 2 a t a c y l i n d e r s e p a r a t i o n o f B = 4 . 0 ( F i g u r e 1 0 . 3 0 ) . I t i s i n t e r e s t i n g t o n o t e t h a t t h e M a t c h i n g T e c h n i q u e p r e d i c t s a b e h a v i o u r o f n e g a t i v e added mass c o e f f i c i e n t s o c c u r r i n g a t c e r t a i n f r e q u e n c i e s f o r a c y l i n d e r s e p a r a t i o n o f B = 3 .47 and B = 4 . 0 ( F i g u r e s 1 0 . 2 9 and 1 0 . 3 0 ) . T h i s phenomenon o f n e g a t i v e a d d e d mass w o u l d s u g g e s t a r e g i o n b y w h i c h t h e f l u i d i m p a r t s an a s s i s t i n g f o r c e on t o t h e body d u r i n g i t s m o t i o n . The r e s u l t s o b t a i n e d e x p e r i m e n t a l l y shows t r e n d l i n e s s i m i l a r t o t h o s e p r e d i c t e d b y t h e M a t c h i n g T e c h n i q u e b u t do n o t show s u c h h i g h d e g r e e o f dependency upon t h e c y l i n d e r s e p a r a t i o n as does t h e M a t c h i n g T e c h n i q u e p r e d i c t i o n s . I n a l l f i v e f i g u r e s a s i m i l a r t r e n d p a t t e r n i s n o t i c e a b l e . A s t h e f r e q u e n c y i s i n c r e a s e d f r o m w = 0 , t h e v a l u e s o f t h e added mass c o e f f i c i e n t r a p i d l y d e c r e a s e t o w a r d s a minimum v a l u e . These minimum v a l u e s do v a r y a c c o r d i n g t o t h e s e p a r a t i o n . They v a r y f r o m a h i g h o f a ^ s 0 . 0 0 4 7 o c c u r r i n g a t w s 1 .05 f o r B = 2 . 0 5 , ( F i g u r e 1 0 . 2 6 ) , t o a = 0 . 0 0 0 5 1 o c c u r r i n g a t to = 0 . 7 4 f o r B = 4 . 0 ( F i g u r e 1 0 . 3 0 ) . The f r e q u e n c y a t w h i c h t h e s e l o c a l minimums o c c u r i s a p p r o x i m a t e l y w = 0 . 7 5 compared t o t h e t h e o r e t i c a l p r e d i c t i o n 81 o f t h e minimum v a l u e s o c c u r r i n g a t w = 0 . 5 . The e x p e r i m e n t a l r e s u l t s show t h a t as t h e f r e q u e n c y i n c r e a s e s b e y o n d w = 0 . 7 5 , t h e added mass c o e f f i c i e n t s i n c r e a s e i n m a g n i t u d e , r e a c h i n g a r e l a t i v e maximum b e f o r e d e c r e a s i n g t o w a r d s a ^ = 0 a t t h e h i g h f r e q u e n c y r a n g e o f w > 3 . 5 . The v a l u e o f t h e s e r e l a t i v e p e a k v a l u e s o f t h e added mass c o e f f i c i e n t s r a n g e f r o m a maximum o f a ^ = 0 . 0 1 0 3 1 a t w = 1 .79 f o r a c y l i n d e r s e p a r a t i o n o f B = 2 . 4 8 , ( F i g u r e 1 0 . 2 7 ) , t o a minimum o f a = 0 . 0 0 7 1 8 a t w = 1 .71 f o r a c y l i n d e r s e p a r a t i o n o f B = 4 . 0 ( F i g u r e 1 0 . 3 0 ) . I t i s i n t e r e s t i n g t o n o t e t h a t t h e t h e s e r e l a t i v e maximum e x p e r i m e n t a l v a l u e s do n o t s t e a d i l y d e c r e a s e as t h e c y l i n d e r s e p a r a t i o n i s i n c r e a s e d , b u t seem i n s t e a d s l i g h t l y e r r a t i c . The s p r e a d o f t h e d a t a due t o d i f f e r i n g d r i v i n g f r e q u e n c i e s i s n o t as g r e a t as one m i g h t e x p e c t . A r e a s o n a b l e amount o f s p r e a d i n t h e d a t a does o c c u r , e s p e c i a l l y a t t h e f r e q u e n c y r a n g e o f w = 1 .25 t o w = 3 . 0 , where t h e added mass c o e f f i c i e n t s r i s e t o a r e l a t i v e maximum b e f o r e d e c r e a s i n g t o & z z = 0 . T h i s s p r e a d i n t h e d a t a i s mos t n o t i c e a b l e i n t h e c a s e o f B = 2 . 4 8 ( F i g u r e 1 0 . 2 7 ) . C o m p a r i n g t h e p r e d i c t i o n o f t h e M a t c h i n g T e c h n i q u e t o t h a t o f t h e e x p e r i m e n t a l r e s u l t s shows t h a t t h e e x p e r i m e n t a l v a l u e s do n o t change w i t h t h e c y l i n d e r s e p a r a t i o n as g r e a t l y as t h e M a t c h i n g T e c h n i q u e p r e d i c t i o n s . The e x p e r i m e n t a l r e s u l t s , w h i l e s h o w i n g a s i m i l a r t r e n d i n t h e d a t a , do n o t ma tch t h e M a t c h i n g T e c h n i q u e p r e d i c t i o n s f o r t h e c y l i n d e r s e p a r a t i o n s o f B = 2 . 0 5 , B = 2 . 4 8 and B = 3 . 0 v e r y w e l l ( F i g u r e s 1 0 . 2 6 , 1 0 . 2 7 , and 1 0 . 2 8 ) . Howeve r , f o r 82 t h e c a s e s o f B = 3 .47 and B = 4 . 0 , ( F i g u r e s 1 0 . 2 9 and 1 0 . 3 0 ) , t h e r e s u l t s a r e b e t t e r c o r r e l a t e d . E x c e p t f o r t h e i n s t a n c e s o f t h e n e g a t i v e a d d e d mass c o e f f i c i e n t s , t h e t r e n d i n b o t h e x p e r i m e n t a l r e s u l t s and n u m e r i c a l c a l c u l a t i o n s show a r e l a t i v e l y h i g h d e g r e e o f s i m i l a r i t y w i t h i n t h e f r e q u e n c y r a n g e o f w = 0 t o to = 2 . 0 . A t t h e h i g h e r f r e q u e n c i e s (w > 2 . 0 ) t h e p r e d i c t i o n and t h e e x p e r i m e n t a l d a t a s e p a r a t e t o t h e i r r e s p e c t i v e h i g h f r e q u e n c y s t e a d y s t a t e v a l u e s . 5 . 4 . 1 . 2 DAMPING COEFFICIENTS F i g u r e s 1 0 . 3 1 t o 1 0 . 3 5 show t h e i n d u c e d damp ing c o e f f i c i e n t s o f t h e s i n g l e c y l i n d e r . The r e s u l t s show v e r y l o w v a l u e s and i t i s s u r p r i s i n g t h a t a g e n e r a l t r e n d was d e t e c t e d b y t h e f o r c e t r a n s d u c e r . The v a l u e s o f t h e i n d u c e d heave damp ing c o e f f i c i e n t s a r e r o u g h l y one o r d e r o f m a g n i t u d e l o w e r t h a n t h e i n d u c e d h e a v e added mass c o e f f i c i e n t s . The M a t c h i n g T e c h n i q u e p r e d i c t s t h a t as t h e f r e q u e n c y i n c r e a s e s f r o m w = 0 , t h e i n d u c e d damp ing c o e f f i c i e n t s d e c r e a s e a t an e x p o n e n t i a l r a t e f r o m v e r y h i g h i n i t i a l v a l u e s t o a s t e a d y s t a t e v a l u e o f b = 0 . V e r y l i t t l e d i f f e r e n c e e x i s t s b e t w e e n 22 J e a c h o f t h e f i g u r e s i n d i c a t i n g t h a t t h e b e h a v i o u r o f t h e i n d u c e d damp ing c o e f f i c i e n t s i s n o t v e r y dependan t upon t h e c y l i n d e r s e p a r a t i o n . The o n l y d i f f e r e n c e b e t w e e n e a c h c u r v e i s t h a t t h e r a t e o f d e c r e a s e becomes s l i g h t l y h i g h e r as t h e c y l i n d e r s e p a r a t i o n i s i n c r e a s e d . A n o t h e r d i f f e r e n c e , i s t h a t as t h e 83 c y l i n d e r s e p a r a t i o n i n c r e a s e s t o t h e B = 3 .47 and B = 4 . 0 c a s e , ( F i g u r e s 1 0 . 3 4 and 1 0 . 3 5 ) , t h e i n d u c e d damping c o e f f i c i e n t s become s l i g h t l y n e g a t i v e f o r h i g h e r f r e q u e n c i e s . I n t h e c a s e o f B = 3 . 4 7 , ( F i g u r e 1 0 . 3 4 ) , a t f r e q u e n c i e s b e y o n d w > 1 . 0 , t h e damping c o e f f i c i e n t s a r e n e g a t i v e , and i n t h e c a s e o f B = 4 . 0 , ( F i g u r e 1 0 . 3 5 ) , t h e v a l u e s a r e n e g a t i v e f o r f r e q u e n c i e s b e y o n d u> > 0 . 8 0 . H o w e v e r , i n b o t h c a s e s t h e damp ing c o e f f i c i e n t s s t i l l d e c r e a s e t o w a r d s b = 0 as t h e f r e q u e n c y i n c r e a s e s . The e x p e r i m e n t a l r e s u l t s show v e r y good ag reemen t w i t h t h e n u m e r i c a l p r e d i c t i o n s made b y t h e M a t c h i n g T e c h n i q u e . The r e s u l t s i n d i c a t e h i g h v a l u e s f o r t h e i n d u c e d damping c o e f f i c i e n t f o r l o w f r e q u e n c i e s (w < 0 . 5 ) . These v a l u e s d e c r e a s e a t a r a p i d r a t e t h e M a t c h i n g T e c h n i q u e , t h e f r e q u e n c y a t w h i c h t h e damp ing i s i n c r e a s e d , i n t h e c a s e o f B = 2 . 0 5 , ( F i g u r e 1 0 . 3 1 ) , t h e damp ing c o e f f i c i e n t s r e a c h a v e r y l o w v a l u e a t a f r e q u e n c y o f w = 1 . 2 6 , compared t o t h e c a s e o f B = 4 . 0 , ( F i g u r e 1 0 . 3 5 ) , where t h e damping c o e f f i c i e n t s r e a c h a v e r y l o w v a l u e a t « = 0 . 7 1 . A t f r e q u e n c i e s g r e a t e r t h a n t h e s e , t h e damping c o e f f i c i e n t s r e m a i n v e r y c l o s e t o b = 0 . 0 and n e v e r e x c e e d b = 0 . 0 0 1 f o r e a c h o f t h e f i v e c a s e s . 2 2 2 2 The t r e n d o f t h e e x p e r i m e n t a l r e s u l t s f o l l o w s v e r y c l o s e l y t h e t r e n d p r e d i c t e d b y t h e M a t c h i n g T e c h n i q u e . The u s e o f d i f f e r e n t d r i v i n g a m p l i t u d e s h a s v e r y m i n i m a l a f f e c t on t h e t r e n d o f t h e d a t a . E a c h t e s t shows t h a t t h e r e e x i s t s v i r t u a l l y no d i f f e r e n c e b e t w e e n p o i n t s o f d i f f e r e n t d r i v i n g 2 2 A s p r e d i c t e d b y c o e f f i c i e n t s r e a c h b = 0 becomes l e s s as t h e c y l i n d e r s e p a r a t i o n 22 84 a m p l i t u d e , e s p e c i a l l y f o r t h e h i g h e r f r e q u e n c i e s (w > 1 .25 ) where t h e d a t a l i e s v e r y n e a r t h e b = 0 a x i s . J 22 5 . 4 . 2 SHALLOW WATER 5 . 4 . 2 . 1 ADDED MASS COEFFICIENT P l o t s o f t h e n u m e r i c a l c a l c u l a t i o n s compu ted f r o m t h e M a t c h i n g T e c h n i q u e and t h e e x p e r i m e n t a l r e s u l t s f o r t h e i n d u c e d a d d e d mass c o e f f i c i e n t s i n s h a l l o w w a t e r a r e f o u n d i n F i g u r e s 1 0 . 3 6 t o 1 0 . 4 0 . The r e s u l t s a r e s u m m a r i z e d on T a b l e 5 . 4 . 2 . 1 - 1 . TABLE 5 . 4 . 2 . 1 - 1 VALUES OF MINIMUM INDUCED ADDED MASS COEFFICIENTS IN SHALLOW WATER C y l . S e p . M a t c h i n g T e c h n i q u e E x p e r i m e n t a l R a t i o R a d i i F r e q . M i n . F r c . F r e q . M i n . F r c . E x p . / T h e o . 2 . 0 5 0 . 6 0 0 . 0 6 2 4 0 . 4 7 0 . 0 0 4 5 0 . 0 7 2 . 4 8 0 . 5 0 0 . 0 1 6 2 0 . 7 4 0 . 0 0 4 2 0 . 2 6 3 . 0 0 0 . 5 0 - 0 . 0 1 6 9 0 . 4 8 0 . 0 0 0 5 - 0 . 0 3 3 . 4 7 0 . 4 0 - 0 . 0 3 6 5 0 . 7 7 0 . 0 0 0 5 - 0 . 0 1 4 . 0 0 0 . 4 0 - 0 . 0 5 1 5 0 . 7 6 0 . 0 0 0 6 - 0 . 0 1 The f i g u r e s o f t h e i n d u c e d h e a v e added mass c o e f f i c i e n t s o f a s i n g l e c y l i n d e r i n a s h a l l o w w a t e r e n v i r o n m e n t a r e a r r a n g e d i n t h e same o r d e r as i n t h e deep w a t e r c a s e . B o t h t h e n u m e r i c a l and e x p e r i m e n t a l r e s u l t s f o l l o w v e r y c l o s e l y t h e r e s u l t s o f t h e deep 85 w a t e r c a s e , b u t t h e m a g n i t u d e o f t h e s h a l l o w w a t e r r e s u l t s a r e h i g h e r t h a n t h o s e o f t h e deep w a t e r c a s e . The M a t c h i n g T e c h n i q u e r e s u l t s a r e , once a g a i n , s i m i l a r i n t r e n d f o r e a c h o f t h e f i v e c y l i n d e r s e p a r a t i o n s , w i t h t h e m a g n i t u d e , o f t h e r e s u l t s d e c r e a s i n g as t h e c y l i n d e r s e p a r a t i o n i n c r e a s e s . The M a t c h i n g T e c h n i q u e t r e n d i n t h e s h a l l o w w a t e r c a s e i s t h e same as t h e deep w a t e r c a s e ; t h e added mass c o e f f i c i e n t s d e c r e a s e r a p i d l y f r o m h i g h i n i t i a l v a l u e s as t h e f r e q u e n c y i n c r e a s e s f r o m w = 0 , t h i s d e c r e a s i n g added mass r e a c h e s a minimum a t a p p r o x i m a t e l y w = 0 . 5 , w i t h t h e m a g n i t u d e o f t h i s minimum d e p e n d i n g on t h e c y l i n d e r s e p a r a t i o n . F o r t h e B = 2 . 0 8 c a s e , ( F i g u r e 1 0 . 3 6 ) , t h e minimum a d d e d mass c o e f f i c i e n t i s a ^ = 0 . 0 6 2 4 , d e c r e a s i n g t o a ^ = - 0 . 0 5 1 5 f o r t h e B = 4 . 0 c a s e ( F i g u r e 1 0 . 4 0 ) . A l s o , as i n t h e deep w a t e r c a s e , t h e M a t c h i n g T e c h n i q u e p r e d i c t s a n o c c u r r e n c e o f n e g a t i v e added mass c o e f f i c i e n t s w h i c h o c c u r i n t h e r e g i o n o f t h e w = 0 . 5 min imum. A s t h e f r e q u e n c y i n c r e a s e s b e y o n d w = 0 . 5 , t h e added mass c o e f f i c i e n t s r a p i d l y i n c r e a s e t o w a r d s a s t e a d y s t a t e v a l u e w h i c h i s d e p e n d a n t upon t h e c y l i n d e r s e p a r a t i o n ; r a n g i n g f r o m a h i g h o f a = 0 . 1 0 7 f o r t h e B = 2 . 0 8 c a s e ( F i g u r e 1 0 . 3 6 ) t o a l o w o f a £ 22 ° 22 0 . 0 1 2 7 f o r t h e B = 4 . 0 c a s e ( F i g u r e 1 0 . 4 0 ) . The e x p e r i m e n t a l d a t a f o r t h e s h a l l o w w a t e r t e s t s a r e v e r y s i m i l a r t o t h o s e o b t a i n e d i n t h e deep w a t e r t e s t s . The s h a l l o w w a t e r d a t a i s , h o w e v e r , s l i g h t l y h i g h e r i n m a g n i t u d e t h a n t h e deep w a t e r r e s u l t s . As i n t h e deep w a t e r c a s e , t h e d a t a s u g g e s t s a b e h a v i o u r i n w h i c h t h e added mass c o e f f i c i e n t s d e c r e a s e as t h e 86 f r e q u e n c y i s i n c r e a s e d f r o m w = 0 . T h i s d e c r e a s i n g added mass r e a c h e s a minimum n e a r a = 0 a t a f r e q u e n c y w i t h i n t h e r a n g e w = 0 . 4 5 t o <j = 0 . 8 0 . A f t e r t h i s p o i n t t h e e x p e r i m e n t a l d a t a s u g g e s t s t h a t t h e added mass c o e f f i c i e n t i n c r e a s e s i n m a g n i t u d e t o w a r d s a r e l a t i v e maximum b e f o r e d e c r e a s i n g t o w a r d s a = 0 as t h e f r e q u e n c y i n c r e a s e s b e y o n d w > 3 . 2 5 . The r e l a t i v e m a g n i t u d e o f t h e s e minimum and maximum v a l u e s do depend on t h e c y l i n d e r s e p a r a t i o n . F o r t h e c a s e o f l e a s t s e p a r a t i o n , B = 2 . 0 8 ( F i g u r e 1 0 . 3 6 ) , t h e minimum added mass c o e f f i c i e n t i s a = 0 . 0 0 4 5 3 22 o c c u r r i n g a t a f r e q u e n c y o f w = 0 . 4 7 and t h e r e l a t i v e maximum i s a 2 2 ~ 0 - 0 1 4 o c c u r r i n g a t w = 1 . 1 9 , w h i l e f o r t h e c a s e o f maximum c y l i n d e r s e p a r a t i o n , B = 4 . 0 ( F i g u r e 1 0 . 4 0 ) t h e minimum v a l u e o f a d d e d mass i s a — 0 . 0 0 0 6 o c c u r r i n g a t co — 0 . 7 6 , and t h e r e l a t i v e maximum i s a ^ = 0 . 0 0 8 o c c u r r i n g a t w = 1 . 7 . J u s t as i n t h e deep w a t e r c a s e , t h e r e i s l i t t l e s p r e a d i n t h e d a t a due t o d i f f e r i n g d r i v i n g f r e q u e n c y . The a r e a c o n t a i n i n g t h e g r e a t e s t amount o f e x p e r i m e n t a l r a n g e o c c u r s i n t h e a r e a o f t h e r e l a t i v e maximum. T h i s s p r e a d i n t h e d a t a s u g g e s t s t h a t f o r t h e h i g h e r d r i v i n g a m p l i t u d e s , t h e r e l a t i v e maximum added mass c o e f f i c i e n t s w i l l be s l i g h t l y h i g h e r t h a n f o r t h e l o w e r d r i v i n g a m p l i t u d e s . C o m p a r i n g t h e r e s u l t s o f t h e M a t c h i n g T e c h n i q u e p r e d i c t i o n t o t h e e x p e r i m e n t a l r e s u l t s shows t h e same g e n e r a l c o n c l u s i o n s as t h e deep w a t e r c a s e . The e x p e r i m e n t a l r e s u l t s do n o t s u g g e s t s u c h a g r e a t v a r i a t i o n i n m a g n i t u d e due t o i n c r e a s i n g c y l i n d e r s e p a r a t i o n as does t h e n u m e r i c a l r e s u l t s . The o v e r a l l t r e n d o f t h e 87 e x p e r i m e n t a l r e s u l t s do show a h i g h d e g r e e o f s i m i l a r i t y t o t h e t r e n d o f t h e n u m e r i c a l r e s u l t s , e s p e c i a l l y w i t h i n t h e r a n g e o f to = 0 t o to = 2 . 0 , t h e same r a p i d d e c r e a s e t o a m i n i m a l v a l u e b e f o r e i n c r e a s i n g once a g a i n i s v i s i b l e i n t h e e x p e r i m e n t a l and n u m e r i c a l r e s u l t s . C o m p a r i s o n o f t h e r e s u l t s o f t h e B = 3 . 0 , B= 3 . 4 8 , and B= 4 . 0 c y l i n d e r s e p a r a t i o n ( F i g u r e 1 0 . 3 8 , 1 0 . 3 9 and 1 0 . 4 0 ) show good ag reemen t w i t h i n t h i s r a n g e e x c e p t f o r t h e r a n g e o f n e g a t i v e added mass c o e f f i c i e n t s p r e d i c t e d b y t h e M a t c h i n g T e c h n i q u e . A t t h e h i g h e r f r e q u e n c i e s , to > 2 . 0 , t h e t h e o r y and e x p e r i m e n t a l r e s u l t s seem t o d i v e r g e away f r o m one a n o t h e r . C o m p a r i s o n b e t w e e n t h e r e s u l t s o f t h e deep w a t e r and s h a l l o w w a t e r s c e n a r i o s show v e r y l i t t l e d i f f e r e n c e b e t w e e n t h e two c a s e s e x c e p t i n r e l a t i v e m a g n i t u d e s . The t r e n d o f t h e n u m e r i c a l p r e d i c t i o n s and t h e e x p e r i m e n t a l r e s u l t s a r e v i r t u a l l y i d e n t i c a l . I n t h e s h a l l o w w a t e r c a s e t h e M a t c h i n g T e c h n i q u e p r e d i c t i o n s show h i g h e r s e n s i t i v i t y t o c y l i n d e r s e p a r a t i o n , and l a r g e r h i g h f r e q u e n c y s t e a d y s t a t e v a l u e s t h a n i n t h e deep w a t e r c a s e . The e x p e r i m e n t a l r e s u l t s a l s o show h i g h e r r e s u l t s i n t h e s h a l l o w w a t e r c a s e t h a n i n t h e deep w a t e r c a s e . I n t e rms o f t h e r e l a t i v e maximum added mass c o e f f i c i e n t s f o r t h e f r e q u e n c i e s n e a r to = 1 . 7 5 , t h e s h a l l o w w a t e r r e s u l t s a r e a p p r o x i m a t e l y 50% h i g h e r f o r t h e l o w c y l i n d e r s e p a r a t i o n (B = 2 . 0 ) d e c r e a s i n g t o a p p r o x i m a t e l y 15% h i g h e r f o r t h e l a r g e c y l i n d e r s e p a r a t i o n (B = 4 . 0 ) . 88 5 . 4 . 2 . 2 DAMPING COEFFICIENTS F i g u r e s 1 0 . 4 1 t o 1 0 . 4 5 show t h e r e s u l t s o f t h e i n d u c e d h e a v e damp ing c o e f f i c i e n t s o f a s i n g l e c y l i n d e r i n a s h a l l o w w a t e r s c e n a r i o . The t r e n d o f t h e s h a l l o w w a t e r d a t a i s v i r t u a l l y t h e same as t h e t r e n d o f t h e d a t a o b t a i n e d i n t h e deep w a t e r t e s t s b u t a r e h i g h e r i n m a g n i t u d e . The M a t c h i n g T e c h n i q u e p r e d i c t s t h a t as t h e f r e q u e n c y i n c r e a s e s f r o m w = 0 , t h e i n d u c e d damping c o e f f i c i e n t s d e c r e a s e a t a n e x p o n e n t i a l r a t e t o w a r d s a s t e a d y s t a t e v a l u e o f b = 0 . The b e h a v i o u r o f t h e c u r v e s i s s l i g h t l y dependan t upon t h e c y l i n d e r s e p a r a t i o n , as t h e c y l i n d e r s e p a r a t i o n i n c r e a s e s t h e s t e e p n e s s o f t h e c u r v e s i n c r e a s e and t h e f r e q u e n c y a t w h i c h t h e c u r v e r e a c h e s t h e b = 0 d e c r e a s e s . F o r t h e c a s e o f t h e c y l i n d e r s e p a r a t i o n b e i n g B = 2 . 