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Hydrodynamic coefficients of compound circular cylinders in surge motion Goodridge, Douglas N. 1986

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HYDRODYNAMIC COEFFICIENTS OF COMPOUND CIRCULAR CYLINDERS SURGE MOTION By DOUGLAS N. GOODRIDGE B.Eng.(Mech), Memorial U n i v e r s i t y of Newfoundland, 1983 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n FACULTY OF GRADUATE STUDIES Department of Mechanical Engineering We accept t h i s thesis as conforming to the required standard i THE UNIVERSITY OF BRITISH COLUMBIA 15 October, 1986 © Douglas N. Goodridge, 1986 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t the 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 , I agree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by t h e head o f my department o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department O f Mechanical Engineering The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date October 15, 1986 ABSTRACT The hydrodynamic c o e f f i c i e n t s , that i s the added mass and damping c o e f f i c i e n t s , are determined experimentally for compound c i r c u l a r cylinders i n surge motion. The surge force induced by forced harmonic o s c i l l a t i o n of c y l i n d e r models i s measured. In a d d i t i o n the induced wave height i n the flow f i e l d i s also measured. The e f f e c t of varying the displacement amplitude, the frequency of o s c i l l a t i o n and the c y l i n d e r d r a f t are investigated for single, double and t r i p l e c y l i n d e r models. The r e s u l t s are compared to the t h e o r e t i c a l p r e d i c t i o n s of the Matching Technique (MT) and the Boundary Element Method (BEM). The Matching Technique formulation uses c o n t i n u i t y of pressures and v e l o c i t i e s between adjacent regions i n the flow f i e l d to solve f o r the v e l o c i t y p o t e n t i a l s and hence to determine the hydrodynamic c o e f f i c i e n t s of the c y l i n d e r models. A p p l i c a t i o n of the Boundary Element Method i s s i m p l i f i e d by the axisymmetry of the problem. In t h i s method a c o n t r o l volume i s defined and d i s c r e t i z e d into r i n g shaped elements. The p o t e n t i a l within each element can then be solved simultaneously using the boundary conditions. In a d d i t i o n experiments were conducted to measure the wave induced e x c i t i n g force on the c y l i n d e r models and the r e s u l t s of these experiments were compared with the t h e o r e t i c a l predictions of the Boundary Element J 1 Method. i i -TABLE OF CONTENTS ABSTRACT i i L I S T OF FIGURES v i i L I S T OF TABLES x i ACKNOWLEDGEMENTS x i i 1. INTRODUCTION 1 2 . EXPERIMENTAL WORK 8 2 . 1 HYDRODYNAMIC TESTS 8 2 . 2 WAVE INDUCED EXCITING FORCE TESTS 11 3 . THEORETICAL MODELS 15 3 . 1 MATCHING TECHNIQUE 17 3 . 1 . 1 GOVERNING EQUATION 18 3 . 1 . 2 BOUNDARY CONDITIONS 20 3 . 1 . 3 DEFIN IT ION OF POTENTIALS 23 3 . 1 . 3 . 1 REGION 1 25 3 . 1 . 3 . 2 REGION 2 25 3 . 1 . 3 . 3 REGION 3 26 3 . 1 . 3 . 4 REGION 4 28 3 . 1 . 4 SOLVING FOR THE UNKNOWN COEFFICIENTS 29 3 . 1 . 4 . 1 CONTINUITY OF PRESSURE BETWEEN REGION 1 AND 2 31 3 . 