0 8 ( F i g u r e 1 0 . 4 1 ) , t h e M a t c h i n g T e c h n i q u e p r e d i c t s t h a t t h e damp ing c o e f f i c i e n t w i l l be e s s e n t i a l l y z e r o a t f r e q u e n c i e s b e y o n d w > 1 . 5 ; f o r t h e c a s e o f B = 4 . 0 ( F i g u r e 1 0 . 4 5 ) , t h i s f r e q u e n c y r a n g e d e c r e a s e s t o w > 0 . 8 . F o r t h e c a s e s where B = 3 . 0 , B = 3 . 4 8 , and B = 4 . 0 ( F i g u r e s 1 0 . 4 3 , 1 0 . 4 4 , and 1 0 . 4 5 ) , t h e n u m e r i c a l p r e d i c t i o n s become n e g a t i v e as t h e d e c r e a s i n g damp ing c o e f f i c i e n t s p a s s t h r o u g h t h e u> = 0 a x i s . H o w e v e r , as i n t h e deep w a t e r c a s e , as t h e f r e q u e n c y i n c r e a s e s t h e damp ing c o e f f i c i e n t s a p p r o a c h w = 0 . The e x p e r i m e n t a l r e s u l t s show good c o r r e l a t i o n w i t h t h e n u m e r i c a l p r e d i c t i o n s , The r e s u l t s behave i n t h e same manner as i n t h e deep w a t e r c a s e - r a p i d l y d e c r e a s i n g damp ing c o e f f i c i e n t s 89 as t h e f r e q u e n c y i s i n c r e a s e d f r o m co = 0 . The e x p e r i m e n t a l l y o b t a i n e d damp ing c o e f f i c i e n t s do n o t show g r e a t v a r i a t i o n due t o c y l i n d e r s e p a r a t i o n , t h e m a g n i t u d e o f t h e c o e f f i c i e n t s r e m a i n s v i r t u a l l y u n c h a n g e d f o r e a c h o f t h e f i v e c y l i n d e r s e p a r a t i o n s . The f r e q u e n c y a t w h i c h t h e damping c o e f f i c i e n t s r e a c h = 0 does d e c r e a s e s l i g h t l y as t h e c y l i n d e r s e p a r a t i o n i n c r e a s e s . I n t h e c a s e o f B = 2 . 0 8 ( F i g u r e 1 0 . 4 1 ) , t h i s f r e q u e n c y i s w = 1 . 2 7 , w h i l e f o r t h e c a s e o f B = 4 . 0 ( F i g u r e 1 0 . 4 5 ) , t h i s f r e q u e n c y i s co = 0 . 7 6 . A t f r e q u e n c i e s b e y o n d t h e s e v a l u e s t h e damp ing c o e f f i c i e n t s r e m a i n v e r y n e a r b = 0 . J 22 The u s e o f d i f f e r e n t d r i v i n g a m p l i t u d e s h a s l i t t l e e f f e c t on t h e s p r e a d o f t h e d a t a . O n l y a t l o w f r e q u e n c i e s o f o s c i l l a t i o n (co < 0 . 5 ) c a n one see any d e v i a t i o n f r o m a g e n e r a l t r e n d due t o d i f f e r e n t d r i v i n g f r e q u e n c i e s . A t t h e h i g h e r f r e q u e n c i e s (w > 1 . 0 ) , t h e r e s p o n s e i s i n s e n s i t i v e t o t h e d r i v i n g f r e q u e n c y . C o m p a r i n g t h e p l o t s o f t h e deep w a t e r and t h e s h a l l o w w a t e r r e s u l t s shows t h a t f o r f r e q u e n c i e s b e l o w w = 1 . 0 , t h e s h a l l o w w a t e r e n v i r o n m e n t p r o d u c e s damping c o e f f i c i e n t s w h i c h a r e a p p r o x i m a t e l y f o u r t i m e s as g r e a t as t h o s e o f t h e deep w a t e r e n v i r o n m e n t . F o r f r e q u e n c i e s h i g h e r t h a n co = 1 . 0 , t h e r e s u l t s a r e e s s e n t i a l l y t h e same (b = 0) f o r b o t h c o n f i g u r a t i o n s . 9 0 5 . 5 COMPARISON OF EXPERIMENTAL VALUES AND THEORETICAL PREDICTIONS A d i s c u s s i o n on t h e p o s s i b l e c a u s e s o f d i s c r e p a n c i e s b e t w e e n e x p e r i m e n t a l v a l u e s and t h e o r e t i c a l p r e d i c t i o n s i s p r e s e n t e d i n o r d e r t o b e t t e r u n d e r s t a n d t h e l i m i t a t i o n s o f t h e t h e o r y and t h e u n c e r t a i n t i e s w i t h i n t h e e x p e r i m e n t a l r e s u l t s . R e s u l t s f r o m t h e h y d r o d y n a m i c c o e f f i c i e n t s o f compound c y l i n d e r e x p e r i m e n t s , t h e i n d u c e d s u r g e f o r c e s on a c y l i n d e r e x p e r i m e n t s , and t h e d e t e r m i n a t i o n o f t h e i n d u c e d h e a v e h y d r o d y n a m i c c o e f f i c i e n t s o f a c y l i n d e r a r e d i s c u s s e d i n t h i s s e c t i o n . 5 . 5 . 1 L IM ITS IN THE MATCHING TECHNIQUE THEORETICAL MODEL When t h e t h e o r i e s were f i r s t i n t r o d u c e d , some a s s u m p t i o n s were made i n o r d e r t o s i m p l i f y t h e p r o b l e m . These i n c l u d e d : a ) The f l u i d was t r e a t e d as an i d e a l i n v i s c i d f l u i d , w h i c h c o n t a i n s no v o r t i c e s . b ) The t h e o r y i s s i m p l i f i e d u s i n g a l i n e a r i z a t i o n t e c h n i q u e w h i c h assumes s m a l l a m p l i t u d e s o f m o t i o n . c ) I n e a c h o f t h e t h r e e c o n d u c t e d e x p e r i m e n t s , t h e w a l l s o f t h e t a n k a r e assumed t o be s u f f i c i e n t l y f a r away as t o n o t a f f e c t t h e r e s u l t s . 91 d) I n t h e c a s e o f t h e i n d u c e d s i d e f o r c e s and t h e i n d u c e d h e a v e h y d r o d y n a m i c c o e f f i c i e n t s on s i n g l e c y l i n d e r i n a s h a l l o w w a t e r t a n k , t h e b o t t o m o f t h e t a n k i s assumed r i g i d w i t h no c r o s s f l o w t a k i n g p l a c e and t h e l e n g t h o f t h e t a n k i s assumed t o be i n f i n i t e . e ) I n t h e c a s e o f t h e i n d u c e d h e a v e h y d r o d y n a m i c c o e f f i c i e n t s on a s i n g l e c y l i n d e r , t h e t h e o r e t i c a l mode l o n l y c o n s i d e r s t h e i n i t i a l i n c o m i n g wave o n l y and n o t any r e f l e c t i o n s r e t u r n i n g t o t h e c y l i n d e r . The t h e o r e t i c a l mode l s do n o t a c c o u n t f o r t h e v i s c o s i t y o f t h e f l u i d and h e n c e t h e p r e s e n c e o f v o r t i c e s s h e d f r o m t h e c y l i n d e r m o d e l s i s n o t t a k e n i n t o a c c o u n t . I n o r d e r t o i n c l u d e t h e s e e f f e c t s , a c o m p l i c a t e d v o r t e x r i n g mode l must be i n c o r p o r a t e d i n t o t h e t h e o r y . From t h e n o n - d i m e n s i o n a l a n a l y s i s , i t was shown t h a t t h e f l o w s e p a r a t i o n e f f e c t s w i l l be much l e s s t h a n t h e d i f f r a c t i o n e f f e c t s f o r l a r g e d i a m e t e r t o wave ( D / L ) r a t i o s . T h e r e f o r e , v i s c o u s e f f e c t s w o u l d be e x p e c t e d t o be l o w e r as t h e f r e q u e n c y i n c r e a s e s . V e n u g o p a l (1984) c o n d u c t e d f l o w v i s u a l i z a t i o n t e s t s on h e a v i n g c y l i n d e r s and c o n c l u d e d t h a t v o r t e x s h e d d i n g does o c c u r a t t h e c o r n e r s o f t h e c y l i n d e r . T h i s v o r t e x s h e d d i n g i n t r o d u c e s v i s c o u s damping i n a d d i t i o n t o t h e wave m a k i n g damp ing m o d e l e d i n t h e t h e o r y . The p r e s e n c e o f v i s c o u s damp ing w h i c h i s n o t m o d e l e d i n t h e t h e o r e t i c a l p r e d i c t i o n s w i l l c a u s e d i s c r e p a n c i e s b e t w e e n t h e o r y and e x p e r i m e n t a l r e s u l t s . The l i n e a r i z a t i o n a s s u m p t i o n assumes t h a t t h e a m p l i t u d e o f 92 m o t i o n i s s m a l l w i t h r e s p e c t t o t h e c y l i n d e r d i a m e t e r . These s m a l l r a t i o s when r a i s e d t o o r d e r s g r e a t e r t h a n u n i t y a r e assumed t o be n e g l i g i b l e . E x c e p t f o r t h e c a s e o f t h e i n d u c e d s u r g e f o r c e s , t h e e x p e r i m e n t a l r e s u l t s show t h a t t h i s l i n e a r i z a t i o n a s s u m p t i o n i s v a l i d , e s p e c i a l l y i n t h e h i g h e r f r e q u e n c y r a n g e co > 1 . 0 . A l l t h e t r e n d l i n e s f o r t h e v a r i o u s a m p l i t u d e s o f m o t i o n show l i t t l e s e p a r a t i o n i n most o f t h e c o n d u c t e d e x p e r i m e n t s . I n t h e c a s e o f t h e i n d u c e d s u r g e f o r c e s e x p e r i m e n t s , a c l e a r s e p a r a t i o n b e t w e e n a m p l i t u d e s o f m o t i o n i s v i s i b l e , e s p e c i a l l y f o r t h e l o w e r f r e q u e n c i e s , w < 1 . 5 . T h i s w o u l d s u g g e s t t h a t t h e l i n e a r i z a t i o n a s s u m p t i o n i s n o t n e c e s s a r i l y v a l i d i n p r e d i c t i n g i n d u c e d s u r g e f o r c e s upon an a d j a c e n t c y l i n d e r . I n t h e p r e d i c t i o n o f t h e v a r i o u s r e s u l t s o f t h e t h r e e i n v e s t i g a t i o n s c o n d u c t e d i n t h i s t h e s i s , t h e n u m e r i c a l mode l assumes t h e t a n k w a l l s a r e s u f f i c i e n t l y f a r away so t h a t t h e y have no e f f e c t on t h e r e s u l t s . I n t h e e x p e r i m e n t s , h o w e v e r , t h e w a l l s we re o b s e r v e d t o r e f l e c t t h e r a d i a t i n g waves d i s s i p a t i n g f r o m t h e c y l i n d e r b a c k t o w a r d s t h e c y l i n d e r . T h i s h a s t h e e f f e c t o f a l t e r i n g t h e m e a s u r e 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 o r i n d u c e d s i d e f o r c e s . A n a b s o r b i n g b e a c h made o f an a b s o r b i n g " h o r s e h a i r " m a t e r i a l was u s e d t o m i n i m i z e t h e s e e f f e c t s . Ma ts o f t h i s " h o r s e h a i r " m a t e r i a l were l i n e d a l o n g t h e t a n k w a l l s i n t h e v i c i n i t y o f t h e e x p e r i m e n t s . T h i s m a t e r i a l h e l p e d i n t h e a b s o r p t i o n o f i n c o m i n g w a v e s , h o w e v e r , i t was o b s e r v e d t h a t t h e r e was s t i l l some r e f l e c t i o n o f waves b a c k t o w a r d s t h e c y l i n d e r , h e n c e t h e t o t a l e l i m i n a t i o n o f t h e " w a l l e f f e c t " i n t h e t o w i n g t a n k i s v i r t u a l l y 93 i m p o s s i b l e . I n t h e e x p e r i m e n t s t o d e t e r m i n e t h e i n d u c e d s i d e f o r c e s and t h e i n d u c e d h e a v e h y d r o d y n a m i c c o e f f i c i e n t s o f a s i n g l e c y l i n d e r i n a s h a l l o w w a t e r e n v i r o n m e n t , t h e s h a l l o w w a t e r f r ame was o b s e r v e d t o o s c i l l a t e as t h e e x p e r i m e n t s were t a k i n g p l a c e . T h i s s h a l l o w w a t e r f rame c o n s t r u c t e d f r o m s h e e t s o f p l y w o o d was s u s p e n d e d i n t h e deep w a t e r t o w i n g t a n k ; t h e r e f o r e , a p o s s i b l e t r a n s f e r o f e n e r g y f r o m t h e s h a l l o w w a t e r f rame t o t h e deep w a t e r b a s i n w h i l e t h e e x p e r i m e n t s were b e i n g c o n d u c t e d w o u l d r e s u l t i n i n a c c u r a t e r e a d i n g s o f t h e i n d u c e d added mass and damping c o e f f i c i e n t s o r i n d u c e d s i d e f o r c e s . I n s p e c t i o n o f t h e r e s u l t s show t h a t w h i l e mos t o f t h e s h a l l o w w a t e r r e s u l t s a r e h i g h e r t h a n t h o s e o f t h e deep w a t e r c a s e f o r most f r e q u e n c i e s ; t h e r e s u l t s o f t h e s h a l l o w w a t e r t e s t s a r e p r o b a b l y l o w e r i n t h i s s u s p e n d e d s h a l l o w w a t e r t a n k t h a n i f t h e e x p e r i m e n t s were p e r f o r m e d i n an a c t u a l s h a l l o w w a t e r t a n k . I n a d d i t i o n t o e n e r g y b e i n g t r a n s f e r r e d t h r o u g h t h e b o t t o m o f t h e t a n k , t h i s s u s p e n d e d s h a l l o w w a t e r t a n k was l i m i t e d i n s i z e t o a p p r o x i m a t e l y 9 . 7 c y l i n d e r d i a m e t e r s . T h i s l i m i t e d s i z e i n l e n g t h may a l s o c o n t r i b u t e t o e n e r g y b e i n g l o s t i n t o t h e deep w a t e r b a s i n t h r o u g h t h e ends o f t h e s h a l l o w w a t e r t a n k . i I n t h e c a s e o f t h e i n d u c e d h e a v e h y d r o d y n a m i c c o e f f i c i e n t s on a s i n g l e c y l i n d e r t h e M a t c h i n g T e c h n i q u e i s s i m p l i f i e d t o o n l y c o n s i d e r t h e f i r s t i n c o m i n g p r e s s u r e wave and does n o t t a k e i n t o c o n s i d e r a t i o n any s u b s e q u e n t r e f l e c t i o n s i n p r e s s u r e w a v e s . T h i s h a s t h e e f f e c t o f n o t a c c u r a t e l y p r e d i c t i n g t h e a c t u a l v a l u e o f 94 t h e i n c o m i n g p r e s s u r e wave and h e n c e , t h e a c t u a l h y d r o d y n a m i c c o e f f i c i e n t s . T h i s p r o b l e m becomes more a c u t e as t h e s e p a r a t i o n b e t w e e n t h e c y l i n d e r s i s d e c r e a s e d . I n t h e c a s e o f l o w c y l i n d e r s e p a r a t i o n s , one w o u l d e x p e c t t h a t many r e f l e c t i o n s o f waves o c c u r b e t w e e n t h e two c y l i n d e r s and t h e m o d e l i n g o f o n l y one r e f l e c t i o n w i l l p r o b a b l y p r o v i d e i n a c c u r a t e r e s u l t s . I n s p e c t i o n o f t h e i n d u c e d h e a v e h y d r o d y n a m i c c o e f f i c i e n t d a t a shows t h a t a t t h e l o w e r c y l i n d e r s e p a r a t i o n s (B = 2 . 0 , B = 2 . 5 and B = 3 . 0 ) t h e r e s u l t s o f t h e e x p e r i m e n t a l r e s u l t s and t h e t h e o r e t i c a l p r e d i c t i o n s i s n o t v e r y g o o d . The r e s u l t s o f t h e M a t c h i n g T e c h n i q u e t h e o r y a l s o p r e d i c t n e g a t i v e a d d e d mass a n d damp ing c o e f f i c i e n t s w h i c h do n o t ma tch t h e e x p e r i m e n t a l r e s u l t s . These n e g a t i v e p r e d i c t i o n s c o u l d be a t t r i b u t e d t o t h e f a c t t h a t s u b s e q u e n t r e f l e c t i o n s o f t h e d i s s i p a t e d p r e s s u r e wave i s n o t m o d e l e d i n t h e t h e o r y . 5 . 5 . 2 L IM ITS IN EXPERIMENTAL RESULTS E x p e r i m e n t a l o r p r o c e d u r a l e r r o r s w i l l a l s o c a u s e d i s c r e p a n c i e s b e t w e e n t h e o r e t i c a l p r e d i c t i o n s and e x p e r i m e n t a l r e s u l t s . T h e s e e r r o r s c o u l d be c a u s e d b y f a u l t y o r i n a c c u r a t e t r a n s d u c e r s , f l e x i b l e members i n t h e s y s t e m , p o o r r e s o l u t i o n i n t h e d a t a a c q u i s i t i o n e q u i p m e n t , o r f a u l t s i n t h e d a t a a n a l y s i s s o f t w a r e . A p r o b l e m o f t e n e n c o u n t e r e d d u r i n g t h e e x p e r i m e n t a l t e s t i n g was t h e l i m i t e d a b i l i t y o f t h e dynamometer t o r e s o l v e t h e s m a l l 95 a m p l i t u d e f o r c e s a t t h e l o w f r e q u e n c i e s o f o s c i l l a t i o n (w < 1 . 0 ) . T h i s p r o b l e m c o u l d be o b s e r v e d b y i n s p e c t i n g t h e raw d a t a t r a c e a t t h e l o w f r e q u e n c i e s w h i c h showed a h i g h p r o p o r t i o n o f s p i k e s and g l i t c h e s i n t h e d a t a as compared t o d a t a c o l l e c t e d a t t h e h i g h e r f r e q u e n c i e s w h i c h seemed t o be b e t t e r b e h a v e d . I t w o u l d be b e t t e r t o h a v e two dynamometer u n i t s , one t o measure t h e l o w r a n g e o f f o r c e s and a n o t h e r one t o measu re t h e h i g h e r r a n g e o f f o r c e s . The e x i s t i n g 2200 Newton dynamometer i s s u f f i c i e n t t o measu re a l l t h e h i g h f o r c e s w h i l e a s m a l l e r u n i t o f p e r h a p s 150 Newtons w o u l d be b e t t e r s u i t e d t o measure t h e h y d r o d y n a m i c f o r c e s a t t h e l o w f r e q u e n c y r a n g e (w < 1 . 0 ) . As m e n t i o n e d p r e v i o u s l y , t h e a p p a r a t u s u s e d t o h o l d t h e m o t i o n g e n e r a t i o n s y s t e m and c y l i n d e r i n p l a c e was o b s e r v e d t o f l e x d u r i n g some o f t h e e x p e r i m e n t s . T h i s phenomenon was mos t n o t i c e a b l e d u r i n g t e s t s t o d e t e r m i n e t h e h e a v e h y d r o d y n a m i c c o e f f i c i e n t s o f a t r i p l e c y l i n d e r a t f r e q u e n c i e s o f m o t i o n i n e x c e s s o f w = 1 . 5 . E l a s t i c i t y o f t h e c y l i n d e r mode l and f a s t e n i n g s c r e a t e r e l a t i v e m o t i o n s b e t w e e n t h e c y l i n d e r and t h e m o t i o n g e n e r a t o r . T h i s r e l a t i v e m o t i o n may c r e a t e t h e added mass and damp ing r e s u l t s t o be i n a c c u r a t e a t t h e h i g h e r f r e q u e n c y r a n g e . D e t e r m i n a t i o n o f t h e r e l a t i v e p h a s e d i f f e r e n c e b e t w e e n t h e v a r i o u s d a t a c o l l e c t i o n c h a n n e l s i s c r i t i c a l i n o r d e r t o c o r r e c t l y measu re t h e v a r i o u s h y d r o d y n a m i c c o e f f i c i e n t s . T e s t s were c o n d u c t e d on t h e equ ipmen t i n o r d e r t o b e s t d e t e r m i n e t h e e x t e n t o f t h e e r r o r s w i t h i n t h e d a t a a n a l y s i s e q u i p m e n t . A h e a v e 96 h y d r o d y n a m i c t e s t c o n d u c t e d o u t o f t h e w a t e r was c a r r i e d o u t t o s e e w h i c h s e t s o f f i l t e r s s h o u l d b e s t be u s e d i n t h e s i g n a l c o n d i t i o n e r d u r i n g a l l t h e d a t a c o l l e c t i o n . The r e s u l t s a r e shown on F i g u r e 1 0 . 4 6 and d e t a i l s c o n c e r n i n g t h e t e s t c a n be f o u n d i n A p p e n d i x A , S e c t i o n 6 . 2 . 3 . 1 . 1 . The r e s u l t s show t h a t l a r g e e r r o r s c a n i n a d v e r t e n t l y be i n t r o d u c e d b y t h e i m p r o p e r s e l e c t i o n o f l o w p a s s f i l t e r s . The r e s u l t s a l s o show t h a t t o t a l e l i m i n a t i o n o f t h e p h a s e d i f f e r e n c e b e t w e e n t h e v a r i o u s c h a n n e l s i s n o t p o s s i b l e e s p e c i a l l y a t a v e r y l o w f r e q u e n c y r a n g e (w < 0 . 3 0 ) . I n c l u s i o n i n t h e d a t a a n a l y s i s s o f t w a r e o f a r o u t i n e t o d e t e r m i n e t h e r e l a t i v e p h a s e d i f f e r e n c e b e t w e e n e a c h o f t h e c h a n n e l s h e l p s i n t h e r e d u c t i o n o f t h e e r r o r . Howeve r , t h e f o r m u l a t i o n t o mode l t h e r e l a t i v e p h a s e l a g i s a l i n e a r e q u a t i o n w h i c h may i m p r o p e r l y mode l t h e p h a s e l a g a t t h e s e v e r y l o w f r e q u e n c i e s . A l s o , s i n c e t h e added mass i s p r o p o r t i o n a l t o t h e c o s i n e o f t h e p h a s e a n g l e and t h e damp ing c o e f f i c i e n t i s p r o p o r t i o n a l t o t h e s i n e o f t h e p h a s e a n g l e , a s m a l l e r r o r i n t h e t h e p h a s e a n g l e w i l l h a v e a g r e a t e r e f f e c t on t h e added mass c o e f f i c i e n t t h a n t h e damping c o e f f i c i e n t . 5 . 5 . 2 . 1 ANALYSIS OF EXPERIMENTAL ERROR As a f i r s t o r d e r a n a l y s i s o f t h e m a g n i t u d e o f e r r o r t h a t may h a v e b e e n a c q u i r e d i n t h e s e s e t s o f e x p e r i m e n t s , a u s e o f p a r t i a l d i f f e r e n t i a l c a l c u l u s c a n be u s e d . I f a q u a n t i t y , U , i s t h e f u n c t i o n o f s e v e r a l m e a s u r e d q u a n t i t i e s , x , y , and z ; U = f ( x , y , z ) 97 t h e n t h e m a g n i t u d e o f t h e e r r o r c a n be e x p r e s s e d u s i n g t h e t o t a l d i f f e r e n t i a l as f o l l o w s : dU T — dx + — j — dy + —— dz 3x ay J dz F i v e s e p a r a t e p a r a m e t e r s h a v e b e e n p l o t t e d as a f u n c t i o n o f 2 t h e n o n - d i m e n s i o n a l f r e q u e n c y , — — . These a r e t h e added mass and damp ing c o e f f i c i e n t s o f a t r i p l e c y l i n d e r i n h e a v e m o t i o n m e a s u r e d w i t h t h e 2200N. f o r c e dynamometer , t h e i n d u c e d s u r g e f o r c e on a s i n g l e c y l i n d e r m e a s u r e d w i t h t h e u n i v e r s a l s h e a r beam, and t h e i n d u c e d h e a v e added mass and damp ing c o e f f i c i e n t s o f a s i n g l e c y l i n d e r u s i n g t h e u n i v e r s a l s h e a r beam. The t h r e e p l o t t e d p a r a m e t e r s a r e n o n - d i m e n s i o n a l i s e d and p l o t t e d as p r e v i o u s l y d e f i n e d : A d d e d M a s s , a F m 22 a 22(nd) _ 2. _ _ pV w ApV pV Damping C o e f f i c i e n t , b b = — 22 ( n d ) pVw wApV I n d u c e d S u r g e F o r c e , F F l(nd) pVg The f o l l o w i n g r a n g e o f p a r a m e t e r s were u s e d d u r i n g t h e t e s t s : 3 p = 1000 Kg /m g = 9 .8067 m / s 2 98 V = 2 4 . 4 x 10 -3 t o 7 4 . 3 x 1 0 " V A = 10 mm. t o 45 mm. = 1 .57 s - l t o 1 5 . 7 1 s - l D u r i n g t h e e x p e r i m e n t s t h e p r i m a r y s o u r c e s o f e r r o r w o u l d t h e f o r c e measuremen ts u s i n g t h e f o r c e dynamometers and t h e measurement o f t h e d i s p l a c e m e n t o f t h e c y l i n d e r u s i n g t h e y o - y o t r a n s d u c e r . A n e s t i m a t e o f t h e o r d e r o f m a g n i t u d e a s s o c i a t e d w i t h e a c h o f t h e p l o t t e d p a r a m e t e r s w i l l be d e r i v e d u s i n g t h e t o t a l d i f f e r e n t i a l f o r m u l a t i o n . From dynamometer f o r c e c a l i b r a t i o n t e s t s , t h e RMS e r r o r i n t h e 2200N. dynamometer was f o u n d t o be ± 6 . 9 Newtons ( S e c t i o n 6 . 2 . 5 . 1 . 2 ) and t h e a c c u r a c y o f t h e u n i v e r s a l s h e a r beams was f o u n d t o be w i t h i n 0.3% o r ± 2 . 6 7 Newton ( S e c t i o n 6 . 2 . 5 . 2 . 1 ) . E r r o r s i n t h e measurement o f d i s p l a c e m e n t was w i t h i n 0 . 0 2 5 mm. and e r r o r i n t h e measurement o f t i m e i s n e g l i g i b l e s i n c e t h e c l o c k s p e e d o f t h e d a t a a c q u i s i t i o n s y s t e m was s e t t o 1 /1000 o f a s e c o n d . U s i n g t h e t o t a l d i f f e r e n t i a l and t h e g i v e n p a r a m e t e r v a l u e s , t h e f o l l o w i n g r a n g e o f e r r o r s a r e d e t e r m i n e d : 99 T a b l e 5 . 4 . 2 . 1 - 1 ESTIMATE OF ERROR FOR PLOTTED PARAMETERS P a r a m e t e r Min imum M i d - R a n g e Maximum E r r o r E r r o r E r r o r a 0 . 0 1 1 0 . 0 4 5 1 1 . 5 22 ( T r i p l e C y l . ) a 0 . 0 0 0 1 0 . 0 0 1 0 . 0 6 8 22 (Single C y l . ) b 0 . 0 1 1 0 . 0 4 5 1 1 . 5 22 ( T r i p l e C y l . ) b 0 . 0 0 0 1 0 . 0 0 2 0 . 1 0 2 22 (Single C y l . ) F ' 0 . 0 0 2 1 W h i l e t h e maximum e r r o r i s q u i t e u n a c c e p t a b l e , i t r e p r e s e n t s t h e w o r s t p o s s i b l e c o m b i n a t i o n o f p a r a m e t e r s w h i c h p h y s i c a l l y may n o t h a v e o c c u r r e d . As an e x a m p l e , t h e g r e a t e s t e r r o r i n a damp ing c o e f f i c i e n t o c c u r s when m e a s u r i n g t h e maximum f o r c e on t h e c y l i n d e r w i t h t h e l o w e s t d r a f t w i t h t h e l a r g e s t o s c i l l a t i o n o f m o t i o n . A more e x h a u s t i v e s t u d y on t h e m a g n i t u d e o f e r r o r w o u l d r e q u i r e c o n s i d e r i n g e a c h t e s t i n d i v i d u a l l y . F o r a f i r s t o r d e r o f m a g n i t u d e o f e r r o r , t h e ' m i d - r a n g e ' e r r o r i s a s u i t a b l e measure o f t h e e r r o r . 100 5.6 U T I L I T Y OF RESULTS One may a s k , a f t e r i n s p e c t i n g a l l t h i s d a t a , how one c a n u s e t h i s d a t a t o s o l v e a r e a l e n g i n e e r i n g p r o b l e m . F o r a n a v a l a r c h i t e c t t h e p r e d i c t i o n o f l o a d i n g c o n d i t i o n s s u c h as w i n d l o a d s , wave l o a d s , c u r r e n t s e t c . on a n a x i s y m m e t r i c f l o a t i n g body i s v e r y i m p o r t a n t . W h i l e many w e l l known p r o c e d u r e s and d a t a b a s e s e x i s t f o r t h e d e s i g n o f s h i p s , t h e p r e d i c t i o n o f l o a d i n g on o f f s h o r e d r i l l i n g p l a t f o r m s i s s t i l l i n i t s i n f a n c y . A w i d e l y a c c e p t e d , s t a n d a r d p r o c e d u r e , f o r t h e d e s i g n o f s u c h s t r u c t u r e s h a s y e t t o be e s t a b l i s h e d . F o r any d e s i g n t o be i m p l e m e n t e d , a good p r e d i c t i o n o f t h e c y c l i c l o a d on t h e s t r u c t u r e w i l l l e a d t o a r e l i a b l e p r e d i c t i o n i n v i b r a t i o n a l l e v e l s w i t h i n t h e s t r u c t u r e . W i t h t h e k n o w l e d g e o f t h e v i b r a t i o n a l l e v e l s w i t h i n t h e s t r u c t u r e as w e l l as t h e f a t i g u e c h a r a c t e r i s t i c s o f t h e m a t e r i a l s u s e d , t h e e x p e c t e d l i f e o f t h e s t r u c t u r e c a n t h e n be r e l i a b l y p r e d i c t e d . W h i l e t h e o c e a n e n v i r o n m e n t i s b a s i c a l l y c h a r a c t e r i z e d as a random p r o c e s s , i t s components a r e s t i l l c l a s s i f i e d as a s u p e r p o s i t i o n o f s i n u s o i d a l waves o f v a r i o u s a m p l i t u d e s and p h a s e s . P r e d i c t i o n o f l o a d s and r e s p o n s e s o f a f l o a t i n g s t r u c t u r e a r e b a s e d on a s i n u s o i d a l wave s p e c t r u m . K n o w i n g t h a t t h e b a s i c componen ts o f t h e b e l o w d e c k p o r t i o n o f a n o f f s h o r e p l a t f o r m a r e c y l i n d r i c a l i n n a t u r e , t h e u n d e r s t a n d i n g o f t h e b e h a v i o u r o f c y l i n d e r s i n f l u i d e n v i r o n m e n t w i l l l e a d t o t h e d e s i g n o f more s e a w o r t h y o f f s h o r e s t r u c t u r e s . W i t h t h e r e s u l t s o f t h i s s t u d y , a 101 n a v a l a r c h i t e c t c a n u s e g e t an i n i t i a l e s t i m a t e f o r t h e h y d r o d y n a m i c c o e f f i c i e n t s o f t h e b e l o w d e c k c y l i n d r i c a l members o f an o f f s h o r e s t r u c t u r e s u b j e c t e d t o h e a v e m o t i o n . 102 CONCLUSIONS 1. By c o m p a r i n g t h e r e s u l t s o f t h e e x p e r i m e n t s t o d e t e r m i n e t h e h e a v e h y d r o d y n a m i c c o e f f i c i e n t s o f compound c y l i n d e r s t o t h e t h e o r e t i c a l p r e d i c t i o n s , t h e f o l l o w i n g c o n c l u s i o n s a r e n o t e d : a ) Good ag reemen t b e t w e e n t h e o r y and e x p e r i m e n t was o b s e r v e d f o r t h e added mass and damp ing c o e f f i c i e n t s f o r n o n - d i m e n s i o n a l f r e q u e n c i e s g r e a t e r t h a n oo = 1 . 0 . b ) The r e s u l t s show t h a t t h e added mass c o e f f i c i e n t o f a compound c y l i n d e r r e m a i n s a p p r o x i m a t e l y c o n s t a n t above n o n - d i m e n s i o n a l f r e q u e n c i e s above w = 1 . 0 . c ) F a u l t y e q u i p m e n t u s e d i n p r e v i o u s e x p e r i m e n t s t o d e t e r m i n e t h e h e a v e h y d r o d y n a m i c c o e f f i c i e n t s r e s u l t e d i n i n a c c u r a t e r e s u l t s . d) V a r y i n g 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 h e a v i n g c y l i n d e r r e s u l t s r e l a t i v e l y no change i n t h e h y d r o d y n a m i c c o e f f i c i e n t s . The l i n e a r i z a t i o n a s s u m p t i o n u s e d i n t h e t h e o r e t i c a l p r e d i c t i o n s i s v a l i d f o r a m p l i t u d e / d i a m e t e r r a t i o s up t o 0 . 1 1 7 . e) A s l i g h t r e d u c t i o n i n t h e heave added mass c o e f f i c i e n t i s o b s e r v e d as t h e d r a f t o f t h e c y l i n d e r i s i n c r e a s e d . 103 2 . Any d i s c r e p a n c i e s b e t w e e n t h e e x p e r i m e n t a l v a l u e s and t h e o r e t i c a l p r e d i c t i o n s f o r t h e h y d r o d y n a m i c c o e f f i c i e n t s o f a compound c y l i n d e r c a n be a t t r i b u t e d t o : a) The i n a b i l i t y o f t h e f o r c e dynamometer t o r e s o l v e l o w f o r c e s . T h i s i s p r e v a l e n t i n t h e r e s u l t s o b t a i n e d f o r n o n - d i m e n s i o n a l f r e q u e n c i e s b e l o w co = 1.0. b ) E f f e c t s o f t h e w a l l on t h e f l o w f i e l d w h i c h a r e n o t m o d e l e d i n t h e t h e o r y . c ) The c y l i n d e r and f rame a p p a r a t u s n o t b e i n g r i g i d e n o u g h . T h i s c o n d i t i o n w o u l d r e s u l t i n t h e c r e a t i o n o f a r e l a t i v e m o t i o n b e t w e e n t h e c y l i n d e r and t h e m o t i o n g e n e r a t i o n s y s t e m . 3. By c o m p a r i n g t h e r e s u l t s o f t h e e x p e r i m e n t s t o d e t e r m i n e t h e i n d u c e d s i d e f o r c e s on a v e r t i c a l c y l i n d e r w i t h t h e p r e d i c t i o n s made b y t h e M a t c h i n g T e c h n i q u e , t h e f o l l o w i n g c o n c l u s i o n s a r e made: a) T h e r e i s r e a s o n a b l y good ag reemen t b e t w e e n t h e o r y and e x p e r i m e n t a l r e s u l t s f o r t h e i n d u c e d s i d e f o r c e s on a c y l i n d e r f o r b o t h t h e deep w a t e r and s h a l l o w w a t e r t e s t c a s e s . b ) The r e s u l t s show t h a t t h e i n d u c e d s i d e f o r c e s q u i c k l y i n c r e a s e and d e c r e a s e as t h e f r e q u e n c y i n c r e a s e s f r o m co = 0 104 w i t h t h e p e a k i n d u c e d s i d e f o r c e s o c c u r r i n g a t a f r e q u e n c y o f a b o u t to = 0 . 5 t o to = 0 . 7 5 f o r b o t h deep w a t e r and s h a l l o w w a t e r e n v i r o n m e n t s . c ) The r e s u l t s show t h a t i n d u c e d s i d e f o r c e s r e m a i n v i r t u a l l y c o n s t a n t f o r f r e q u e n c i e s above to > 2 . 0 . d) F o r b o t h t h e deep w a t e r and s h a l l o w w a t e r c a s e , t h e m a g n i t u d e o f t h e i n d u c e d s i d e f o r c e s d e c r e a s e as t h e c y l i n d e r s e p a r a t i o n i s i n c r e a s e d . e ) V a r y i n g 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 o s c i l l a t i n g c y l i n d e r does v a r y t h e i n d u c e d s i d e f o r c e f o r l o w f r e q u e n c i e s , to < 1 . 0 . A t t h e h i g h e r f r e q u e n c i e s t h e e f f e c t i s n o t so p r e v a l e n t . f ) The i n d u c e d s u r g e f o r c e s a r e h i g h e r f o r a s h a l l o w w a t e r c a s e t h a n f o r a deep w a t e r c a s e , t h e d i f f e r e n c e i s o f t h e o r d e r o f 20%. 4 . By c o m p a r i n g t h e r e s u l t s o f t h e e x p e r i m e n t s t o d e t e r m i n e t h e h e a v e i n d u c e d heave h y d r o d y n a m i c c o e f f i c i e n t s o f a v e r t i c a l c y l i n d e r w i t h t h e p r e d i c t i o n s made b y t h e M a t c h i n g T e c h n i q u e , t h e f o l l o w i n g c o n c l u s i o n s a r e made: 105 a) F o r t h e c a s e o f t h e h e a v e i n d u c e d added mass c o e f f i c i e n t s , t h e d e g r e e o f ag reemen t b e t w e e n t h e o r y and e x p e r i m e n t a l r e s u l t s i s n o t v e r y h i g h , e s p e c i a l l y f o r t h e c a s e o f t h e l o w c y l i n d e r s e p a r a t i o n s , B = 2 . 0 , B = 2 . 5 , and B = 3 . 0 . T h i s r e s u l t seems t r u e f o r b o t h t h e deep and s h a l l o w w a t e r c a s e s . b ) The e x p e r i m e n t a l r e s u l t s show t h a t t h e i n d u c e d a d d e d mass c o e f f i c i e n t s d e c r e a s e as t h e f r e q u e n c y i n c r e a s e s f r o m w = 0 , r e a c h i n g a minimum v a l u e a t a f r e q u e n c y n e a r w = 0 . 7 5 f o r b o t h t h e deep and s h a l l o w w a t e r c a s e s . c ) The r e s u l t s show t h a t t h e i n d u c e d added mass c o e f f i c i e n t s t e n d t o w a r d s a ^ = 0 as t h e f r e q u e n c y i n c r e a s e s b e y o n d u> = 2 . 0 f o r b o t h deep and s h a l l o w w a t e r . d) The e f f e c t o f i n c r e a s i n g t h e c y l i n d e r s e p a r a t i o n h a s t h e e f f e c t o f d e c r e a s i n g t h e o v e r a l l v a l u e s o f t h e i n d u c e d added mass c o e f f i c i e n t s . e) The i n d u c e d added mass c o e f f i c i e n t s a r e g e n e r a l l y h i g h e r i n t h e s h a l l o w w a t e r c a s e t h a n i n t h e deep w a t e r c a s e . The d i f f e r e n c e i s o f t h e o r d e r o f 20 %. f ) F o r t h e c a s e o f t h e h e a v e i n d u c e d damp ing c o e f f i c i e n t s , t h e e x p e r i m e n t a l r e s u l t s and t h e p r e d i c t e d t h e o r y a r e q u i t e s i m i l a r a t a l l f i v e c y l i n d e r s e p a r a t i o n s f o r b o t h t h e deep w a t e r 106 and s h a l l o w w a t e r c a s e s . g) The r e s u l t s show t h a t as t h e f r e q u e n c y i n c r e a s e s f r o m co = 0 , t h e i n d u c e d damping c o e f f i c i e n t s r a p i d l y d e c r e a s e t o w a r d s b = 0 f o r b o t h t h e deep and s h a l l o w w a t e r c a s e s . h ) The r e s u l t s show t h a t t h e i n d u c e d damping c o e f f i c i e n t s a r e n o t as s e n s i t i v e t o c y l i n d e r s e p a r a t i o n as t h e i n d u c e d a d d e d mass c o e f f i c i e n t s . W h i l e t h e damping c o e f f i c i e n t s do t e n d t o w a r d s b = 0 a t a l o w e r f r e q u e n c y as t h e c y l i n d e r s e p a r a t i o n i n c r e a s e s , t h e t r e n d i s n o t a f f e c t e d b y c y l i n d e r s e p a r a t i o n . i ) The e f f e c t o f s h a l l o w w a t e r on t h e i n d u c e d damp ing c o e f f i c i e n t s i s t o c a u s e them t o be a p p r o x i m a t e l y t w i c e as l a r g e t h a n i n t h e deep w a t e r c a s e f o r t h e l o w f r e q u e n c i e s , co < 1 . 0 . j ) The e f f e c t o f u s i n g d i f f e r e n t d r i v i n g a m p l i t u d e s o f t h e h e a v i n g c y l i n d e r h a s v i r t u a l l y no e f f e c t on t h e i n d u c e d damp ing c o e f f i c i e n t s . I n t h e c a s e o f i n d u c e d added m a s s , t h e r e i s some s l i g h t s p r e a d i n d a t a due t o d i f f e r i n g d r i v i n g f r e q u e n c i e s , e s p e c i a l l y a r o u n d t h e peak added mass v a l u e s , b u t e s s e n t i a l l y t h e v a l u e s a r e t h e same f o r e a c h d r i v i n g f r e q u e n c y . T h i s p r o v e s t h a t t h e l i n e a r i z a t i o n a s s u m p t i o n u s e d b y t h e t h e o r y i s i n d e e d v a l i d f o r p r e d i c t i o n o f i n d u c e d damping c o e f f i c i e n t s and t h e i n d u c e d a d d e d mass c o e f f i c i e n t s . 107 5. F o r t h e c a s e o f t h e i n d u c e d s i d e f o r c e s t e s t s and t h e i n d u c e d h e a v e h y d r o d y n a m i c c o e f f i c i e n t s t e s t s , d i s c r e p a n c i e s b e t w e e n t h e o r e t i c a l p r e d i c t i o n s and e x p e r i m e n t a l r e s u l t s c a n p r o b a b l y be a t t r i b u t e d t o : a ) L i m i t s i n t h e t h e o r e t i c a l m o d e l , s u c h as t h e e x c l u s i o n o f t h e e f f e c t s o f v i s c o s i t y , t h e s i m p l i f i e d l i n e a r i z a t i o n a s s u m p t i o n f o r t h e m o t i o n s o f t h e c y l i n d e r , and t h e c o n s i d e r a t i o n o f o n l y one r e f l e c t i o n o f waves b e t w e e n t h e two c y l i n d e r s . b ) E x p e r i m e n t a l l i m i t a t i o n s i n c l u d e , i n h e r e n t e r r o r s due t o t h e i n a b i l i t y o f t h e f o r c e dynamometers t o r e s o l v e v e r y l o w f o r c e s , t h e i n t r o d u c t i o n o f p h a s e s h i f t s b e t w e e n t h e v a r i o u s c h a n n e l s o f d a t a a c q u i s i t i o n , f l e x i n g o f t h e c y l i n d e r s u p p o r t s y s t e m o r t h e s u s p e n d e d s h a l l o w w a t e r b a s i n , o r f a u l t s on t h e d a t a a n a l y s i s s o f t w a r e . 108 RECOMMENDATIONS 1. I f f u t u r e wo rk i s t o be p e r f o r m e d i n t h i s t o p i c , an e f f o r t s h o u l d be made t o i m p r o v e t h e a c c u r a c y o f t h e r e s u l t s o b t a i n e d i n t h e e x p e r i m e n t s . The e f f o r t s h o u l d be c o n c e n t r a t e d i n t h e s e s p e c i f i c a r e a s : 1) A s e c o n d a r y f o r c e dynamometer be c o n s t r u c t e d w h i c h c a n measu re l o w f o r c e s more a c c u r a t e l y t h a n t h e one i n c u r r e n t u s e f o r t h e d e t e r m i n a t i o n o f t h e h e a v e h y d r o d y n a m i c c o e f f i c i e n t s o f a t r i p l e c y l i n d e r . T h i s s e c o n d f o r c e t r a n s d u c e r w o u l d h e l p t o d e t e r m i n e t h e h y d r o d y n a m i c c o e f f i c i e n t s a t l o w f r e q u e n c i e s where v e r y l o w f o r c e s a r e m e a s u r e d . 2) The e x p e r i m e n t a l a p p a r a t u s be made more r i g i d , w i t h e m p h a s i s on t h e members h o l d i n g t h e c y l i n d e r t o t h e m o t i o n g e n e r a t i o n s y s t e m . 3) A n a t t e m p t s h o u l d be made t o i n v e s t i g a t e new d a t a a c q u i s i t i o n p a c k a g e s on t h e m a r k e t w h i c h have l e s s p r o b l e m s i n t e rms o f i n t r o d u c e d p h a s e s h i f t s i n t h e e x p e r i m e n t a l d a t a b e i n g c o l l e c t e d . 4 ) R e d u c t i o n i n e x p e r i m e n t a l e r r o r f o r s h a l l o w w a t e r t e s t s c e n a r i o s b y c o n d u c t i n g t h e e x p e r i m e n t s i n a s h a l l o w w a t e r t a n k w h i c h i s b e t t e r a t m o d e l i n g s h a l l o w w a t e r c o n d i t i o n s t h a n t h e s u s p e n d e d s h a l l o w w a t e r f rame t h a t i s c u r r e n t l y b e i n g u s e d . 109 2. Improvement o f t h e M a t c h i n g T e c h n i q u e i n t h e a r e a o f i n d u c e d h e a v e h y d r o d y n a m i c c o e f f i c i e n t s c a n be made i f more t h a n o n l y t h e f i r s t d i f f r a c t i o n p o t e n t i a l i s c o n s i d e r e d i n t h e f o r m u l a t i o n . 110 BIBLIOGRAPHY A b r a m o w i t z , M. and I . A . S t e g u n , , 1 9 6 4 , Handbook o f M a t h e m a t i c a l F u n c t i o n s , W a s h i n g t o n , DC: N a t i o n a l B u r e a u o f S t a n d a r d s . A u b a n e l , E r i c E . , and K e i t h B . O ldham, 1 9 8 5 , Fourier Smoothing Without the Fast Fourier Transform, B y t e , Vo lume 1 0 , No . 2 , p p . 2 0 7 - 2 1 8 . B r e b b i a , C . A . , 1 9 7 8 , The B o u n d a r y E l e m e n t Me thod f o r E n g i n e e r s , P e n t e c h P r e s s . C a l i s a l , S . M. and T . S a b u n c u , 1 9 8 1 , Hydrodynamic Coefficients for Vertical Cylinders at Finite Depth, Ocean E n g i n e e r i n g , Vo lume 1 1 , p p . 2 5 - 6 3 . C a l i s a l , S . M. and T . S a b u n c u , 1 9 8 4 , Hydrodynamic Coefficients for Vertical Composite Cylinders, Ocean E n g i n e e r i n g , Vo lume 1 1 , p p . 5 2 9 - 5 4 2 . C a l i s a l , S . M. and T . S a b u n c u , 1986 , Heave Motion Induced Side Forces on Two Vertical Cylinders, ( t o be p u b l i s h e d ) . C a l i s a l , S . M. and T . S a b u n c u , 1 9 8 8 , A Study of Heaving Vertical Cylinders in a Towing Tank, J o u r n a l o f S h i p R e s e a r c h ( t o be p u b l i s h e d ) . C a l i s a l , S . M. and J . L . K. C h a n , 1 9 8 5 , Hydrodynamics of Vertical Cylinders, p a p e r p r e s e n t e d a t M a r i n t e c C h i n a '85, S h a n g h a i , D e c . 2 - 8 , 1 9 8 5 . C a l i s a l , S . M. 1 9 8 6 , Mech 540 C l a s s N o t e s , U n i v e r s i t y o f B r i t i s h C o l u m b i a G r a d u a t e C o u r s e , V a n c o u v e r B . C . C h a n , J . L . K . , 1 9 8 4 , H y d r o d y n a m i c C o e f f i c i e n t s F o r A x i s y m m e t r i c B o d i e s , M . A . S c . T h e s i s , U n i v e r s i t y o f B r i t i s h C o l u m b i a , V a n c o u v e r B . C . C h a n , J . L . K . , 1 9 8 8 , N u m e r i c a l P r o c e d u r e F o r P o t e n t i a l F l o w P r o b l e m s W i t h a F r e e S u r f a c e , P h d . T h e s i s , U n i v e r s i t y o f B r i t i s h C o l u m b i a , V a n c o u v e r B . C . G a r r i s o n , C . J . , 1974 , Hydrodynamics of Large Objects in the Sea -Part 1: Hydrodynamic Analysis, J o u r n a l o f H y d r o d y n a m i c s , Vo lume 8 , No . 1 . , p p . 5 - 1 2 . G a r r i s o n , C . J . , 1 9 7 5 , Hydrodynamics of Large Objects in the Sea -Part 2: Motion of Free-Floating Bodies, J o u r n a l o f 111 H y d r o d y n a m i c s , Vo lume 9 , No . 2 . , p p . 5 8 - 6 3 . G o o d r i d g e , D. N . , 1 9 8 6 , H y d r o d y n a m i c C o e f f i c i e n t s o f Compound C i r c u l a r C y l i n d e r s i n S u r g e M o t i o n , M . A . S c . T h e s i s , U n i v e r s i t y o f B r i t i s h C o l u m b i a , V a n c o u v e r B . C . I s a a c s o n , M. and T l S a r p k a y a , 1 9 8 1 , M e c h a n i c s o f Wave F o r c e s on O f f s h o r e S t r u c t u r e s , New Y o r k : V a n N o s t r a n d R e i n h o l d Company. I s a a c s o n , M. 1 9 8 6 , C I v l 540 C l a s s N o t e s , U n i v e r s i t y o f B r i t i s h C o l u m b i a G r a d u a t e C o u r s e , V a n c o u v e r B . C . Newman, J . N . , 1 9 7 7 , M a r i n e H y d r o d y n a m i c s , C a m b r i d g e , M a s s a c h u s e t t s : The MIT P r e s s . R a m i r e z , R. W . , 1 8 8 5 , The FFT F u n d a m e n t a l and C o n c e p t s , E n g l e w o o d C l i f f s , New J e r s e y : P r e n t i c e - H a l l I n c . S t r e e t e r V . J . and E . B. W y l i e , 1 9 8 1 , F l u i d M e c h a n i c s , T o r o n t o , O n t a r i o : M c G r a w - H i l l R y e r s o n L i m i t e d . Wehausen , J . V . , 1 9 7 1 , The Motion of Floating Bodies, A n n u a l R e v i e w o f F l u i d M e c h a n i c s , Vo lume 5 , p p . 2 3 7 - 2 6 8 . V e n u g o p a l , M . , 1 9 8 4 a , H y d r o d y n a m i c C o e f f i c i e n t s o f S i n g l e and D o u b l e C y l i n d e r s , T e c h n i c a l R e p o r t P r e p a r e d F o r D r . S . C a l i s a l , D e p a r t m e n t o f M e c h a n i c a l E n g i n e e r i n g , U n i v e r s i t y o f B r i t i s h C o l u m b i a , V a n c o u v e r B . C . V e n u g o p a l , M . , 1984b , H y d r o d y n a m i c C o e f f i c i e n t s o f Compound C i r c u l a r C y l i n d e r s i n Heave M o t i o n , M . A . S c . T h e s i s , U n i v e r s i t y o f B r i t i s h C o l u m b i a , V a n c o u v e r B . C . 112 APPENDIX A 6 EXPERIMENTAL SET-UP 6 . 1 EXPERIMENTAL F A C I L I T I E S A l l o f t h e e x p e r i m e n t s were c o n d u c t e d i n t h e t o w i n g t a n k o f t h e Ocean E n g i n e e r i n g C e n t r e ( O . E . C . ) o f B r i t i s h C o l u m b i a R e s e a r c h C o r p o r a t i o n , w h i c h i s l o c a t e d on t h e campus o f t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , V a n c o u v e r , B . C . A l l t e s t s were c o n d u c t e d b e t w e e n M a r c h 1987 and A p r i l 1 9 8 8 . A d e s c r i p t i o n o f t h e s e f a c i l i t i e s f o l l o w s . 6 . 1 . 1 TOWING TANK Two m a i n e x p e r i m e n t a l t a n k s a r e l o c a t e d w i t h i n t h e Ocean E n g i n e e r i n g C e n t r e . A s h i p mode l m a n e u v e r i n g b a s i n m e a s u r i n g 2 0 . 4 8 m . x 2 6 . 8 2 m . x 2 .44m. and a t o w i n g t a n k m e a s u r i n g 67 .10m x 3 .66m. x 2 .44m. P r i o r t o J a n u a r y 1 9 8 8 , an a l u m i n i u m b u l k h e a d w i t h a d r a f t o f 1 .22m. s e p a r a t e d t h e two t a n k s . T h i s a l l o w e d f o r f r e e c o m m u n i c a t i o n b e t w e e n t h e two t a n k s b e l o w t h e d e p t h o f t h e b u l k h e a d . T h i s s i t u a t i o n may h a v e r e s u l t e d i n s l i g h t l y e r r o n e o u s measu remen ts o f v a r i a b l e s d u r i n g e x p e r i m e n t s w i t h i n t h e t o w i n g t a n k . I n J a n u a r y 1 9 8 8 , a c o n c r e t e b u l k h e a d was i n s t a l l e d w h i c h c o m p l e t e l y s e p a r a t e d t h e two t a n k s . F i g u r e 9 . 3 shows t h e b a s i n s a t t h e O . E . C . p r i o r t o J a n u a r y 1 9 8 8 . 113 The t o w i n g t a n k ' s p r i m a r y f u n c t i o n i s s h i p mode l r e s i s t a n c e t e s t i n g . S h i p m o d e l s a r e towed t h e l e n g t h o f t h e t a n k b y a t o w i n g c a r r i a g e w h i c h i s e q u i p p e d w i t h t h e d a t a a c q u i s i t i o n e q u i p m e n t . The c a r r i a g e t r a v e r s e s t h e l e n g t h o f t h e t a n k on r a i l s and i s d r i v e n b y a n e x t e r n a l h y d r a u l i c mo to r c a b l e s y s t e m . A p h o t o g r a p h o f t h e c a r r i a g e s y s t e m i s shown i n F i g u r e 9 . 4 . A t one end o f t h e t o w i n g t a n k i s a h i n g e d f l a p p e r t y p e wave maker and a wave a b s o r b i n g b e a c h on t h e o t h e r e n d . T h i s wave TM maker i s h y d r a u l i c a l l y d r i v e n and c o n t r o l l e d b y a n IBM PC p e r s o n a l c o m p u t e r , e q u i p p e d w i t h a n d i g i t a l t o a n a l o g u e c o n v e r t e r ( D / A c o n v e r t e r ) . The i n p u t s i g n a l f r o m t h e c o m p u t e r i s f e d t o a wave s y n t h e s i z e r w h i c h compares t h e i n p u t s i g n a l w i t h a s i g n a l f r o m a p o s i t i o n t r a n s d u c e r f i x e d t o t h e wave p a d d l e . A f t e r c o m p a r i n g t h e two s i g n a l s t h e s y n t h e s i z e r s e n d s a c o r r e c t i o n s i g n a l t o t h e h y d r a u l i c a c t u a t o r w h i c h i n t u r n c o n t r o l s t h e h y d r a u l i c p i s t o n . The wave p a d d l e i s a b l e t o g e n e r a t e q u a s i - r a n d o m waves as w e l l as a r a n g e o f r e g u l a r r e p e a t e d wave fo rms u s u a l l y b e t w e e n 0 . 3 H e r t z and 1 .5 H e r t z . T h i s t o w i n g t a n k h a s a l s o t h r e e u n d e r w a t e r w indows l o c a t e d a t t h e m i d p o i n t o f t h e t a n k f o r f l o w v i s u a l i z a t i o n e x p e r i m e n t s . L o c a t e d a l s o a t t h e m i d p o i n t o f t h e t a n k i s an o v e r h e a d h o i s t i n g m e c h a n i s m . T h i s o v e r h e a d h o i s t h a s a maximum c a p a c i t y s a f e l o a d o f 1 3 0 0 n . and i s f r e e t o move a p p r o x i m a t e l y 4 . 8 m . a c r o s s t h e t a n k and 2 .4m. a l o n g t h e l e n g t h . T h i s h o i s t was t h e p r i m a r y l i f t i n g d e v i c e u s e d f o r p o s i t i o n i n g e q u i p m e n t u s e d i n t h e s e e x p e r i m e n t s . T h i s h o i s t i s shown i n F i g u r e 9 . 5 . 114 A l l o f t h e t e s t s c o n d u c t e d f o r t h i s r e s e a r c h were c a r r i e d o u t a t t h e m i d p o i n t o f t h e t o w i n g t a n k , b e l o w t h e o v e r h e a d h o i s t . The t o w i n g c a r r i a g e was l o c a t e d n e a r t h e work a r e a and c o n t a i n e d a l l t h e d a t a a c q u i s i t i o n h a r d w a r e . D u r i n g many o f t h e t e s t s t e m p o r a r y a r t i f i c i a l b e a c h e s made f r o m b a t e s o f s y n t h e t i c h o r s e h a i r m a t e r i a l were p l a c e d a t v a r i o u s s t r a t e g i c p o i n t s i n t h e t o w i n g t a n k . T h e s e f l o a t i n g b e a c h e s , a l o n g w i t h t h e pe rmanen t a r t i f i c i a l b e a c h e s i n t h e t a n k , h e l p e d t o dampen t h e waves g e n e r a t e d b y t h e m o t i o n o f t h e c y l i n d e r and t o p r e v e n t unwan ted wave r e f l e c t i o n s f r o m t h e w a l l s r e a c h i n g t h e c y l i n d e r . Many o f t h e e x p e r i m e n t s p e r f o r m e d were done w i t h a s h a l l o w w a t e r f a l s e b o t t o m s u s p e n d e d i n t h e t a n k . T h i s f a l s e b o t t o m was 3 c o n s t r u c t e d w i t h t h r e e 4 ' x 8 ' x - " p l y w o o d s h e e t s on a s l o t t e d a n g l e i r o n f r ame w h i c h was s u s p e n d e d i n t h e deep w a t e r t a n k . T h i s a l l o w e d s h a l l o w w a t e r e x p e r i m e n t s t o be p e r f o r m e d i n a deep w a t e r t a n k . A l o n g w i t h t h e f a l s e b o t t o m , p l y w o o d s i d e w a l l s were f a s t e n e d t o t h e s h a l l o w w a t e r f rame i n o r d e r t o n a r r o w t h e t a n k w i d t h . These s i d e w a l l s c o u l d be moved t o c r e a t e v a r i o u s t a n k w i d t h s i n o r d e r t o i n v e s t i g a t e t h e ' w a l l e f f e c t ' on h e a v e h y d r o d y n a m i c c o e f f i c i e n t s o f a s i n g l e c y l i n d e r . The e n t i r e s h a l l o w w a t e r s y s t e m was 12 f e e t l o n g x 12 f e e t w i d e x 3 f e e t d e e p . The s h a l l o w w a t e r f rame and s i d e w a l l s a r e shown i n F i g u r e 9 . 6 . 115 6 . 2 EXPERIMENTAL EQUIPMENT 6 . 2 . 1 MOTION GENERATION SYSTEM The m o t i o n g e n e r a t i o n s y s t e m u s e d f o r t h e s e e x p e r i m e n t s c o n s i s t e d o f a s c o t c h - y o k e mechan ism s u p p o r t e d on a n a l u m i n i u m f r a m e . F i g u r e 9 . 7 shows a p h o t o g r a p h o f t h e s y s t e m . The m o t i o n g e n e r a t o r was o r i g i n a l l y d e s i g n e d b y K i e S z e t o and c o n s t r u c t e d b y t h e M e c h a n i c a l E n g i n e e r i n g M a c h i n e Shop i n 1 9 8 2 . Two r e s e a r c h e r s h a v e p r e v i o u s l y a p p l i e d t h e e q u i p m e n t t o t h e i r r e s e a r c h . Madan V e n u g o p a l u s e d t h e equ ipmen t i n 1984 f o r t e s t i n g o f h e a v e m o t i o n c h a r a c t e r i s t i c s o f c y l i n d e r s and Doug G o o d r i d g e i n 1986 u s e d t h e e q u i p m e n t f o r t e s t s o f s u r g e m o t i o n c h a r a c t e r i s t i c s o f c y l i n d e r s . The m o t i o n g e n e r a t i o n s y s t e m c o n s i s t s o f a h o r i z o n t a l f i x e d f r ame 12 f e e t x 4 f e e t made up o f two a l u m i n i u m I - b e a m s . A r i g h t a n g l e d t r i a n g u l a r f rame made f r o m a l u m i n i u m i s b o l t e d on t o p o f t h e I - b e a m s . The f rame s u p p o r t s a h y d r a u l i c m o t o r , a s c o t c h - y o k e m e c h a n i s m , and a s l i d e r p l a t e t o w h i c h t h e dynamometer and c y l i n d e r i s a t t a c h e d . The e n t i r e a r r a n g e m e n t o f h y d r a u l i c m o t o r / s c o t c h - y o k e mechan ism c a n be f a s t e n e d a t v a r i o u s p o s i t i o n s on t h e f rame t o v a r y t h e d r a f t o f t h e o s c i l l a t i n g c y l i n d e r . The s c o t c h - y o k e mechan ism i s d r i v e n b y a r a d i a l p i s t o n h y d r a u l i c mo to r w i t h a d i s p l a c e m e n t o f 0 . 2 0 8 L / r e v o l u t i o n and i s c a p a b l e o f p r o d u c i n g 680 N-m. o f o u t p u t t o r q u e . U n f o r t u n a t e l y , due t o l i m i t a t i o n s o f t h e a v a i l a b l e e l e c t r i c power s u p p l y , t h i s r a t e d t o r q u e c o u l d o n l y be d e l i v e r e d up t o a r o t a t i o n a l f r e q u e n c y 116 o f 0 . 8 3 H e r t z . The o u t p u t t o r q u e d i m i n i s h e d t o 225 N-m. a t a f r e q u e n c y o f 2 . 5 H e r t z . A c l o s e d l o o p t r a n s m i s s i o n c i r c u i t e n s u r e d t i g h t c o n t r o l o f t h e mo to r s p e e d o v e r a s p e e d r a n g e o f 3rpm t o 150rpm. The h y d r a u l i c power u n i t c o n s i s t s o f a 1750rpm, 3 .7kW, 4 4 0 v . e l e c t r i c m o t o r d r i v i n g a v a r i a b l e d i s p l a c e m e n t a x i a l p i s t o n pump w i t h a maximum d i s p l a c e m e n t o f 41 m L / r e v o l u t i o n . A c r o s s - p o r t r e l i e f v a l v e s e t a t 3 0 0 0 p s i i s u s e d t o p r e v e n t e x c e s s i v e c i r c u i t p r e s s u r e s and m e c h a n i c a l s t o p s p r e v e n t o v e r l o a d i n g o f t h e m o t o r . The h y d r a u l i c power u n i t i s shown i n F i g u r e 9 . 8 . 6 . 2 . 2 SECONDARY CYLINDER FRAME A s e c o n d f rame s y s t e m was c o n s t r u c t e d t o h o l d a s e c o n d a r y c y l i n d e r u s e d i n t h e e x p e r i m e n t s t h a t d e t e r m i n e d i n d u c e d s u r g e and h e a v e f o r c e s i n one c y l i n d e r when a n a d j a c e n t c y l i n d e r i s i n h e a v e m o t i o n . The f i x e d f rame m e a s u r e d 12 f e e t x 4 f e e t and was made up o f 6 i n c h a l u m i n i u m I -beams b r i d g i n g t h e t o w i n g t a n k . A n a l u m i n i u m s u p p o r t f rame was b o l t e d on t o t h e beams and t h i s i n t u r n s u p p o r t e d t h e c y l i n d e r t h r o u g h an a l u m i n i u m box -beam c a n t i l e v e r arm a t t a c h e d t o t h e c y l i n d e r and f i x e d t o t h e f r a m e . TM C o n n e c t e d t o t h e c a n t i l e v e r arm were two U n i v e r s a l S h e a r Beams w h i c h m e a s u r e d t h e i n d u c e d s u r g e and h e a v e f o r c e s as w e l l as t h e c y l i n d e r s u p p o r t b r a c k e t . The c y l i n d e r s u p p o r t b r a c k e t s a t i n s i d e t h e c y l i n d e r and c o n s i s t e d o f a s t e e l r o d f i x e d a t e a c h end t o s t e e l b a r s w h i c h f a s t e n e d t o t h e c y l i n d e r w a l l s . The s t e e l r o d 117 was a t t a c h e d t o an a l u m i n i u m b r a c k e t i n t o w h i c h t h e s h e a r beams were f a s t e n e d . The c y l i n d e r was b a l l a s t e d so t h a t t h e s h e a r beams s a t a t t h e a p p r o x i m a t e c e n t r e o f g r a v i t y o f t h e n e a r n e u t r a l l y b u o y a n t c y l i n d e r . F i g u r e 9 . 9 shows t h e c a n t i l e v e r a rm , s h e a r beams , and c y l i n d e r s u p p o r t b r a c k e t . I t was i m p o r t a n t t o d e s i g n a s e c o n d a r y f rame w h i c h was i n no way a t t a c h e d t o t h e M o t i o n G e n e r a t i o n S y s t e m f rame so t h a t v i b r a t i o n a l f o r c e s were n o t t r a n s f e r r e d t o t h e f o r c e t r a n s d u c e r s . A l s o , s i n c e i t was i n d e p e n d e n t o f t h e more h e a v y m o t i o n g e n e r a t i o n s y s t e m , t h e s e c o n d a r y c y l i n d e r f r ame c o u l d be moved t o t h e d e s i r e d l o c a t i o n w i t h o u t u s e o f t h e o v e r h e a d h o i s t . F i g u r e 9 . 1 0 shows a p h o t o g r a p h o f t h e S e c o n d a r y C y l i n d e r Frame S y s t e m . 6 . 2 . 3 DATA COLLECTION EQUIPMENT The d a t a c o l l e c t i o n equ ipmen t was p r o v i d e d b y B . C . R e s e a r c h and t h e D e p a r t m e n t o f M e c h a n i c a l E n g i n e e r i n g o f U . B . C . The two m a i n componen ts o f t h e d a t a c o l l e c t i o n s y s t e m a r e t h e ST41B™ TM s i g n a l c o n d i t i o n e r , and t h e MINC 11 m i n i c o m p u t e r . TM 6 . 2 . 3 . 1 ST41B SIGNAL CONDITIONER The s i g n a l c o n d i t i o n e r p r o v i d e d b o t h t h e r e q u i r e d e x c i t a t i o n v o l t a g e t o t h e m e a s u r i n g t r a n s d u c e r s as w e l l as a m p l i f i c a t i o n o f t h e r e t u r n e d s i g n a l . t h e s i g n a l c o n d i t i o n e r was d e s i g n e d and m a n u f a c t u r e d b y T e r r a s c i e n c e Sys tems L i m i t e d o f V a n c o u v e r , B . C . 118 The u n i t was o r i g i n a l l y p u r c h a s e d f o r u s e i n a mode l f i s h i n g b o a t t o s i m u l a t e c a p s i z i n g o f o v e r l o a d e d f i s h b o a t s i n v a r i o u s s e a s t a t e s . The p r i n c i p l e f e a t u r e s o f t h e e i g h t c h a n n e l u n i t a r e : - I n d e p e n d e n t v a r i a b l e r e g u l a t e d e x c i t a t i o n f o r e a c h o f e i g h t c h a n n e l s (2 t o 10 V D C ) . - I n d e p e n d e n t r e g u l a t e d p o s i t i v e and n e g a t i v e e x c i t a t i o n p e r d u a l c h a n n e l u n i t (±13 V D C ) . - I n d e p e n d e n t s w i t c h s e l e c t a b l e g a i n (1 t o 1000) f o r e a c h c h a n n e l . - P r o v i s i o n f o r b r i d g e c o m p l e t i o n components t o a c c e p t ^, ^ o r f u l l b r i d g e i n p u t s b r i d g e i n p u t s t o e a c h c h a n n e l . - F o u r p o l e B u t t e r w o r t h l o w p a s s f i l t e r on e a c h c h a n n e l . - A l l s u p p l i e s and o u t p u t s f u l l y s h o r t c i r c u i t p r o t e c t e d . F u r t h e r i n f o r m a t i o n and s p e c i f i c a t i o n s c a n be f o u n d i n t h e owners manua l s u p p l i e d b y t h e m a n u f a c t u r e r , a v a i l a b l e a t B . C . R e s e a r c h . The a m p l i f i e r was g e n e r a l l y s e t - u p as f o l l o w s : 119 T a b l e 6 . 2 . 3 . 1 A M P L I F I E R CONFIGURATION A m p l i f i e r T r a n s d u c e r E x c i t a t i o n G a i n C h a n n e l N o . V o l t a g e 1. Compound C y l i n d e r Heave T e s t / Heave I n d u c e d R e s o n a n c e T e s t 1 Dynamometer S u r g e 1 0 . 0 1000 2 Dynamometer P i t c h 1 0 . 0 1000 3 Dynamometer Heave 1 0 . 0 1000 4 Y o - Y o P o s . T r a n s d u c e r 1 0 . 0 2 . 0 2 . Heave I n d u c e d S u r g e and P i t c h F o r c e T e s t 1 S h e a r Beam S u r g e 1 0 . 0 1000 2 S h e a r Beam Heave 1 0 . 0 1000 3 Y o - Y o P o s . T r a n s d u c e r 1 0 . 0 2 . 0 A f t e r t h e s i g n a l c o n d i t i o n e r was s e t - u p as s p e c i f i e d , e a c h c h a n n e l was b a l a n c e d t o g i v e an a p p r o x i m a t e mean s i g n a l o f z e r o . I t was n o t c r i t i c a l t h a t t h e o u t p u t be s e t t o e x a c t l y z e r o s i n c e t h e d a t a p r o c e s s i n g s o f t w a r e c o r r e c t s f o r any DC o f f s e t . The e x c i t a t i o n as w e l l as t h e r e t u r n e d s i g n a l t r a v e l l e d t h r o u g h f o u r - c o n d u c t o r s h i e l d e d i n s t r u m e n t a t i o n w i r e and c o n n e c t e d t o t h e a p p r o p r i a t e i n p u t p l u g on t h e f a c e o f t h e s i g n a l c o n d i t i o n e r . A l s o on t h e f a c e o f t h e s i g n a l c o n d i t i o n e r i s a n a r r a y o f BNC p l u g s f r o m w h i c h t h e o u t p u t s i g n a l s f o r e a c h c h a n n e l a r e a v a i l a b l e . C o a x i a l c a b l e was u s e d t o c o n n e c t t h e s e BNC t e r m i n a l s I TM t o f o u r BNC t e r m i n a l s l o c a t e d on t h e f r o n t o f t h e MINC 11 m i n i c o m p u t e r . 120 6 . 2 . 3 . 1 . 1 SIGNAL CONDITIONER PHASE LAG TESTS Of c o n s i d e r a b l e i m p o r t a n c e i n t h e s e e x p e r i m e n t s was t h e r e l a t i v e p h a s e a n g l e b e t w e e n e a c h c h a n n e l . G o o d r i d g e (1986) i n v e s t i g a t e d f u l l y t h e m a g n i t u d e o f t h i s p h a s e s h i f t and i t s e f f e c t on t h e d a t a c o l l e c t i o n . H i s r e s u l t s showed t h a t t h e p h a s e l a g o f e a c h c h a n n e l was l i n e a r l y d e p e n d e n t on t h e i n p u t f r e q u e n c y , and c o u l d be e x p r e s s e d a s : <f> = - 2 . 7 6 3 f - 0 . 5 6 8 6 . 2 . 3 . 1 . 1 - 1 where <t> = l a g a n g l e b e t w e e n o u t p u t and i n p u t o f t h e s i g n a l c o n d i t i o n e r ( d e g r e e s ) f = f r e q u e n c y o f s i g n a l ( H e r t z ) He c o n c l u d e d t h a t t h e p h a s e s h i f t was n o t a f f e c t e d b y t h e c h a n n e l number o r t h e g a i n s e t t i n g . H i s i n v e s t i g a t i o n a l s o c o n c l u d e d t h a t s i n c e e a c h c h a n n e l h a s a p h a s e l a g o f t h e same amount , t h e r e l a t i v e p h a s e a n g l e r e m a i n e d u n a f f e c t e d b y t h e s i g n a l c o n d i t i o n e r . Howeve r , t h i s r e s e a r c h e r d e c i d e d t o r e - e v a l u a t e t h e s i g n a l c o n d i t i o n e r u s i n g v a r i o u s l o w p a s s f i l t e r s i n t h e s i g n a l c o n d i t i o n e r i n o r d e r t o d e c i d e i f a p h a s e s h i f t i s i n t r o d u c e d t h r o u g h t h e u s e o f d i f f e r e n t f i l t e r s . I t was d e c i d e d t h a t a d r y t e s t r u n on t h e o s c i l l a t i n g c y l i n d e r be d o n e ; t h a t i s , t h e c y l i n d e r be o s c i l l a t e d i n a i r and .121 n o t i n w a t e r . S i n c e a i r adds v e r y l i t t l e v i s c o u s r e s i s t a n c e on t h e o s c i l l a t i n g c y l i n d e r compared t o w a t e r , i t c o u l d be assumed t h a t t h e damp ing c o e f f i c i e n t i n a i r s h o u l d a l w a y s r e m a i n z e r o f o r a l l f r e q u e n c i e s . F i g u r e 1 0 . 4 6 shows t h e r e s u l t s o f t h e s e t e s t s . F i v e s e t s o f l o w p a s s f i l t e r s were u s e d i n c o m b i n a t i o n s o f 2 , 1 0 , and 50 H e r t z . The r e s u l t s show t h a t m i x i n g d i f f e r e n t f r e q u e n c y f i l t e r s g i v e e r r o n e o u s r e s u l t s . A 2 H e r t z and 10 H e r t z f i l t e r were i n s t a l l e d i n t h e dynamometer s u r g e c h a n n e l and i n t h e y o - y o t r a n s d u c e r c h a n n e l w h i c h r e s u l t e d i n damping c o e f f i c i e n t s o f up t o 0 . 8 . These r e s u l t s f a r e x c e e d e d t h e e x p e c t e d v a l u e s and t h e r e f o r e p r o v e t h a t m i x i n g o f l o w p a s s f i l t e r s s h o u l d be a v o i d e d . I t i s e v i d e n t t h a t f i l t e r s o f t h e same f r e q u e n c y l o w e r t h e s e v a l u e s o f damp ing c o e f f i c i e n t t o more r e s p e c t a b l e l e v e l s . H o w e v e r , t h e c u r v e s o f t h e 10 H e r t z and 50 H e r t z f i l t e r s s t i l l g i v e r e l a t i v e l y h i g h v a l u e s o f damping c o e f f i c i e n t s , e s p e c i a l l y a t f r e q u e n c i e s b e l o w 1 .0 H e r t z . The 10 and 50 H e r t z f i l t e r s p r o d u c e d damp ing c o e f f i c i e n t s i n e x c e s s o f 0 . 1 a t l o w f r e q u e n c i e s . T h i s , a l s o , was deemed u n a c c e p t a b l e . O n l y t h e 2 H e r t z l o w p a s s f i l t e r seemed t o g i v e r e s u l t s w h i c h r e s e m b l e d t h e e x p e c t e d v a l u e s . A l l t h e damp ing c o e f f i c i e n t s e x c e p t a t v e r y l o w f r e q u e n c i e s were b e l o w 0 . 0 5 . I f one compares t h e s e v a l u e s t o t h e n o r m a l damping c o e f f i c i e n t v a l u e s t h a t were o b t a i n e d f r o m t h e t e s t s , i t w o u l d show t h a t t h e maximum e r r o r p r o d u c e d i s n o t more t h a n 5% a t t h e r a n g e o f f r e q u e n c i e s t e s t e d . These r e s u l t s p r o v e t h a t o n l y 2 H e r t z l o w p a s s f i l t e r s s h o u l d be u s e d i n t h e s i g n a l c o n d i t i o n e r t o 122 g i v e t h e b e s t r e s u l t s i n d a t a c o l l e c t i o n . 6 . 2 . 3 . 2 MINC™ 11 MINI COMPUTER TM A D i g i t a l E l e c t r o n i c s MINC 11 m i n i compu te r was u s e d f o r a l l t h e d a t a c o l l e c t i o n . The s y s t e m h a s two f l o p p y d i s k d r i v e s w h i c h r e q u i r e 8 i n c h s i n g l e s i d e d d o u b l e d e n s i t y d i s k e t t e s . I n n o r m a l o p e r a t i o n a l l t h e d a t a c o l l e c t i o n p r o g r a m s a r e c o n t a i n e d on one d i s k e t t e and t h e o t h e r i s u s e d t o s t o r e t h e raw d a t a . The c o m p u t e r u s e s R T - 1 1 f o r m a t t i n g so t h a t i t i s c o m p a t i b l e w i t h t h e TM D i g i t a l PDP 11 s e r i e s c o m p u t e r s . TM The h e a r t o f t h e MINC 11 i s c o n t a i n e d i n a s e p a r a t e m o d u l e . T h i s modu le c o n t a i n s a l l o f t h e a n a l o g u e t o d i g i t a l ( A / D ) c o n v e r s i o n h a r d w a r e w h i c h c o n v e r t s v o l t a g e o u t p u t s f r o m t h e d a t a c o l l e c t i o n t r a n s d u c e r s i n t o d i g i t a l v a l u e s t o b e s t o r e d on t h e d i s k e t t e s . The A / D c o n v e r t e r h a s 16 a n a l o g u e p o r t s and c a n a c c e p t i n p u t s i g n a l s b e t w e e n - 5 . 1 2 v o l t s and +5 .12 v o l t s w i t h a r e s o l u t i o n o f 2 . 5 mV. The A / D c o n v e r t e r c o n v e r t s t h e s e v o l t a g e s i n t o a n i n t e g e r number b e t w e e n 0 and 4096 b y t h e f o l l o w i n g r e l a t i o n : I INT (400xV) + 2048 where I i n t e g e r v a l u e V a n a l o g u e i n p u t ( v o l t ) INT(#) g r e a t e s t i n t e g e r o f a rgument i n p a r e n t h e s i s 123 The MINC 11 u t i l i z e s a D i g i t a l VT105 G r a p h i c s V i d e o t e r m i n a l t o a l l o w one t o v i s u a l l y d i s p l a y r e c o r d e d s i g n a l s . A T e k t r o n i x S c r e e n Dump P r i n t e r was u s e d t o g e n e r a t e h a r d c o p i e s o f t h e v i d e o s c r e e n c o n t e n t s . 6 . 2 . 4 CYLINDER MODELS Two t y p e s o f c y l i n d e r mode l s were u s e d i n t h e h y d r o d y n a m i c s r e s e a r c h : a compound o r t r i p l e c y l i n d e r and two s i n g l e c y l i n d e r s . The mode l s were b u i l t i n t h e M a c h i n e Shop o f t h e M e c h a n i c a l E n g i n e e r i n g D e p a r t m e n t o f U . B . C . These same m o d e l s were u s e d b y V e n u g o p a l (1984) and G o o d r i d g e (1986) i n e a r l i e r h y d r o d y n a m i c r e s e a r c h . The geome t r y o f e a c h o f t h e c y l i n d e r s i s shown i n F i g u r e s 8 . 6 and 8 . 7 , and p h o t o g r a p h s o f t h e c y l i n d e r s may be f o u n d i n F i g u r e s 9 . 1 0 and 9 . 1 1 . E a c h o f t h e s i n g l e c y l i n d e r s were c o n s t r u c t e d o f PVC t u b i n g w i t h a 455mm. (15 i n c h e s ) d i a m e t e r . End p l a t e s made f r o m a l u m i n i u m were u s e d as a b o t t o m and a t o p o f t h e c y l i n d e r s The t r i p l e c y l i n d e r u t i l i z e d t h e s i n g l e c y l i n d e r as t h e m i d d l e s e c t i o n o f a compound c y l i n d e r . The t o p s e c t i o n o f t h e t r i p l e c y l i n d e r i s c o m p r i s e d o f a n a l u m i n i u m c y l i n d r i c a l s e c t i o n 613mm. h i g h and 219mm. ( 8 - i n c h e s ) i n d i a m e t e r ; t h e b o t t o m s e c t i o n i s c o m p r i s e d o f 8 a s i m i l a r a l u m i n i u m c y l i n d r i c a l s e c t i o n 312mm. h i g h and 219mm. ( 8 -i n c h e s ) i n d i a m e t e r . The t o t a l h e i g h t o f t h e a s s e m b l e d c y l i n d e r u n i t i s 1 .38 m e t r e s . TM The c y l i n d e r s were k e p t w a t e r t i g h t u s i n g B u n a - N 0 - r i n g 124 m a t e r i a l w h i c h f i t t e d i n t o mach ine g r o o v e s l o c a t e d b e t w e e n a l l submerged m a t i n g s u r f a c e s . F o u r t h r e a d e d r o d s l o c a t e d i n s i d e t h e c y l i n d e r we re u s e d t o f a s t e n t h e c y l i n d e r s t o g e t h e r . T h e s e r o d s s e r v e d as f a s t e n e r s f o r l e a d w e i g h t s w h i c h were u s e d t o b a l l a s t t h e c y l i n d e r s t o t h e d e s i r e d d r a f t . The t h r e a d e d r o d s e x t e n d e d an e x t r a 160mm. b e y o n d t h e t o p p l a t e o f t h e c y l i n d e r so t h a t a s t i f f e n i n g c o l l a r c o u l d be added t o f i n e t une t h e d r a f t o f t h e c y l i n d e r as w e l l as t o add e x t r a s t i f f n e s s t o t h e s y s t e m . V a r i o u s s i z e d c o l l a r s m e a s u r i n g 2 . 5 , 5 . 0 , and 7 . 5 cm. c o u l d be e a s i l y i n s e r t e d b e t w e e n t h e c y l i n d e r and t h e a d a p t e r b l o c k t o r a i s e o r l o w e r t h e c y l i n d e r , d r a f t as r e q u i r e d . An a l u m i n i u m a d a p t e r b l o c k was u s e d t o c o n n e c t t h e c y l i n d e r t o t h e dynamometer w h i c h i n t u r n was c o n n e c t e d t o t h e m o t i o n g e n e r a t o r . I n t h e c a s e o f t h e s i n g l e c y l i n d e r u s e d t o s t u d y i n d u c e d s u r g e and h e a v e f o r c e s due t o an a d j a c e n t s i n g l e c y l i n d e r i n heave m o t i o n , s l i g h t m o d i f i c a t i o n s were made. I n t h i s c a s e , t h e t o p p l a t e o f t h e c y l i n d e r was removed i n o r d e r t o a l l o w t h e i n s t r u m e n t e d c a n t i l e v e r arm t o e x t e n d i n t o t h e c y l i n d e r . T h i s r e m o v a l was r e q u i r e d t o a l l o w t h e U n i v e r s a l S h e a r Beams™ t o be p l a c e d as c l o s e t o t h e c e n t e r o f g r a v i t y as p o s s i b l e . To a c c o m p l i s h t h i s p r e c i s e p l a c e m e n t o f t h e s h e a r beams , h o l e s were d r i l l e d i n t o t h e s i d e s o f t h e c y l i n d e r i n o r d e r t o b o l t t h e c y l i n d e r o n t o t h e c y l i n d e r s u p p o r t b r a c k e t , w h i c h , i n t u r n , was f a s t e n e d t o t h e s h e a r beams and t h e c a n t i l e v e r a rm . Two t h r e a d e d r o d s e c t i o n s p a s s e d f r o m t h e b o t t o m p l a t e t o two a l u m i n i u m b a r s l a y e d o v e r t h e t o p o f t h e c y l i n d e r . The t h r e a d e d r o d s h e l d t h e 125 c y l i n d e r w a t e r t i g h t and f o r m e d a n e c e s s a r y c o n t a c t p o i n t f o r t h e b a l l a s t . Howeve r , t h e c a n t i l e v e r arm d i d n o t t o u c h t h i s t h r e a d e d r o d i n any way ; a l l i n d u c e d f o r c e s p a s s e d t h r o u g h t h e s h e a r beams . 6 . 2 . 5 INSTRUMENTATION USED 6 . 2 . 5 . 1 LOAD CELL DYNAMOMETER A dynamometer f i r s t u s e d b y G o o d r i d g e (1986) was u s e d b y t h i s r e s e a r c h e r t o d e t e r m i n e added mass and damp ing c o e f f i c i e n t s o f a t r i p l e c y l i n d e r i n heave m o t i o n as w e l l as t o d e t e r m i n e t h e a d d e d mass and damp ing c o e f f i c i e n t s o f a s i n g l e c y l i n d e r i n h e a v e m o t i o n w i t h i n a n a r r o w c h a n n e l . T h i s l o a d c e l l was f i r s t c o n s t r u c t e d t o r e p l a c e l o a d c e l l s u s e d b y V e n u g o p a l (1984) t h a t were l a t e r f o u n d d e f e c t i v e upon i n s p e c t i o n b y G o o d r i d g e (1986) . The l o a d c e l l was d e s i g n e d a n d w i r e d b y W e i r - J o n e s E n g i n e e r i n g C o n s u l t a n t s L i m i t e d o f V a n c o u v e r and m a c h i n e d and f a b r i c a t e d b y t h e M a c h i n e Shop o f t h e M e c h a n i c a l E n g i n e e r i n g D e p a r t m e n t a t U . B . C . Two i d e n t i c a l l o a d c e l l s were c o n s t r u c t e d t o a l l o w f o r a b a c k - u p i n c a s e o f a m e c h a n i c a l b r e a k d o w n . The l o a d c e l l s a r e o f d u a l c a n t i l e v e r c o n f i g u r a t i o n w i t h t h r e e e l e c t r i c a l l y i s o l a t e d s t r a i n gauge b r i d g e s f o r t h e measurement o f s u r g e , p i t c h , and h e a v e . The dynamometer i s shown i n F i g u r e s 9 . 1 2 and 9 . 1 3 as w e l l as a r e d u c e d c o p y o f t h e e n g i n e e r i n g d r a w i n g i s p r o v i d e d i n F i g u r e 9 . 1 5 . The s t r a i n gauges a r e a l l epoxy s u b s t a t e c o n s t a n t r o s e t t e s t y p e s w h i c h a r e epoxy 126 b o n d e d t o a m a c h i n e d 7075-T6 a l u m i n i u m b l a n k . The l o a d c e l l s a r e p r o t e c t e d a g a i n s t m o i s t u r e and i m p a c t damage. More d e t a i l e d s p e c i f i c a t i o n s o f t h e u n i t a r e as f o l l o w s . T a b l e 6 . 2 . 5 . 1 - 1 DYNAMOMETER SPECIF ICATIONS S u r g e C a p a c i t y P i t c h C a p a c i t y R a t e d O u t p u t N o n - L i n e a r i t y Max . S a f e O v e r l o a d s S u r g e P i t c h Heave Sway Yaw R o l l C r o s s S e n s i t i v i t y T h e r m a l C o e f f . o f Z e r o T h e r m a l C o e f f . o f S e n s i t i v i t y C r e e p Maximum E x c i t a t i o n V o l t a g e R e s i s t a n c e I n p u t / O u t p u t 2220 N . 2030 N-m. 1 mV/V e a c h b r i d g e 1% FS e a c h b r i d g e 3330 N. c o m b i n e d 3050 N-m. 1 2 , 4 5 0 N. ' i n d i v i d u a l 4 4 , 4 7 0 N. i n d i v i d u a l 2 7 , 1 1 0 N-m. i n d i v i d u a l 2030 N - m . . i n d i v i d u a l S u r g e / P i t c h 2% 1% F S / 4 C a l l b r i d g e s 1% p e r 2 2 . 2 C C 0.1% p e r 20 m i n u t e s 10 VDC 350O. 6 . 2 . 5 . 1 . 1 DYNAMOMETER ORIENTATION S i n c e t h e dynamometer h a d b e e n u s e d p r e v i o u s l y and shown t o g i v e a c c u r a t e r e s u l t s , i t was d e c i d e d t h a t t h i s dynamometer s h o u l d be u s e d t o d e t e r m i n e h e a v e h y d r o d y n a m i c c o e f f i c i e n t s . Howeve r , i t was f o u n d t h a t t h i s l o a d c e l l c o u l d n o t be u s e d i n r e a d i n g a c c u r a t e l y p u r e c o m p r e s s i v e o r t e n s i l e l o a d s . T h i s t y p e o f l o a d i n g s h o u l d h a v e b e e n m e a s u r e d b y t h e h e a v e s t r a i n gauges moun ted i n t h e l o a d c e l l b u t t h e y were f o u n d i n o p e r a t i v e . F o r t u n a t e l y , t h e l o a d c e l l f i t i n t o a n a d a p t e r c r a d l e w h i c h 127 a l l o w e d t e n s i l e and c o m p r e s s i v e l o a d s t o be t r a n s f e r r e d t o t h e l o a d c e l l as a s h e a r l o a d and moment. T h i s a d a p t e r c r a d l e a l l o w e d t h e l o a d c e l l t o o p e r a t e as a c a n t i l e v e r beam and p r o v i d e d an a c c u r a t e and s e n s i t i v e s i g n a l f r o m t h e l o a d c e l l . The dynamometer and a d a p t e r b r a c k e t i s shown i n F i g u r e 9 . 1 4 . 6 . 2 . 5 . 1 . 2 DYNAMOMETER CALIBRATION A t h o r o u g h i n v e s t i g a t i o n o f t h e l o a d c e l l was c o n d u c t e d b y G o o d r i d g e (1986) t o c a l i b r a t e t h e dynamometer . S t a t i c as w e l l as d y n a m i c c a l i b r a t i o n s were c a r r i e d o u t on t h e l o a d c e l l . H i s r e s u l t s r e v e a l e d t h a t b o t h o f t h e two l o a d c e l l s gave a c c u r a t e r e s u l t s , u s u a l l y t o w i t h i n l e s s t h a n 1% o f f u l l s c a l e . Upon r e a d i n g h i s r e c o m m e n d a t i o n s , i t was d e c i d e d t h a t dynamometer number 1 be u s e d f o r a l l e x p e r i m e n t s s i n c e i t h a d t h e l e a s t amount o f e r r o r o f t h e two dynamomete rs . Howeve r , i t was d e c i d e d t h a t a r e c a l i b r a t i o n o f t h e dynamometer be done b e f o r e e a c h s e t o f e x p e r i m e n t s t o v e r i f y t h e a c c u r a c y o f t h e l o a d c e l l . T h i s c a l i b r a t i o n c o n s i s t e d o f s t a t i c l o a d s b e i n g a p p l i e d on t o t h e l o a d c e l l a d a p t e r c r a d l e and t h e v o l t a g e r e a d i n g s o f t h e l o a d c e l l b e i n g r e c o r d e d . A c a l i b r a t i o n p r o g r a m e x i s t i n g on t h e O . E . C d a t a c o l l e c t i o n s y s t e m was u s e d t o o b t a i n t h e s l o p e and i n t e r c e p t o f t h e c a l i b r a t i o n d a t a . From t h e s e r e s u l t s , i t was f o u n d t h a t t h e dynamometer h a d a n RMS e r r o r o f a p p r o x i m a t e l y ± 6 . 9 N e w t o n s . The t e s t s showed t h a t t h e l o a d c e l l p r o v i d e d a v e r y l i n e a r r e s p o n s e f o r a l l l o a d s a p p l i e d t o i t . 128 6 . 2 . 5 . 2 UNIVERSAL SHEAR BEAMS The e x i s t i n g l o a d c e l l u s e d f o r t h e o t h e r e x p e r i m e n t s was t o o l a r g e f o r t h o s e t h a t d e t e r m i n e t h e i n d u c e d s u r g e and h e a v e f o r c e s due t o a n i d e n t i c a l a d j a c e n t c y l i n d e r i n h e a v e m o t i o n . Two TM U n i v e r s a l S h e a r Beams (USB) m a n u f a c t u r e d b y H o t t i n g e r B a l d w i n M e a s u r e m e n t s I n c . o f F ram ingham, M a s s . , were b o t h compac t enough t o f i t e a s i l y i n s i d e t h e c y l i n d e r mode l and s e n s i t i v e enough t o measu re s m a l l f o r c e s a c c u r a t e l y . The s p e c i f i c a t i o n s o f t h e TM U n i v e r s a l S h e a r Beams a r e as f o l l o w s : T a b l e 6 . 2 . 5 . 2 - 1 UNIVERSAL SHEAR BEAMS™ SPECIF ICATIONS M o d e l R a t e d C a p a c i t y R a t e d O u t p u t Non L i n e a r i t y H y s t e r i s i s N o n - R e l i a b i l i t y Max . E x c i t a t i o n V o l t a g e S i d e L o a d R e j e c t i o n Maximum S a f e L o a d USB200 8 8 9 . 9 N ( 2 0 0 1 b s . ) 1 .9925 mV/V 0.02% o f r a t e d o u t p u t 0 .01% o f r a t e d o u t p u t 0 .01% o f r a t e d o u t p u t 18 V 5 0 0 : 1 150% U n f o r t u n a t e l y , one o f t h e u n i t s t u r n e d o u t t o be d e f e c t i v e and h a d t o be s e n t b a c k t o t h e m a n u f a c t u r e r f o r r e p a i r s . A b r a c k e t was f a b r i c a t e d i n o r d e r t o o r i e n t a t e t h e s h e a r beams i n t h e c o r r e c t manner : r e s u l t i n g i n one s h e a r beam i n t h e h o r i z o n t a l p o s i t i o n i n o r d e r t o measure i n d u c e d s u r g e f o r c e s and t h e o t h e r i n a v e r t i c a l p o s i t i o n t o measure i n d u c e d h e a v e f o r c e s . A p h o t o g r a p h o f t h e a r r a n g e m e n t i s shown i n F i g u r e 9 . 1 5 . The e n t i r e a p p a r a t u s was l o w e r e d i n t o t h e c y l i n d e r and b o l t e d i n t o 129 p l a c e . 6 . 2 . 5 . 2 . 1 UNIVERSAL SHEAR BEAM™ CALIBRATION P r i o r t o i n s t a l l a t i o n o f t h e s h e a r beams , a s t a t i c c a l i b r a t i o n o f t h e i n s t r u m e n t s was c o n d u c t e d . I n o r d e r t o c a l i b r a t e t h e s h e a r beams, w e i g h t s were p l a c e d on them and v o l t a g e r e a d i n g s were t a k e n . A g a i n , a c a l i b r a t i o n p r o g r a m e x i s t i n g on t h e O . E . C d a t a c o l l e c t i o n s y s t e m was u s e d t o o b t a i n t h e s l o p e and i n t e r c e p t o f t h e c a l i b r a t i o n d a t a . B o t h U S B ' s p r o v i d e d a v e r y l i n e a r r e s p o n s e f o r a l l a p p l i e d l o a d s w i t h a maximum e r r o r o f l e s s t h a n 0 .3% o r a p p r o x i m a t e l y 2 . 6 7 N e w t o n s . A f t e r t a l k i n g t o t h e t e c h n i c i a n s a t t h e Ocean E n g i n e e r i n g C e n t r e i t was d e c i d e d t h a t a dynamic c a l i b r a t i o n was n o t r e q u i r e d . T h i s d e c i s i o n was r e a c h e d b e c a u s e t h e s e e x a c t t y p e o f s h e a r beams h a v e b e e n u s e d i n numerous dynamic t e s t s o f mode l s h i p s o v e r many y e a r s . The r e s p o n s e o f t h e s h e a r beams h a s a l w a y s b e e n a c c u r a t e and i n s t a n t a n e o u s f o r a f u l l r a n g e o f f r e q u e n c i e s . I t was d e c i d e d TM t h a t t h e s e t y p e s o f U n i v e r s a l S h e a r Beams a r e w e l l s u i t e d t o p r o v i d e a c c u r a t e r e s u l t s f o r i n d u c e d f o r c e t e s t s . 6 . 2 . 5 . 3 YO-YO POSITION TRANSDUCER I n o r d e r t o measure t h e d i s p l a c e m e n t o f t h e c y l i n d e r m o d e l s u s e d i n e a c h o f t h e t h r e e s e t s o f e x p e r i m e n t s , a y o - y o t y p e p o s i t i o n t r a n s d u c e r was u s e d . S i n c e t h e m o t i o n o f t h e c y l i n d e r 130 m o d e l s was a l w a y s s i n u s o i d a l , one c o u l d u s e t h e d i s p l a c e m e n t r e a d i n g s t o d e t e r m i n e t h e a m p l i t u d e o f t h e v e l o c i t y (V = wX) and 2 o f t h e a c c e l e r a t i o n (A = -co X ) . The y o - y p p o s i t i o n t r a n s d u c e r was f i x e d o n t o t h e m o t i o n g e n e r a t o r f r a m e ; i t s r e c i p r o c a t i n g c a b l e was a t t a c h e d t o t h e s l i d i n g p l a t e o f t h e s c o t c h - y o k e mechan ism w h i c h a t t a c h e d d i r e c t l y t o t h e o s c i l l a t i n g c y l i n d e r . The c a l i b r a t i o n was c o n d u c t e d b y s l i d i n g t h e p l a t e and m e a s u r i n g t h e s i g n a l as a f u n c t i o n o f d i s p l a c e m e n t . The c a l i b r a t i o n showed t h a t t h e y o - y o p o s i t i o n t r a n s d u c e r p r o v i d e d v e r y l i n e a r r e s p o n s e f o r a l l d i s p l a c e m e n t s w i t h v i r t u a l l y no e r r o r . The t y p e o f y o - y o p o t e n t i o m e t e r u s e d was a L o c k h e e d E l e c t r o n i c s Company M o d e l #1119 and t h e s p e c i f i c a t i o n s a r e as f o l l o w s : T a b l e 6 . 2 . 5 . 3 - 1 YO-YO POSITION TRANSDUCER SPECIF ICATIONS Maximum E x c i t a t i o n L i n e a r i t y Max . A c c e l e r a t i o n o f A c t u a t i o n C a b l e 15 VDC ±0.065% o f f u l l s c a l e 4 g ' s 131 APPENDIX B 7 SOFTWARE USED IN THE EXPERIMENTS Two d i s t i n c t g r o u p s o f s o f t w a r e were u s e d f o r t h e s e e x p e r i m e n t s : d a t a a c q u i s i t i o n s o f t w a r e and d a t a a n a l y s i s s o f t w a r e . The d a t a a c q u i s i t i o n s o f t w a r e was d e v e l o p e d p r i m a r i l y b y p e r s o n n e l o f B . C . R e s e a r c h C o r p o r a t i o n and i s u s e d i n c o n j u n c t i o n w i t h t h e TM d a t a a c q u i s i t i o n s y s t e m on t h e MINC 11 m i n i c o m p u t e r . T h i s s o f t w a r e i s u s e d f o r a l l d a t a c o l l e c t i o n on a l l t y p e s o f e x p e r i m e n t s c o n d u c t e d a t t h e Ocean E n g i n e e r i n g C e n t r e . The d a t a a n a l y s i s s o f t w a r e was o r i g i n a l l y d e v e l o p e d b y Doug G o o d r i d g e (1986) e t . a l . f o r d a t a a n a l y s i s o f raw d a t a o b t a i n e d i n e x p e r i m e n t s o f h y d r o d y n a m i c e f f e c t s on c y l i n d e r s i n s u r g e m o t i o n . T h i s a u t h o r a d a p t e d t h e r o u t i n e s w i t h a number o f m o d i f i c a t i o n s i n o r d e r t o a n a l y z e d a t a c o l l e c t e d f r o m t h e v a r i o u s e x p e r i m e n t s t h a t were c o n d u c t e d . The d a t a a n a l y s i s s o f t w a r e was w r i t t e n i n o r d e r t o p r o c e s s d a t a f r o m i t s o r i g i n a l raw b i n a r y code t h r o u g h t o p r e p a r i n g a t a b u l a t i o n o f t h e f i n a l r e s u l t s . A l l o f t h e r o u t i n e s TM were w r i t t e n i n F o r t r a n code t o r u n on t h e D i g i t a l VAX 1 1 / 7 5 0 compu te r i n t h e M e c h a n i c a l E n g i n e e r i n g Depa r tmen t o f U . B . C . T h r e e c o m p l e t e s e t s o f d a t a a n a l y s i s s o f t w a r e were d e v e l o p e d . The f i r s t was d e s i g n e d t o p r o c e s s r e s u l t s f r o m h y d r o d y n a m i c h e a v e t e s t s o f a compound c y l i n d e r ; t h e s e c o n d was d e s i g n e d t o p r o c e s s r e s u l t s f r o m i n d u c e d s u r g e and h e a v e h y d r o d y n a m i c f o r c e s due t o an a d j a c e n t , i d e n t i c a l c y l i n d e r i n heave m o t i o n ; and t he t h i r d was d e s i g n e d t o 132 p r o c e s s r e s u l t s f r o m h y d r o d y n a m i c h e a v e t e s t s o f a s i n g l e c y l i n d e r i n a n a r r o w c h a n n e l . A l l t h r e e s o f t w a r e p a c k a g e s a r e q u i t e s i m i l a r and s h a r e many o f t h e same s u b r o u t i n e s . 7 . 1 DATA ACQUISIT ION SOFTWARE TM A l l o f t h e d a t a a c q u i s i t i o n s o f t w a r e was r u n on t h e MINC 11 m i n i c o m p u t e r d e s c r i b e d i n S e c t i o n 6 . 2 . 3 . 2 . A l l raw d a t a was s t o r e d on e i g h t i n c h d i s k e t t e s and l a t e r t r a n s f e r r e d t o t h e VAX™ 1 1 / 7 5 0 c o m p u t e r f o r d a t a a n a l y s i s . The f o l l o w i n g e x p l i c a t i o n i s n o t i n t e n d e d as a u s e r s manua l f o r t h e d a t a a c q u i s i t i o n s o f t w a r e b u t as a b r i e f i n s i g h t t o e a c h o f t h e p r o g r a m s . F u r t h e r i n f o r m a t i o n on t h e p rog rams may be o b t a i n e d f r o m d o c u m e n t a t i o n a v a i l a b l e a t B . C . R e s e a r c h . 7 . 1 . 1 ADCAL PROGRAM The p r o g r a m ADCAL i s t h e i n i t i a l c a l i b r a t i o n p r o g r a m w h i c h i s r u n p r i o r t o any a c t u a l d a t a a c q u i s i t i o n . The MINC™ 11 m i n i c o m p u t e r r e a d s v o l t a g e s i g n a l s and c o n v e r t s t h e s e s i g n a l s t o a d i g i t a l number b y a p r o c e s s d e s c r i b e d i n S e c t i o n 6 . 2 . 3 . 2 . The p r o g r a m c r e a t e s a f i l e w h i c h d e f i n e s t h e c a l i b r a t i o n f a c t o r s r e l a t i n g t h e d i g i t a l t r a n s d u c e r s i g n a l s t o t h e p h y s i c a l q u a n t i t i e s t h a t t h e y a r e m e a s u r i n g . I n o r d e r t o r u n t h i s p r o g r a m , i t i s e s s e n t i a l t o s e t - u p t h e i n s t r u m e n t a t i o n e x a c t l y as i t w o u l d be d u r i n g t h e t e s t p h a s e . The c a l i b r a t i o n i s c a r r i e d o u t b y e n t e r i n g 133 t h e a p p r o p r i a t e c h a n n e l number when i t i s p r o m p t e d f o r b y t h e p r o g r a m . The o u t p u t o f ADCAL i s a 16 x 5 m a t r i x w i t h e a c h row c o r r e s p o n d i n g t o e a c h o f 16 c h a n n e l s . The c o l u m n p a r a m e t e r s c o n t a i n s l o p e , y - i n t e r c e p t , v a r i a n c e , d e l t a - y , and t h e number o f s a m p l e s t a k e n f o r e a c h c h a n n e l . ADCAL a l l o w s t h e u s e r t o l o o k a t a g r a p h o f e a c h c a l i b r a t i o n i m m e d i a t e l y a f t e r i t i s c o n d u c t e d i n o r d e r t o v e r i f y t h e a c c u r a c y o f t h e c a l i b r a t i o n . 7 . 1 . 2 ADMAIN PROGRAM f-The p r o g r a m ADMAIN i s t h e a c t u a l d a t a a c q u i s i t i o n p r o g r a m u s e d f o r t h e t e s t s . I t i s c a p a b l e o f s a m p l i n g up t o 16 c h a n n e l s s i m u l t a n e o u s l y a t a s a m p l i n g r a t e b e t w e e n 1 and 1000 H e r t z . I n r u n n i n g t h i s p r o g r a m t h e u s e r i s p r o m p t e d f o r a s e r i e s o f i n p u t s : t h e s e i n c l u d e c a l i b r a t i o n f i l e name as p r o d u c e d b y ADCAL, comments , d a t a f i l e name, number o f c h a n n e l s t o be s a m p l e d , t h e samp le p e r i o d , and t h e d a t a f i l e s i z e ( w h i c h l i m i t s t h e d u r a t i o n o f t h e a c t u a l d a t a a c q u i s i t i o n ) . A f t e r a s i n g l e t e s t i s c o m p l e t e d t h e p r o g r a m a u t o m a t i c a l l y t e r m i n a t e s ; i n o r d e r t o be r u n a g a i n , a l l o f t h e p a r a m e t e r s must be r e - e n t e r e d . A l l t h e s a m p l e d d a t a i s s t o r e d i n a b i n a r y m u l t i p l e x e d f o r m i n o r d e r t o c o n s e r v e d i s k s p a c e . I n o r d e r t o d e m u l t i p l e x t h e d a t a i n t o s t a n d a r d A S C I I c o d e , t h e p r o g r a m ADMUX must be r u n . The p r o g r a m ADMAIN h a s t h e c a p a b i l i t y t o a l l o w r e a l t i m e v i e w i n g o f e a c h c h a n n e l p r i o r t o a c t u a l s a m p l i n g . T h i s i s u s e f u l t o a l l o w v e r i f i c a t i o n t h a t a l l c o n n e c t i o n s h a v e b e e n made p r o p e r l y 134 and t h a t e a c h c h a n n e l i s r e c o r d i n g . Howeve r , t h e g r a p h i c s i g n a l o u t p u t o f ADMAIN h a s v e r y p o o r r e s o l u t i o n and i s somet imes i n s u f f i c i e n t i n i n d i c a t i n g w h e t h e r o r n o t a s i g n a l i s i n f a c t p r e s e n t . I f t h i s i s t h e c a s e , i t i s p o s s i b l e t o r u n a samp le t e s t and d e m u l t i p l e x t h e c h a n n e l a f t e r d a t a h a s b e e n a c q u i r e d t o c h e c k t h e o u t p u t . 7 . 1 . 3 ADMUX PROGRAM The p r o g r a m ADMUX i s u s e d t o d e m u l t i p l e x t h e d a t a c o l l e c t e d b y t h e p r o g r a m ADMAIN and s t o r e t h e d a t a i n s p e c i f i e d f i l e s . B e f o r e ADMUX i s r u n t h e c a l i b r a t i o n f a c t o r s c r e a t e d b y ADCAL a r e i n c o r p o r a t e d w i t h t h e raw d a t a f i l e s t o g i v e t h e o u t p u t i n t h e s p e c i f i e d u s e r u n i t s . I n o r d e r t o b e g i n d e m u l t i p l e x i n g t h e u s e r i s p r o m p t e d f o r t h e c h a n n e l number t o be d e m u l t i p l e x e d and t h e f i l e name i n w h i c h t h e d a t a i s t o be s t o r e d . The t i m e i t t a k e s t o r u n ADMUX p r o g r a m on t h e MINC™ c o m p u t e r i s e x t r e m e l y l e n g t h y . T h e r e f o r e , t h i s p r o g r a m was o n l y r u n p e r i o d i c a l l y on t h e d a t a f i l e s d u r i n g t h e a c t u a l t e s t i n g i n o r d e r t o q u i c k l y c h e c k d a t a f i l e s t o e n s u r e t h a t e a c h c h a n n e l was r e c e i v i n g a n o u t p u t s i g n a l . D u r i n g t h e a c t u a l d a t a a n a l y s i s a TM p r o g r a m s i m i l a r t o ADMUX i s r u n on t h e VAX 1 1 / 7 5 0 t o d e m u l t i p l e x t h e d a t a . 135 7 . 1 . 4 ADVIEW PROGRAM The ADVIEW p r o g r a m g r a p h i c a l l y d i s p l a y s t h e o u t p u t o b t a i n e d f r o m ADMUX on t h e v i d e o t e r m i n a l . T h i s a l l o w s v i s u a l i n s p e c t i o n o f t h e q u a l i t y o f d a t a f r o m any p a r t i c u l a r c h a n n e l . T h i s p r o g r a m p l o t s t h e r e s u l t s on a r e a l t i m e a x i s and a l l o w s t h e u s e r t o s c a l e t h e r e s u l t s . ADVIEW t a k e s c o n s i d e r a b l e t i m e t o r u n on t h e MINC™ 11 c o m p u t e r a n d , t h e r e f o r e , was o n l y r u n o c c a s i o n a l l y t o i n s p e c t t h e s a m p l e d r e s u l t s . 7 . 2 DATA ANALYSIS SOFTWARE The d a t a a n a l y s i s s o f t w a r e was u s e d t o a n a l y z e a l l d a t a c o l l e c t e d f r o m t h e t h r e e e x p e r i m e n t s . Two s e p a r a t e b u t v e r y s i m i l a r p r o g r a m s were d e v e l o p e d f o r e a c h e x p e r i m e n t : a ) P r o g r a m QW t o a n a l y z e d a t a c o l l e c t e d f r o m t h e t e s t s t o d e t e r m i n e h y d r o d y n a m i c c o e f f i c i e n t s o f compound c y l i n d e r s i n h e a v e m o t i o n . b ) P r o g r a m DF t o a n a l y z e d a t a c o l l e c t e d f r o m t h e i n d u c e d s u r g e and h e a v e f o r c e s on a s i n g l e c y l i n d e r due t o a n a d j a c e n t i d e n t i c a l c y l i n d e r i n heave m o t i o n . E a c h p r o g r a m and s u b r o u t i n e i s i n d i v i d u a l l y documented b u t a g e n e r a l o v e r v i e w i s e s t a b l i s h e d t o g i v e an e x p l a n a t i o n o f how t h e e n t i r e p a c k a g e w o r k s . R e f e r t o F i g u r e 8 . 8 f o r a f l o w c h a r t o f t h e p r o g r a m p a c k a g e . 136 7 . 2 . 1 QW PROGRAM P r o g r a m QW i s t h e m a i n p r o g r a m t o a n a l y z e d a t a c o l l e c t e d f r o m h e a v e h y d r o d y n a m i c e x p e r i m e n t s on compound c y l i n d e r s . The p r o g r a m i s b a s e d on t h e p r o g r a m DS u s e d b y G o o d r i d g e (1986) i n a n a l y z i n g e x p e r i m e n t a l d a t a o f s u r g e h y d r o d y n a m i c t e s t s o f c y l i n d e r s . QW p r o m p t s t h e u s e r f o r v a r i o u s i n p u t s and c a l l s t h e v a r i o u s s u b r o u t i n e s u s e d i n t h e d a t a a n a l y s i s . The u s e r i s p r o m p t e d f o r t h e f o l l o w i n g : 1) How many t e s t s a r e t o be i n p u t . 2) Raw d a t a f i l e s t o be p r o c e s s e d . 3) Name o f t h e f i l e t o c o n t a i n r e s u l t s . 4 ) C y l i n d e r mode l t y p e ( s i n g l e , d o u b l e , o r t r i p l e ) . 5) C y l i n d e r s t e p d r a f t . 6) Mass o f c y l i n d e r and h a r d w a r e b e l o w t h e dynamometer . 7) F i l t e r f a c t o r , o p t i o n t o change f i l t e r f a c t o r f r o m d e f a u l t v a l u e o f f o u r (See S e c t i o n 7 . 2 . 1 . 3 f o r d e t a i l s ) . 8) C h a n n e l a s s i g n m e n t s , o p t i o n t o change f r o m d e f a u l t a s s i g n m e n t s . T h i s r e q u i r e d i n p u t a l o n g w i t h t h e d a t a c o n t a i n e d i n t h e raw d a t a f i l e i s s u f f i c i e n t t o a l l o w t h e c o m p u t e r t o a u t o m a t i c a l l y c a l c u l a t e t h e h y d r o d y n a m i c c o e f f i c i e n t s and t h e o t h e r u s e f u l i n f o r m a t i o n . The p r o g r a m i s c a p a b l e o f a n a l y z i n g up t o e i g h t i n p u t c h a n n e l s o f d a t a c o l l e c t e d a t t h e t o w i n g t a n k . Howeve r , i n t h e s e e x p e r i m e n t s o n l y f o u r c h a n n e l s o f d a t a were c o l l e c t e d . I t i s i m p o r t a n t t o be s u r e t h a t t h e c h a n n e l a s s i g n m e n t s u s e d w h i l e c o l l e c t i n g d a t a on t h e MINC™ 11 m i n i compu te r a t B . C . R e s e a r c h m a t c h t h e d e f a u l t c h a n n e l a s s i g n m e n t s w r i t t e n i n t o t h e p r o g r a m . 137 The c h a n n e l a s s i g n m e n t s a r e as f o l l o w s : T a b l e 7 . 2 . 1 - 1 T r a n s d u c e r C h a n n e l A s s i g n m e n t s and U s e r U n i t s Assumed b y "QW" Computer C a l i b r a t i o n C h a n n e l # T r a n s d u c e r Q u a n t i t y M e a s u r i n g U n i t s 3 Y o - Y o P o s i t i o n C y l i n d e r P o s i t i o n cm 2 Dynamometer S u r g e S u r g e F o r c e N 1 Dynamometer P i t c h P i t c h F o r c e N-m 0 Dynamometer Heave Heave F o r c e N I f t h e u s e r h a s u s e d d i f f e r e n t c h a n n e l a s s i g n m e n t s , a s i m p l e t a s k o f e n t e r i n g t h e new c h a n n e l a s s i g n m e n t s c a n be done p r i o r t o t h e a n a l y s i s o f t h e d a t a . E v e n t h o u g h t h e h e a v e c h a n n e l was r e c o r d i n g d a t a i n t h e s e e x p e r i m e n t s , t h e d a t a was e x t r e m e l y n o i s y and deemed u n a c c e p t a b l e f o r a n a l y s i s ; t h e r e f o r e , t h e p r o g r a m d i d n o t u t i l i z e t h e h e a v e c h a n n e l d a t a . The p r o g r a m c o n v e r t s a l l d a t a t o S I u n i t s f r o m t h e i m p e r i a l u n i t s t h e d a t a c o l l e c t i o n s y s t e m u s e s a t t h e t o w i n g t a n k . Howeve r , i f a c o n v e r s i o n change i s r e q u i r e d , a s i m p l e change o f s c a l i n g w i t h i n t h e f i n a l s u b r o u t i n e i s a l l t h a t i s n e e d e d t o a c c o m p l i s h t h i s . A s t h e p r o g r a m i s r u n , QW w i l l a u t o m a t i c a l l y c r e a t e f i l e s t o s t o r e d a t a u s e d i n t h e i n t e r m e d i a t e s t a g e s . T h e s e f i l e s a r e a s s i g n e d names w h i c h a r e g e n e r a l e x t e n s i o n s t o t h e o r i g i n a l f i l e i name. T a b l e 7 . 2 . 1 - 2 g i v e s an examp le o f t h e name e x t e n s i o n s e q u e n c e c a r r i e d o u t on a samp le raw d a t a f i l e c a l l e d DATA.DAT as p e r f o r m e d b y p r o g r a m QW: 138 T a b l e 7 . 2 . 1 - 2 F i l e Name E x t e n s i o n s U s e d b y "QW" DATA.DAT Raw d a t a f i l e name assumed f o r t h i s t a b l e . DATA_DISP.DAT DATA_SURG.DAT DATA_PITC.DAT DATA HEAV.DAT D e m u l t i p l e x e d o u t p u t f r o m y o - y o t r a n s d u c e r . D e m u l t i p l e x e d s u r g e o u t p u t f r o m dynamometer . D e m u l t i p l e x e d p i t c h o u t p u t f r o m dynamometer . D e m u l t i p l e x e d h e a v e o u t p u t f r o m dynamometer . DATA_DIS P_RAW.DAT DATA_SURG_RAW.DAT o o DATA_DIS P_FFT.DAT DATA_SURG_FFT.DAT o o Raw u n f i l t e r e d d a t a p r i o r t o d e m u l t i p l e x i n g . Raw u n f i l t e r e d d a t a p r i o r t o d e m u l t i p l e x i n g . S i m i l a r l y f o r t h e o t h e r f i l e s l i s t e d . R e s u l t s o f F a s t F o u r i e r T r a n s f o r m a t i o n o f t i m e doma in d a t a c o n t a i n e d i n DATA_DISP.DAT. R e s u l t s o f F a s t F o u r i e r T r a n s f o r m a t i o n o f t i m e doma in d a t a c o n t a i n e d i n DATA_SURG.DAT. S i m i l a r l y f o r t h e o t h e r f i l e s l i s t e d . DATA T IME.DAT F i l e c o n t a i n i n g t h e r e s u l t s o f a R e a l T ime A n a l y s i s o f t h e d a t a , where t h e componen ts o f s u r g e and h e a v e f o r c e a r e i n p h a s e w i t h t h e a c c e l e r a t i o n and v e l o c i t y . DATA FFT .DAT S i m i l a r l y f o r t h e o t h e r f i l e s l i s t e d . Summary o f t h e r e s u l t s o f t h e F a s t F o u r i e r T r a n s f o r m a t i o n , c o n t a i n i n g t h e maximum r e c o r d e d a m p l i t u d e i n a F o u r i e r S p e c t r u m , t h e f r e q u e n c y a t w h i c h i t o c c u r s , and t h e p h a s e a n g l e a s s o c i a t e d w i t h t h i s componen t . S i m i l a r l y f o r t h e o t h e r f i l e s l i s t e d . 7 . 2 . 1 . 1 DEMUX SUBROUTINE The s u b r o u t i n e DEMUX was d e v e l o p e d f r o m t h e ADMUX p r o g r a m ( S e c t i o n 7 . 1 . 3 ) u s e d w i t h t h e d a t a a c q u i s i t i o n p a c k a g e a t t h e 139 t o w i n g t a n k . The p r o g r a m was t r a n s f e r r e d t o t h e VAX 1 1 / 7 5 0 and c o n v e r t e d t o a s u b r o u t i n e . DEMUX d e m u l t i p l e x e s t h e o r i g i n a l raw d a t a f i l e s one c h a n n e l a t a t i m e . The d a t a f r o m e a c h c h a n n e l i s w r i t t e n i n A S C I I code i n s e p a r a t e f i l e s w h i c h a r e g i v e n names l i k e DATA_DISP.DAT e t . a l . as shown i n T a b l e 7 . 2 . 1 - 2 . E a c h new f i l e c o n t a i n s d a t a t h a t h a s b e e n d e m u l t i p l e x e d b y i n c o r p o r a t i n g c a l i b r a t i o n v a l u e s t h a t f i l e ADCAL ( S e c t i o n 7 . 1 . 1 ) c r e a t e d p r i o r t o d a t a c o l l e c t i o n a t t h e Ocean E n g i n e e r i n g C e n t r e . The d e m u l t i p l e x e d d a t a f i l e c o n t a i n s t h e f o l l o w i n g i n f o r m a t i o n : t h e number o f samp le p o i n t s , t h e samp le p e r i o d i n m s e c , and t h e a c t u a l m e a s u r e d t r a n s d u c e r o u t p u t m u l t i p l i e d b y t h e c a l i b r a t i o n f a c t o r . The d a t a i s e x p r e s s e d as a n a r r a y o f d a t a s a m p l e d a t e q u a l t i m e i n t e r v a l s . 7 . 2 . 1 . 2 TREND SUBROUTINE P r i o r t o any e x p e r i m e n t s t a k i n g p l a c e , a l l i n i t i a l t r a n s d u c e r s i g n a l s we re a d j u s t e d t o z e r o i n o r d e r t h a t , i d e a l l y , a l l o f t h e s a m p l e d d a t a s h o u l d h a v e a mean v a l u e v e r y n e a r z e r o — p r o v i d e d , o f c o u r s e , t h a t t h e s a m p l i n g t i m e was s u f f i c i e n t l y l o n g . Howeve r , i n most c a s e s i t i s n e a r l y i m p o s s i b l e t o s e t t h e i n i t i a l o u t p u t t o TM e x a c t l y z e r o , b e c a u s e t h e z e r o b a l a n c e on t h e ST41B s i g n a l c o n d i t i o n e r i s n o t v e r y a c c u r a t e . T h e r e f o r e , t h e s u b r o u t i n e TREND was d e v e l o p e d t o remove a l l t r e n d s f r o m t h e d a t a . T h i s i s a c c o m p l i s h e d b y d e t e r m i n i n g t h e l e a s t s q u a r e s f i t f o r a s t r a i g h t l i n e t h r o u g h a l l t h e d a t a p o i n t s and s u b t r a c t i n g t h i s 140 l i n e f r o m t h e s e same d a t a p o i n t s . T h i s t r e n d l i n e s h o u l d v e r y n e a r l y be a s t r a i g h t l i n e , so what one i s i n f a c t d o i n g i s r e m o v i n g a "DC o f f s e t " v a l u e f r o m t h e d a t a . Any d e v i a t i o n f r o m a h o r i z o n t a l l i n e may be a t t r i b u t e d t o end e f f e c t s , s i n c e t h e r e w i l l be some e r r o r i n t r o d u c e d b y n o t s t a r t i n g and s t o p p i n g s a m p l i n g a t t h e same p o i n t i n a c y c l e . H o w e v e r , t h e m a g n i t u d e o f t h i s e r r o r may be r e d u c e d b y i n c r e a s i n g t h e number o f c y c l e s w i t h i n a s a m p l e . The m a g n i t u d e o f t h e e r r o r i s o f t h e o r d e r e q u a l t o t h e a m p l i t u d e o f t h e s i g n a l d i v i d e d b y t h e number o f c o m p l e t e c y c l e s . T h e r e may be a l s o be some d e v i a t i o n f r o m a s t r a i g h t l i n e "DC o f f s e t " due t o a t r a n s i e n t phenomenon, b u t t h e s e e f f e c t s a r e n o t l i k e l y t o be s i g n i f i c a n t . 7 . 2 . 1 . 3 F I L T E R SUBROUTINE The s u b r o u t i n e F I L T E R i s u s e d t o smooth p e r i o d i c d a t a . The s u b r o u t i n e was d e v e l o p e d b y A u d a n e l and Oldham who p u b l i s h e d t h e p r o g r a m i n t h e O c t o b e r 1985 i s s u e o f BYTE Magazine. D e t a i l s c o n c e r n i n g t h e p r o c e d u r e i m p l e m e n t e d w i t h i n t h e p r o g r a m c a n be f o u n d i n t h i s r e f e r e n c e . G e r r y R o h l i n g (1986) c o n v e r t e d t h e p r o g r a m f r o m B a s i c i n t o F o r t r a n c o d e , and Doug G o o d r i d g e (1986) made f u r t h e r m o d i f i c a t i o n s t o c o n v e r t i t i n t o a F o r t r a n s u b r o u t i n e . The F I L T E R s u b r o u t i n e r e q u i r e s t h a t t h e i n p u t d a t a f i l e be t h e same f o r m a t as t h e o u t p u t f r o m t h e DEMUX s u b r o u t i n e . A l s o , t h e s u b r o u t i n e u t i l i z e s a f i l t e r f a c t o r r a n g i n g f r o m 2 t o N , where 141 N i s t h e number o f d a t a p o i n t s . A f i l t e r f a c t o r o f 2 r e p r e s e n t s no f i l t e r i n g and N r e p r e s e n t s t h e maximum f i l t e r i n g . The d e g r e e o f f i l t e r i n g i s r o u g h l y l i n e a r l y p r o p o r t i o n a l t o t h e q u o t i e n t N , d i v i d e d b y t h e f i l t e r f a c t o r . The b e s t way t o d e t e r m i n e a s u i t a b l e f i l t e r f a c t o r i s b y e x p e r i m e n t a t i o n w i t h t h e t i m e doma in t r a c e s t o v i s u a l l y i n s p e c t t h e smoo thness o f t h e c u r v e . I t was f o u n d t h r o u g h t h i s ; m e t h o d t h a t a v a l u e o f b e t w e e n 3 and 8 was mos t a p p r o p r i a t e . F o r t h e s e e x p e r i m e n t s , i n o r d e r t o s a t i s f y t h e s t r o n g n e e d f o r f i l t e r i n g when t h e s i g n a l o u t p u t was weak , i t was d e c i d e d t o s e t a d e f a u l t f i l t e r f a c t o r e q u a l t o 4 . Howeve r , f o r t h e c a l c u l a t i o n o f h e a v e h y d r o d y n a m i c c o e f f i c i e n t s o f a c y l i n d e r i n a n a r r o w c h a n n e l t h e f i l t e r f a c t o r was i n c r e a s e d t o 8 t o compensa te f o r t h e more p r o n o u n c e d n e e d f o r s m o o t h i n g o f d a t a . A f i l t e r f a c t o r e q u a l t o 4 i s a p p r o x i m a t e l y e q u i v a l e n t t o a 5 H e r t z l o w p a s s f i l t e r f o r d a t a o f t h e t y p e c o l l e c t e d i n t h e s e e x p e r i m e n t s . The o u t p u t f r o m t h e s u b r o u t i n e F I L T E R was w r i t t e n t o a f i l e o f t h e same name as t h e i n p u t f i l e , b u t w i t h a h i g h e r v e r s i o n number . The o u t p u t d a t a was w r i t t e n i n t h e same f o r m a t as t h e i n p u t d a t a f i l e . 7 . 2 . 1 . 4 FOURT SUBROUTINE The s u b r o u t i n e FOURT u s e s a F a s t F o u r i e r T r a n s f o r m a t i o n (FFT) t o c a l c u l a t e t h e d i s c r e t e F o u r i e r S p e c t r u m f r o m p e r i o d i c d a t a . The p r o g r a m was o r i g i n a l l y w r i t t e n b y Norman B r e n n e r o f MIT L i n c o l n L a b o r a t o r y i n J u n e 1 9 6 8 . The s u b r o u t i n e was t r a n s f e r r e d 142 f r o m t h e U . B . C . MTS m a i n f rame s y s t e m t o t h e VAX 1 1 / 7 5 0 c o m p u t e r i n t h e M e c h a n i c a l E n g i n e e r i n g D e p a r t m e n t . The s o u r c e code o f t h e p r o g r a m i s w e l l documented and a more d e t a i l e d e x p l a n a t i o n o f t h e p r o g r a m may be f o u n d i n a p a p e r e n t i t l e d UBC Fourt, a v a i l a b l e f r o m t h e U . B . C . C o m p u t i n g C e n t r e . The F a s t F o u r i e r T r a n s f o r m a t i o n i s a s h o r t c u t method f o r o b t a i n i n g t h e f r e q u e n c y s p e c t r u m o f a t i m e s e r i e s . The FFT assumes a f i n i t e t i m e s e r i e s t o be r e p r e s e n t a t i v e o f an i n f i n i t e t i m e s e r i e s . F o r e x a m p l e , i f a samp le were o f d u r a t i o n 5 s e c o n d s , t h e FFT w o u l d assume t h a t t h e t r a c e o v e r t h e p o r t i o n 0 s e c o n d t o 5 s e c o n d s i s t h e f i r s t p o r t i o n o f an i n f i n i t e l y l o n g s a m p l e . E a c h s u b s e q u e n t 5 s e c o n d i n t e r v a l i s assumed i d e n t i c a l t o t h e f i r s t . T h e r e f o r e , i t i s i m p o r t a n t t o r e a l i z e t h a t i f t h e end p o i n t s a r e n o t e q u a l , t h e n jump d i s c o n t i n u i t i e s w i l l e x i s t a t t h e s e p o i n t s and i n t r o d u c e e r r o r s i n t o t h e t r a n s f o r m a t i o n . I f a f i n i t e samp le i s r e p r e s e n t a t i v e o f a r e p e a t e d i n f i n i t e t i m e s e r i e s , t h e n i t c a n be p r o v e n m a t h e m a t i c a l l y t h a t t h e F a s t F o u r i e r T r a n s f o r m a t i o n w i l l g e n e r a t e a n i d e n t i c a l s p e c t r u m t o t h e more i n v o l v e d F i n i t e F o u r i e r T r a n s f o r m a t i o n . A d i s c u s s i o n o f t h e FFT and i t s many a p p l i c a t i o n s may be f o u n d i n t h e r e f e r e n c e d t i t l e b y R a m i r e z ( 1 9 8 5 ) . I n o r d e r t o s o l v e t h e p r o b l e m o f jump d i s c o n t i n u i t i e s e x i s t i n g a t t h e end p o i n t s , a t e c h n i q u e known as " w i n d o w i n g " i s u t i l i z e d . The t e c h n i q u e i n v o l v e s a p r o c e d u r e whe reby t h e d a t a i s m u l t i p l i e d b y a f u n c t i o n so t h a t t h e d a t a t a p e r s t o z e r o a t b o t h e n d s . The e f f e c t o f d o i n g t h i s i s t o d e - e m p h a s i z e t h e end d a t a a n d , c o n v e r s e l y , p u t t i n g g r e a t e r e m p h a s i s on t h e d a t a l o c a t e d 143 t o w a r d s t h e c e n t e r o f t h e t i m e s e r i e s . T h i s w indowed d a t a i s t h e n t r a n s f o r m e d . The r e s u l t i n g s p e c t r u m i s t h e c o n v o l u t i o n o f t h e o r i g i n a l d a t a and t h e window f u n c t i o n . S c a l i n g f a c t o r s a r e t h e n a p p l i e d t o t h i s s p e c t r u m t o c o r r e c t f o r t h e e f f e c t s o f w i n d o w i n g . The r e s u l t i n g s p e c t r u m c l o s e l y a p p r o x i m a t e s t h e a c t u a l F i n i t e F o u r i e r S p e c t r u m o f t h e o r i g i n a l d a t a . A t a b u l a t i o n o f many o f t h e w i n d o w i n g f u n c t i o n s commonly u s e d and t h e c o r r e s p o n d i n g e f f e c t e a c h one h a s on t h e r e s u l t i n g s p e c t r u m c a n be f o u n d i n R a m i r e z ( 1 9 8 5 ) . I f t h e g a t h e r e d e x p e r i m e n t a l samp le h a s an i n t e g e r number o f c y c l e s , i t i s n o t n e c e s s a r y t o window t h e d a t a s i n c e t h e b e g i n n i n g and end p o i n t s w i l l h a v e t h e same a m p l i t u d e . To o b t a i n a n i n t e g e r number o f c y c l e s f r o m d a t a w i t h one dom inan t f r e q u e n c y i n v o l v e s a s i m p l e m a t h e m a t i c a l p r o c e d u r e o f o b s e r v i n g z e r o c r o s s i n g s , c o u n t i n g t h e c y c l e s , and t r u n c a t i n g any d a t a w h i c h doe.s n o t f o r m a c o m p l e t e c y c l e . T h i s s e c o n d method o f a v o i d i n g jump d i s c o n t i n u i t i e s w o r k s w e l l o n l y f o r d a t a w i t h one dom inan t f r e q u e n c y . I f more t h a n one dom inan t f r e q u e n c y e x i s t s , t h e p r o c e d u r e i s l e s s e f f e c t i v e and t h e w i n d o w i n g p r o c e d u r e s h o u l d be u s e d i n s t e a d . 7 . 2 . 1 . 5 FFT SUBROUTINE The s u b r o u t i n e FFT p r e p a r e s d a t a f o r u s e b y t h e FOURT s u b r o u t i n e , c a l l s t h e FOURT s u b r o u t i n e t o a n a l y z e t h e d a t a , and t h e n w r i t e s t h e r e s u l t i n g d i s c r e t e F o u r i e r s p e c t r u m t o a s p e c i f i e d 144 o u t p u t f i l e . The i n p u t f i l e must c o n t a i n d a t a f o r m a t t e d i n t h e same manner as t h a t o f an o u t p u t t e d DEMUX f i l e . S i n c e t h e f i l t e r e d d a t a f i l e s o f t h e s e e x p e r i m e n t s a l l h a v e one p r e d o m i n a n t f r e q u e n c y , i t i s n o t n e c e s s a r y t o emp loy a w i n d o w i n g t e c h n i q u e d e s c r i b e d i n t h e p r e v i o u s s e c t i o n . I n s t e a d , t h e d a t a i s t r u n c a t e d so t h a t an i n t e g e r number o f c o m p l e t e c y c l e s i s c o n s i d e r e d . T h i s i s done b y f i n d i n g t h e l o c a t i o n o f t h e z e r o c r o s s i n g s and t r u n c a t i n g d a t a b e y o n d t h e f u r t h e s t z e r o c r o s s i n g . T h i s d a t a i s t h e n t r a n s f o r m e d u s i n g FOURT, w h i c h o u t p u t s t h e s p e c t r u m as an a r r a y o f c o m p l e x numbe rs . T h i s a r r a y i s t h e n c o n v e r t e d i n t o p o l a r f o r m , w h i c h e x p r e s s e s t h e a m p l i t u d e and p h a s e a n g l e as a f u n c t i o n o f f r e q u e n c y . The p h a s e a n g l e must be f u r t h e r c o r r e c t e d f o r t h e l e a d i n t r o d u c e d b y t h e t r u n c a t i o n o f t h e o r i g i n a l d a t a . F i n a l l y , t h e o u t p u t i s w r i t t e n t o a s p e c i f i e d o u t p u t f i l e where i t i s t a b u l a t e d i n t h r e e c o l u m n s : f r e q u e n c y ( H e r t z ) , a m p l i t u d e ( u s e r u n i t s ) , and p h a s e a n g l e ( d e g r e e s ) . 7 . 2 . 1 . 6 REALTIME SUBROUTINE The s u b r o u t i n e REALTIME i s a n o t h e r p r o g r a m w r i t t e n b y G o o d r i d g e ( 1 9 8 6 ) . The s u b r o u t i n e d e t e r m i n e s t h e m a g n i t u d e o f t h e f o r c e i n p h a s e w i t h t h e maximum a c c e l e r a t i o n and maximum v e l o c i t y f r o m t h e i n p u t f i l e s c o n t a i n i n g f o r c e t r a c e s as w e l l as d i s p l a c e m e n t t r a c e . S i n c e t h e m o t i o n i s p u r e l y s i n u s o i d a l w i t h o n l y one p r e d o m i n a n t f r e q u e n c y , i t i s c l e a r l y p o s s i b l e f r o m t h e d i s p l a c e m e n t t r a c e t o d e t e r m i n e p o i n t s o f maximum a c c e l e r a t i o n and 145 v e l o c i t y . A t t h e s e p o i n t s t h e a c c e l e r a t i o n i s a t maximum when t h e v e l o c i t y i s a t minimum a n d , c o n v e r s e l y , t h e v e l o c i t y i s a t maximum when t h e a c c e l e r a t i o n i s a t min imum. From t h e d i s p l a c e m e n t d a t a f i l e , t h e s u b r o u t i n e l o c a t e s a l l z e r o c r o s s i n g s w h i c h c o r r e s p o n d t o maximum v e l o c i t y ; t h e n i t r e a d s t h e c o r r e s p o n d i n g v a l u e s f r o m t h e s u r g e , p i t c h , o r h e a v e d a t a f i l e s — d e p e n d i n g on w h i c h i s t h e f o r c e f i l e o f i n t e r e s t . I n a s i m i l a r manner , t h e s u b r o u t i n e l o c a t e s f r o m t h e d i s p l a c e m e n t f i l e p o i n t s o f maximum and minimum d i s p l a c e m e n t w h i c h c o r r e s p o n d t o maximum o r minimum a c c e l e r a t i o n and r e a d s t h e c o r r e s p o n d i n g v a l u e s i n t h e f o r c e d a t a f i l e s . I n o r d e r t o i n c r e a s e a c c u r a c y , t h e s u b r o u t i n e i n t e r p o l a t e s b e t w e e n p o i n t s t o be a b l e t o l o c a t e t h e p o i n t s o f i n t e r e s t more p r e c i s e l y . A s w e l l , t h e s u b r o u t i n e i s a b l e t o d e t e r m i n e t h e maxima o r m i n i m a i n t h e d i s p l a c e m e n t u s i n g an i n t e r p o l a t i o n o f a f i r s t o r d e r a p p r o x i m a t i o n o f t h e d e r i v a t i v e on e i t h e r s i d e o f t h e p o i n t o f i n t e r e s t . These methods p r o v i d e good r e s u l t s e v e n when t h e component i n p h a s e w i t h t h e v e l o c i t y i s n e a r z e r o . 7 . 2 . 1 . 7 B IG SUBROUTINE From t h e o u t p u t f i l e s o f t h e FFT s u b r o u t i n e , s u b r o u t i n e BIG d e t e r m i n e s t h e p o i n t w i t h t h e l a r g e s t a m p l i t u d e and t h e f r e q u e n c y a n d p h a s e a n g l e c o r r e s p o n d i n g t o t h a t p o i n t . T h i s p r o c e d u r e i s c a r r i e d o u t f o r e a c h o u t p u t f i l e and a l l t h e r e s u l t s a r e s u m m a r i z e d and p l a c e d i n one f i l e o f t h e f o r m DATA_FFT.DAT. 146 7 . 2 . 1 . 8 COEF SUBROUTINE The s u b r o u t i n e COEF i s u s e d t o d e t e r m i n e t h e a c t u a l h y d r o d y n a m i c c o e f f i c i e n t s . COEF u t i l i z e s t h e r e s u l t s o f s u b r o u t i n e s REALTIME and BIG as w e l l as p romp ts i n p u t b y t h e u s e r t o c a l c u l a t e t h e added mass and damping c o e f f i c i e n t s u s i n g t h e f o l l o w i n g f o r m u l a e : F a 1 - m 7 . 2 . 1 . 8 - 1 22 2 w X F b = — - 7 . 2 . 1 . 8 - 2 where a = added mass c o e f f i c i e n t 22 b 2 2 = damping c o e f f i c i e n t F = f o r c e i n p h a s e w i t h a c c e l e r a t i o n a F = f o r c e i n p h a s e w i t h v e l o c i t y co = o p e r a t i n g f r e q u e n c y d e t e r m i n e d f r o m t h e d i s p l a c e m e n t t r a c e X = i n p u t d i s p l a c e m e n t a m p l i t u d e m = i n p u t c y l i n d e r mass T h e s e h y d r o d y n a m i c c o e f f i c i e n t s a r e t h e n n o n - d i m e n s i o n a l i s e d i n t h e s t a n d a r d a c c e p t e d manner : a b 22 , 2 2 and 147 where p = d e n s i t y o f f l u i d medium V = vo lume o f d i s p l a c e d f l u i d The o p e r a t i n g f r e q u e n c y i s o b t a i n e d f r o m t h e F a s t F o u r i e r a n a l y s i s o f t h e d i s p l a c e m e n t t r a c e s i n c e i t i s v i r t u a l l y n o i s e f r e e . The f r e q u e n c y i s t h e n n o n - d i m e n s i o n a l i s e d i n a a c c e p t e d manner where t h e n o n - d i m e n s i o n a l f r e q u e n c y , to , i s g i v e n b y : nd 2 to a to = nd g where a = r a d i u s o f t h e c y l i n d e r g = g r a v i t a t i o n a l c o n s t a n t The o u t p u t o f s u b r o u t i n e COEF i s a t a b l e c o n t a i n i n g a number o f p a r a m e t e r s . These i n c l u d e t h e n o n - d i m e n s i o n a l i s e d a d d e d mass and damp ing c o e f f i c i e n t s ; t h e p r i n c i p l e f r e q u e n c y o f e a c h o f t h e f o r c e t r a c e s as c a l c u l a t e d b y t h e F a s t F o u r i e r T r a n s f o r m a t i o n ; t h e f o r c e v a l u e s o f e a c h o f t h e f o r c e t r a c e s w h i c h c o r r e s p o n d t o p o i n t s o f maximum a c c e l e r a t i o n and maximum v e l o c i t y ; t h e n o n - d i m e n s i o n a l i s e d f r e q u e n c y ; and a summary o f t h e i n p u t p romp ts a t t h e b e g i n n i n g o f t h e p r o g r a m . T h i s t a b l e e n a b l e s t h e u s e r t o v i s u a l l y i n s p e c t t h e d a t a o f e a c h p o i n t t o see i f t h e r e a r e any l a r g e d i s c r e p a n c i e s i n any o f t h e c a l c u l a t e d v a l u e s . Such d i s c r e p a n c i e s s u g g e s t a p o s s i b l e e r r o r a s s o c i a t e d w i t h t h a t p o i n t . 148 APPENDIX C FIGURES 149 FIGURE 8.1 COORDINATE SYSTEM AND DEFINITION OF MOTIONS 1 5 0 FIGURE 8.2 GEOMETRY OF TRIPLE CYLINDER FLUID DOMAIN USED FOR MATCHING TECHNIQUE THEORY 151 i a Interior Exterior 0 FIGURE 8.3 GEOMETRY OF SINGLE CYLINDER FLUID DOMAIN USED FOR MATCHING TECHNIQUE THEORY 152 FIGURE 8.4 DEFINITION OF TWIN CYLINDER COORDINATES USED FOR MATCHING TECHNIQUE THEORY 153 FIGURE 8.5 GEOMETRY OF SINGLE CYLINDER MODEL 154 [START V INPUT RAW DATA AND PARAMETERS ( " D S " / 1 DEMULTIPLEX RAW DATA ("DEMUX") CONVERT DATA TO DESIRED UNITS ( "CAL IB" ) REMOVE TREND IN DATA ("TREND") F I L T E R DATA ( " F I L T E R " ) PREPARE DATA FOR FFT -REMOVE INCOMPLETE CYCLES ( "FFT" ) I PERFORM FFT ("FOURT") PERFORM REALTIME ANALYSIS TO DETERMINE MAGNITUDE OF SURGE FORCE I N PHASE WITH MAX VELOCITY AND ACCELERATION ( "REALTIME") FIND MAX -AMPLITUDE AND CORRESPONDING FREQ. AND PHASE ANGLE FROM FFT RESULTS FOR EACH CHANNEL ( "BIG") DETERMINE HYDRODYNAMIC COEFFIC IENTS ("COEF") N STOP FIGURE 8.7 FLOWCHART OF THE DATA ANALYSIS SOFTWARE 156 APPENDIX D PHOTOGRAPHS 157 FIGURE 9.1 GULF CANADA'S 'KULLAC A n A x i s y m m e t r i c F l o a t i n g D r i l l i n g U n i t U s e d i n t h e B e a u f o r t Sea 158 159 F I G U R E 9.3 I N T E R I O R V I E W O F T H E O C E A N E N G I N E E R I N G C E N T R E Showing the Towing Tank and the Manoeuvering Basin 160 F I G U R E 9.4 O V E R V I E W O F T O W I N G C A R R I A G E 161 F I G U R E 9.5 O V E R H E A D H O I S T 162 FIGURE 9.6 HYDRAULIC POWER UNIT 163 F I G U R E 9.7 S I N G L E C Y L I N D E R M O D E L W I T H M O T I O N G E N E R A T I O N C O N N E C T I O N Showing Adapter Block, Force Dyraamometer, and Treaded Rod Stick-up 164 F I G U R E 9.8 T R I P L E C Y L I N D E R M O D E L With Adapter Block Attached 165 F I G U R E 9.9 D Y N A M O M E T E R A N D A D A P T E R B L O C K D E T A I L 166 FIGURE 9.10 DYNAMOMETER WITHOUT PROTECTIVE PLATES 167 APPENDIX E GRAPHICAL REPRESENTATION OF RESULTS 168 UNFILTERED D I S P L A C E M E N T T R A C E 0 TIME (sec.) FIGURE 10.1 UNFILTERED DISPLACEMENT TRACE FOR TYPICAL HYDRODYNAMIC TEST SAMPLE PLOT 169 FILTERED D I S P L A C E M E N T T R A C E - 4 T l 1 l l l i I 0 0 . 5 1 1.5 2 2 . 5 3 3 . 5 TIME (sec.) FIGURE 10.2 FILTERED DISPLACEMENT TRACE FOR TYPICAL HYDRODYNAMIC TEST SAMPLE PLOT 170 D I S P L A C E M E N T F R E Q U E N C Y S P E C T R U M 5 10 15 FREQUENCY (HZ.) 20 FIGURE 10.3 FILTERED DISPLACEMENT SPECTRUM SAMPLE PLOT 171 UNFILTERED S U R G E F O R C E T R A C E 40 T 1 : : : : - 1 0 0 4 0 0.5 1 1.5 2 2.5 TIME (sec.) 3 3.5 FIGURE 10.4 UNFILTERED SURGE CHANNEL TRACE FOR TYPICAL HYDRODYNAMIC TEST SAMPLE PLOT 172 FILTERED S U R G E F O R C E T R A C E 60 (sec.) FIGURE 10.5 FILTERED SURGE CHANNEL TRACE FOR TYPICAL HYDRODYNAMIC TEST SAMPLE PLOT S U P E R P O S I T I O N O F FILTERED A N D UNFILTERED S U R G E F O R C E DATA 60 40-20 -20--40--60-f r-0 0.5 I I Legend • FILTERED DATA • UNFILTERED DATA 1 1.5 2 2.5 TIME (sec.) 3.5 FIGURE 10.6 SUPERPOSITION OF FILTERED AND UNFILTERED SURGE FORCE TRACE SAMPLE PLOT 174 S U R G E F O R C E F R E Q U E N C Y S P E C T R U M (UNFILTERED) 6 0 5 0 H 10H 0 i i 10 15 FREQUENCY (HZ.) 2 0 FIGURE 10.7 UNFILTERED SURGE FORCE SPECTRUM SAMPLE PLOT 175 H E A V E A D D E D M A S S COEFFICIENT C O M P O U N D CYLINDER D R A F T = 9 0 c m 0.55-1 0.50-O > * O CM O 0.45-0.40-O O 0.35 (/) < O £J 0.30 O < o I 0.25-0.20 0.15 • X L Legend • MIKKELSEN(1988) AMP=1cm • MIKKELSEN(1988) AMP=2.5cm • MIKKELSEN(1988) AMP=1.5cm O MATCHING TECH. CALISAL (1984) 2 3 4 5 NON-DIM. FREQUENCY (om*om*a/g) FIGURE 10.8 HEAVE ADDED MASS COEFFICIENT VERSUS FREQUENCY TRIPLE CYLINDER RESULTS, DRAFT 90CM. MIKKELSEN (1988) 176 H E A V E A D D E D M A S S COEFFICIENT C O M P O U N D CYLINDER D R A F T = 9 0 c m 0.55 0 . 5 0 -O 0.45 * o CM CM >5, 0.40 O t 0 . 3 5 -L J o o t o < 0 .30 0.15-0.10-• B • • • • • Legend • VENUG0PAL(1984) AMP=1cm • VENUG0PAL(1984) AMP=1.5cm • MATCHING TECH. CALISAL (1984) 1 2 3 4 5 NON-DIM. FREQUENCY (om*om*a/g) 1 6 FIGURE 10.9 HEAVE ADDED MASS COEFFICIENT VERSUS FREQUENCY TRIPLE CYLINDER RESULTS, DRAFT 90CM. VENUGOPAL (1984) 177 H E A V E A D D E D M A S S COEFFICIENT C O M P O U N D CYLINDER D R A F T = 1 2 0 c m . 0.50-. 0.45 0.40-rr 0.35 0.30-±i 0.25 Q 0.20-0.15-• m m Legend • MIKKELSEN(1988) AMP=1cm • MIKKELSEN(1988) AMP=2.5cm • MIKKELSEN(1988) AMP=1.5cm O MATCHING TECH. CALISAL(1984) • i 2 3 T 4 5 NON-DIM. FREQUENCY (om*om*a/g) FIGURE 10.10 HEAVE ADDED MASS COEFFICIENT VERSUS FREQUENCY TRIPLE CYLINDER RESULTS, DRAFT 120CM. MIKKELSEN (1988) 178 H E A V E A D D E D M A S S COEFFICIENT C O M P O U N D CYLINDER D R A F T = 1 2 0 c m . 0.50-, 0.45-0.40-0.35-O O 0.30-Legend • VENUG0PAL(1984) AMP=1cm. • MATCHING TECH. CAUSAL(1984) o ^ 0.25-O < 0.20-0.15-0.10-1 2 3 4 NON-DIM. FREQUENCY (om*om*a/g) FIGURE 10.11 HEAVE ADDED MASS COEFFICIENT VERSUS FREQUENCY TRIPLE CYLINDER RESULTS. DRAFT 120CM. VENUGOPAL (1984) 179 H E A V E D A M P I N G COEFFICIENT C O M P O U N D CYLINDER D R A F T = 9 0 c m 1 • Legend • MIKKELSEN(1988) AMP=1cm • MIKKELSEN(1988) AMP=2.5cm • MIKKELSEN(1988) AMP=1.5cm NON-DIM. FREQUENCY (om*om*a/g) FIGURE 10.12 HEAVE DAMPING COEFFICIENT VERSUS FREQUENCY TRIPLE CYLINDER RESULTS, DRAFT 90CM. MIKKELSEN (1988) 180 H E A V E D A M P I N G COEFFICIENT C O M P O U N D C Y L I N D E R D R A F T = 9 0 c m 0.09 0.00 Legend • VENUG0PAL(1984) AMP=1cm • VENUG0PAL(1984) AMP=1.5cm • MATCHING TECH. CALISALQ984) • • 1 2 3 4 5 NON-DIM. FREQUENCY (om*om*a/g) 6 FIGURE 10.13 HEAVE DAMPING COEFFICIENT VERSUS FREQUENCY TRIPLE CYLINDER RESULTS, DRAFT 90CM. VENUGOPAL (1984) 181 H E A V E D A M P I N G COEFFICIENT C O M P O U N D CYLINDER D R A F T = 1 2 0 c m . NON-DIM. FREQUENCY (om*om*a/g) FIGURE 10.14 HEAVE DAMPING COEFFICIENT VERSUS FREQUENCY TRIPLE CYLINDER RESULTS. DRAFT 120CM. MIKKELSEN (1988) 182 H E A V E D A M P I N G COEFFICIENT C O M P O U N D C Y L I N D E R D R A F T = 1 2 0 c m . 0.07 Legend • VENUG0PAL(1984) AMP=1cm • MATCHING TECH. CALI5ALQ984) I 0.02-z o NON-DIM. FREQUENCY (om*om*a/g) FIGURE 10.15 HEAVE DAMPING COEFFICIENT VERSUS FREQUENCY TRIPLE CYLINDER RESULTS, DRAFT 120CM. VENUGOPAL (1984) 183 INDUCED SURGE FORCE DEEP WATER B=2.05 radii O . O U - i NON-DIM. FREQUENCY (om*om*a/g) FIGURE 10.16 INDUCED SURGE FORCE VERSUS FREQUENCY, DEEP WATER, B=2.05 184 INDUCED SURGE FORCE DEEP WATER B=2.48 radii 0.014-1 0 1 2 3 4 5 NON-DIM. FREQUENCY (om*om*a/g) FIGURE 10.17 INDUCED SURGE FORCE VERSUS FREQUENCY. DEEP WATER, B=2.48 185 INDUCED SURGE FORCE DEEP WATER B=3.0 radii o.ou-r NON-DIM. FREQUENCY (om*om*a /g ) FIGURE 10.18 INDUCED SURGE FORCE VERSUS FREQUENCY, DEEP WATER, B=3.00 186 INDUCED SURGE FORCE DEEP WATER B=3.47 radii 0.014 -1 0.012 -0 . 0 1 0 -> u 1. l£ 0 . 0 0 B o o o tn 0 . 0 0 6 -to u z> o z 0 .004 0 . 0 0 2 0 . 0 0 0 -Legend • AMPLITUDE=1.5cm • AMPLITUDE=2.5cm • AMPLITUDE=3.5cm O AMPLITUDE=4.5 cm A Moiching Technique * O 1 2 3 4 5 NON-DIM. FREQUENCY (om*om*a/g) FIGURE 10.19 INDUCED SURGE FORCE VERSUS FREQUENCY, DEEP WATER, B=3.47 187 0.012 0.010 XL V> 0 . 0 0 8 > u £ 0 . 0 0 6 P 0 . 0 0 4 -0 . 0 0 2 -0 . 0 0 0 INDUCED SURGE FORCE DEEP WATER B=4.0 radii Legend • AMPLITUDE=2.5cm • AMPLITUDE=3.5cm • AMPLITUDE=4.5cm O Matching Technique 4 • i r I I 0 0.5 1 1.5 2 2.5 3 3.5 4 NON-DIM. FREQUENCY (om*om*a/g) FIGURE 10.20 INDUCED SURGE FORCE VERSUS FREQUENCY, DEEP WATER, B=4.00 188 INDUCED SIDE FORCE SHALLOW WATER B=2.08 radii 0.016-1 NON-DIM. FREQUENCY (om*om*a /g ) FIGURE 10.21 INDUCED SURGE FORCE VERSUS FREQUENCY, SHALLOW WATER, B=2.08 189 INDUCED SURGE FORCE SHALLOW WATER B=2.47 radii 0.016 NON-DIM. FREQUENCY (om*om*a /g ) FIGURE 10.22 INDUCED SURGE FORCE VERSUS FREQUENCY, SHALLOW WATER, B=2.47 190 INDUCED SURGE FORCE SHALLOW WATER B=3.0 radii 0.016 0 . 0 0 0 1 2 3 4 NON-DIM. FREQUENCY (om*om*a/g) FIGURE 10.23 INDUCED SURGE FORCE VERSUS FREQUENCY, SHALLOW WATER, B=3.00 191 INDUCED SURGE FORCE SHALLOW WATER B=3.48 radii Legend NON-DIM. FREQUENCY (om*om*a/g) FIGURE 10.24 INDUCED SURGE FORCE VERSUS FREQUENCY. SHALLOW WATER. B=3.48 192 INDUCED SURGE FORCE SHALLOW WATER B=4.0 radii O.OM - i NON-DIM. FREQUENCY (om*om*a /g ) FIGURE 10.25 INDUCED SURGE FORCE VERSUS FREQUENCY, SHALLOW WATER, B=4.00 193 HEAVE ADDED MASS COEFFICIENT DEEP WATER B=2.05 radii 0 .07 - r 0 . 0 6 -O > CM O 0 . 0 5 -z U J t_> 0 . 0 4 Legend • AMPLITUDE=1.5cm • AMPLITUDE=4.5cm # Matching Technique o o t o t o < o Q < 0 . 0 3 -I 0 . 0 2 -z o 0 . 0 1 -• • • • • ID • 0 . 0 0 T 2 1 2 3 4 NON-DIM. FREQUENCY (om*om*a /g ) 5 FIGURE 10.26 HEAVE ADDED MASS COEFFICIENT VERSUS FREQUENCY DEEP WATER, B = 2.05 194 HEAVE ADDED MASS COEFFICIENT DEEP WATER B=2.48 radii Legend • AMPLITUDES.5cm • AMPLITUDE=2.5cm • AMPLITUDE=3.5cm O AMPLITUDE=4.5cm A Matching Technique O CD m o • o T 1 2 3 4 NON-DIM. FREQUENCY (om*om*a /g ) — 5 FIGURE 10.27 HEAVE ADDED MASS COEFFICIENT VERSUS FREQUENCY DEEP WATER, B = 2.48 195 HEAVE ADDED MASS COEFFICIENT DEEP WATER B=3.0 radii 0.030-1 0 . 0 2 5 -O > CM o Z U J o 0 . 0 2 0 -Legend • AMPLITUDE=2.5cm • AMPLITUDE=3.5cm • AMPLITUDE=4.5cm O Matching Technique o <_> 0 .015 -V) in < 2 o o < 2 0 . 0 1 0 -o I z o z 0 . 0 0 5 - • • o.ooo-i • t ] • 1 2 3 4 NON-DIM. FREQUENCY (om*om*a/g) FIGURE 10.28 HEAVE ADDED MASS COEFFICIENT VERSUS FREQUENCY DEEP WATER, B =3.00 196 HEAVE ADDED MASS COEFFICIENT DEEP WATER B=3.47 radii 0.020-r 0 .018 -0 .016-- 0.014-1 > , ° , 0 .012 -z IjJ o t 0 . 0 1 0 -o o (/) < 0 . 0 0 8 -o o o 0 .006 -1 O 0 . 0 0 4 -z 0 . 0 0 2 -0.000-- 0 . 0 0 2 Legend • AMPLITUDE=1.5cm • AMPLITUDE=2.5cm • AMPLITUDE=3.5cm O AMPLITUDE=4.5cm A Matching Technique 4 O O • i 1 2 3 4 NON-DIM. FREQUENCY (om*om*a/g) FIGURE 10.29 HEAVE ADDED MASS COEFFICIENT VERSUS FREQUENCY DEEP WATER, B = 3.47 1 9 7 HEAVE ADDED MASS COEFFICIENT DEEP WATER B=4.0 radii 0.014 -1 0.012 - 0 . 0 0 8 -0 0.5 1 1.5 2 2.5 3 3.5 4 NON-DIM. FREQUENCY (om*om*a/g) FIGURE 10.30 HEAVE ADDED MASS COEFFICIENT VERSUS FREQUENCY DEEP WATER, B = 4.00 198 HEAVE DAMPING COEFFICIENT DEEP WATER B=2.05 radii 0.050-1 NON-DIM. FREQUENCY (om*om*a/g) FIGURE 10.31 HEAVE DAMPING COEFFICIENT VERSUS FREQUENCY DEEP WATER. B = 2.05 199 HEAVE DAMPING COEFFICIENT DEEP WATER B=2.48 radii 0 . 0 5 0 -0 .045 -0 . 0 4 0 -O 0 . 0 3 5 -E CM 2 0 . 0 3 0 -t 0 . 0 2 5 -O U O 0 . 0 2 0 -< o a I 0 .015 -z o z 0 . 0 1 0 -Legend • AMPLITUDE=1.5cm • AMPLITUDE=2.5cm • AMPLITUDE=3.5cm O AMPLITUDE=4.5cm A Matching Technique 0 . 0 0 5 -o.ooo- 1 2 3 4 NON-DIM. FREQUENCY (om*om*a/g) — 5 FIGURE 10.32 HEAVE DAMPING COEFFICIENT VERSUS FREQUENCY DEEP WATER, B = 2.48 200 HEAVE DAMPING COEFFICIENT DEEP WATER B=3.0 radii -o.oos * I I i i 1 2 3 4 NON-DIM. FREQUENCY (om*om*a/g) FIGURE 10.33 HEAVE DAMPING COEFFICIENT VERSUS FREQUENCY DEEP WATER, B = 3.00 201 HEAVE DAMPING COEFFICIENT DEEP WATER B=3.47 radii 0.045 - 0 . 0 0 5 Legend • AMPLITUDES.5cm • AMPLITUDE=2.5cm • AMPLITUDE=3.5cm O AMPLITUDE=4.5cm A Matching Technique y A % — — * 1 2 3 . 4 NON-DIM. FREQUENCY (om*om*a/g) FIGURE 10.34 HEAVE DAMPING COEFFICIENT VERSUS FREQUENCY DEEP WATER, B = 3.47 5 202 HEAVE DAMPING COEFFICIENT DEEP WATER B=4.0 radii 0.045 - 1 - 0 . 0 0 5 -0.5 1 1.5 2 2.5 3 3.5 NON-DIM. FREQUENCY (om*om*a /g ) FIGURE 10.35 HEAVE DAMPING COEFFICIENT VERSUS FREQUENCY DEEP WATER, B = 4.00 203 HEAVE ADDED MASS COEFFICIENT SHALLOW WATER B=2.08 radii 0.12 0.10-o > £j 0.08-o z o o O 0.06-00 t o < o a < Q I Z o 0.04-Legend • AMPLITUDE=1.5cm • AMPLITUDE=3.5cm • AMPLITUDE=4.5cm O Matching Technique 0.02-• 1 * a 0.00 ID D i 1 2 3 4 5 NON-DIM. FREQUENCY (om*om*a/g) — 6 FIGURE 10.36 HEAVE ADDED MASS COEFFICIENT VERSUS FREQUENCY SHALLOW WATER, B = 2.08 204 HEAVE ADDED MASS COEFFICIENT SHALLOW WATER B=2.48 radii 0.07 0.06 O CM CM O z LJ O 0.04 O (J) <S> < 2 0.03 Q LJ O O < Q I 0.02 Z o z 0.01 0.00-• e • Legend • AMPUTUDE=1.5cm • AMPLITUDE=3.5cm • AMPLITUDE=4.5cm O Matching Technique 1 2 3 4 5 NON-DIM. FREQUENCY (om*om*a/g) 6 FIGURE 10.37 HEAVE ADDED MASS COEFFICIENT VERSUS FREQUENCY SHALLOW WATER, B = 2.48 205 HEAVE ADDED MASS COEFFICIENT SHALLOW WATER B=3.0 radii 0.04-1 NON-DIM. FREQUENCY (om*om*a /g) FIGURE 10.38 HEAVE ADDED MASS COEFFICIENT VERSUS FREQUENCY SHALLOW WATER, B = 3.00 206 HEAVE ADDED MASS COEFFICIENT SHALLOW WATER B=3.48 radii Legend NON-DIM. FREQUENCY (om*om*a/g) FIGURE 10.39 HEAVE ADDED MASS COEFFICIENT VERSUS FREQUENCY SHALLOW WATER. B = 3.48 207 HEAVE ADDED MASS COEFFICIENT SHALLOW WATER B=4.0 radii 0.02-1 NON-DIM. FREQUENCY (om*om*a/g) FIGURE 10.40 HEAVE ADDED MASS COEFFICIENT VERSUS FREQUENCY SHALLOW WATER, B = 4.00 208 HEAVE DAMPING COEFFICIENT SHALLOW WATER B=2.08 radii 0.16 Legend • AMPLITUDE 1.5cm • AMPLITUDE=3.5cm • AMPLITUDE=4.5cm O Matching Technique o.oo 1 2 3 4 NON-DIM. FREQUENCY (om*om*a /g ) FIGURE 10.41 HEAVE DAMPING COEFFICIENT VERSUS FREQUENCY SHALLOW WATER, B = 2.08 209 HEAVE DAMPING COEFFICIENT SHALLOW WATER B=2.48 radii O.I6-1 NON-DIM. FREQUENCY (om*om*a/g) FIGURE 10.42 HEAVE DAMPING COEFFICIENT VERSUS FREQUENCY SHALLOW WATER, B = 2.48 210 HEAVE DAMPING COEFFICIENT SHALLOW WATER £3=3.0 radii Legend • AMPLITUDE=2.5cm • AMPLITUDE=3.5cm • AMPLITUDE=1.5cm i i i i i i 0 1 2 3 4 5 NON-DIM. FREQUENCY (om*om*a/g) FIGURE 10.43 HEAVE DAMPING COEFFICIENT VERSUS FREQUENCY SHALLOW WATER, B = 3.00 211 HEAVE DAMPING COEFFICIENT SHALLOW WATER B=3.48 radii 1 1 I 1 1 1 0 1 2 3 4 5 NON-DIM. FREQUENCY (om*om*a /g ) FIGURE 10.44 HEAVE DAMPING COEFFICIENT VERSUS FREQUENCY SHALLOW WATER, B = 3.48 212 HEAVE DAMPING COEFFICIENT SHALLOW WATER B=4.0 radii 0.12 - 0 . 0 2 - 1 1 2 3 NON-DIM. FREQUENCY (om*om*a /g) FIGURE 10.45 HEAVE DAMPING COEFFICIENT VERSUS FREQUENCY SHALLOW WATER, B = 4.00 213 HEAVE DAMPING COEFFICIENT DRY TEST AMPLITUDE=4.5cm. 0.8- ! NON-DIM. FREQUENCY (om*om*a/g) FIGURE 10.46 HEAVE DAMPING COEFFICIENT EQUIPMENT ACCURACY TEST 214 APPENDIX F 1 1 . BESSEL FUNCTIONS AND RELATED FORMULAE i ) B e s s e l f u n c t i o n o f t h e f i r s t k i n d o f o r d e r n , j U ) . £ <- 1 > k < * / 2 ) 2 k - 1 0 - i n k=o k! T ( k + l - n ) where t h e Gamma f u n c t i o n i s d e f i n e d a s : T ( n ) = f t 1 1" 1 e _ t d t 1 0 - 2 o i i ) B e s s e l f u n c t i o n o f t h e s e c o n d k i n d o f o r d e r n , J (x ) c o s (pjr) - J ( x ) Y (x ) = l i r a — 5 12 1 0 - 3 n p->n . . . s i n (prc) i i i ) M o d i f i e d B e s s e l F u n c t i o n o f t h e f i r s t k i n d o f o r d e r n , I ( x ) - - i J (x ) = e " i 7 r / 2 J ( x ) 1 0 - 4 i v ) M o d i f i e d B e s s e l F u n c t i o n o f t h e s e c o n d k i n d o f o r d e r n , K (x ) = l i m [ I ( x ) - I ( x ) ] 1 0 - 5 n p->n _ . . . -p p 2 s i n (pjr) 215 v ) H a n k e l F u n c t i o n o f t h e f i r s t k i n d o f o r d e r n , H ( x ) = J ( x ) + i Y ( x ) n n v i ) D e r i v a t i v e s o f B e s s e l F u n c t i o n s B ( x ) = B (x) - ^ B ( x ) where 8 d e n o t e s J , Y , I, K, o r H. v i i ) C o m p l e t e e l l i p t i c i n t e g r a l o f t h e f i r s t k i n d , 7 T / 2 F ( k , 7 r / 2 ) I o 1 - k 2 s i n 2 6 7T r . r i i 2 , 2 r i 3 i 2 , 4 r i 3 512, e = 2 { 1 + [lJ k + [2 4J k + [2 4 6J k v i i i ) C o m p l e t e e l l i p t i c i n t e g r a l o f t h e s e c o n d k i n d , 7T/2 E(k,w /2 ) V 1 - k 2 s i n 2 0 d9 I 2 1 b J L 2 4 J 3 L2 4 6 J 5 216 

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