1 . 4 . 2 CONTINUITY OF VELOCITY BETWEEN REGION 1 AND 2 33 3 . 1 . 4 . 3 CONTINUITY OF PRESSURE BETWEEN REGION 2 AND 4 34 3 . 1 . 4 . 4 CONTINUITY OF PRESSURE BETWEEN REGION 3 AND 4 35 3 . 1 . 4 . 5 CONTINUITY OF VELOCITY BETWEEN REGIONS 2 , 3 AND 4 37 3 . 1 . 4 . 6 SOLVING FOR THE COEFFICIENTS OF THE SERIES 40 3 . 1 . 5 CALCULATION OF THE HYDRODYNAMIC COEFFICIENTS 41 3 . 1 . 6 CALCULATION OF THE SURGE EXCITING FORCE 44 3 . 1 . 7 CYL3 PROGRAM 44 3 . 2 BOUNDARY ELEMENT METHOD 45 3 . 2 . 1 SOLVING FOR THE POTENTIAL FUNCTION 46 - i i i -3 . 2 . 2 DETERMINATION OF THE MOTION INDUCED POTENTIAL 47 3 . 2 . 3 DETERMINATION OF THE HYDRODYNAMIC COEFFICIENTS 50 3 . 2 . 4 DETERMINATION OF THE SURGE EXCITING FORCE 50 3 . 2 . 5 A l l PROGRAM 52 4 . PRESENTATION AND ANALYSIS OF RESULTS 53 4 . 1 SAMPLE DATA PLOTS 55 4 . 2 HYDRODYNAMIC TEST RESULTS 57 4 . 2 . 1 ADDED MASS COEFFICIENTS 57 4 . 2 . 1 . 1 SINGLE CYLINDER. . 57 4 . 2 . 1 . 2 DOUBLE CYLINDER 58 4 . 2 . 1 . 3 T R I P L E CYLINDER 59 4 . 2 . 2 DAMPING COEFFICIENTS 61 4 . 2 . 2 . 1 SINGLE CYLINDER 61 4 . 2 . 2 . 2 DOUBLE. CYLINDER 62 4 . 2 . 2 . 3 T R I P L E CYLINDER : 62 4 . 3 INCIDENT WAVE TEST RESULTS 63 4 . 3 . 2 WAVE INDUCED EXCITING FORCE 64 4 . 3 . 2 . 1 SINGLE CYLINDER 64 4 . 3 . 2 . 2 DOUBLE CYLINDER 65 4 . 3 . 2 . 3 T R I P L E CYLINDER 65 4 . 3 . 2 INDIRECTLY DETERMINED DAMPING COEFFICIENTS 66 4 . 3 . 2 . 1 SINGLE CYLINDER 66 4 . 3 . 2 . 2 DOUBLE CYLINDER 67 4 . 3 . 2 . 3 T R I P L E CYLINDER 67 4 . 4 ANALYSIS OF RESULTS 67 4 . 5 ANALYSIS OF ERROR 71 4 . 6 U T I L I T Y OF RESULTS 75 CONCLUSIONS 76 RECOMMENDATIONS 79 BIBLIOGRAPHY -. .. 80 5 . APPENDIX A - EXPERIMENTAL SET UP 82 5 .1 EXPERIMENTAL F A C I L I T I E S 82 - i v -5 . 1 . 1 TOWING TANK 82 5 . 1 . 2 WAVE MAKER 83 5 . 2 EXPERIMENTAL EQUIPMENT 85 5 . 2 . 1 MOTION GENERATOR 85 5 . 2 . 2 DATA COLLECTION EQUIPMENT 86 5 . 2 . 2 . 1 ST41B™ SIGNAL CONDITIONER 87 5 . 2 . 2 . 1 . 1 SIGNAL CONDITIONER PHASE LAG TESTS 89 5 . 2 . 2 . 2 MINC™ 11 MINI COMPUTER 91 5 . 2 . 3 CYLINDER MODELS 92 5 . 2 . 4 INSTRUMENTATION USED 94 5 . 2 . 4 . 1 LOAD CELL DYNAMOMETER 94 5 . 2 . 4 . 1 . 1 DYNAMOMETER CALIBRATION 95 5 . 2 . 4 . 2 YO-YO POSITION TRANSDUCER 99 5 . 2 . 4 . 3 PRESSURE TRANSDUCERS 100 5 . 2 . 4 . 4 TWO WIRE WAVE PROBE 101 6 . APPENDIX B - SOFTWARE USED IN EXPERIMENTS 103 6 . 1 DATA ACQUISIT ION SOFTWARE 104 6 . 1 . 1 ADMAIN PROGRAM 104 6 . 1 . 2 AD CAL PROGRAM 105 6 . 1 . 3 ADMUX PROGRAM 106 6 . 1 . 4 GRAPH PROGRAM 107 6 . 2 DATA ANALYSIS SOFTWARE 107 6 . 2 . 1 DS PROGRAM 108 6 . 2 . 1 . 1 DEMUX SUBROUTINE I l l 6 . 2 . 1 . 2 CAL IB SUBROUTINE I l l 6 . 2 . 1 . 3 DYNO SUBROUTINE 112 6 . 2 . 1 . 4 TREND SUBROUTINE 112 6 . 2 . 1 . 5 COPY SUBROUTINE 113 6 . 2 . 1 . 6 F I L T E R SUBROUTINE 114 6 . 2 . 1 . 7 FOURT SUBROUTINE 115 6 . 2 . 1 . 8 FFT SUBROUTINE 117 6 . 2 . 1 . 9 REALTIME SUBROUTINE 117 6 . 2 . 1 . 1 0 BIG SUBROUTINE 118 6 . 2 . 1 . 1 1 BIGWAVE SUBROUTINE 119 6 . 2 . 1 . 1 2 DELAY SUBROUTINE 119 6 . 2 . 1 . 1 3 COEF SUBROUTINE 119 - v -6 . 2 . 1 . 1 4 EXCITE SUBROUTINE 121 6 . 2 . 2 DSEXF PROGRAM 122 6 . 2 . 2 . 1 B IGEXF SUBROUTINE 124 6 . 2 . 2 . 2 COEFEXF SUBROUTINE 125 6 . 2 . 3 OTHER SOFTWARE USED 125 6 . 2 . 3 . 1 ANGLE SUBROUTINE 125 6 . 2 . 3 . 2 EZ PROGRAM 126 6 . 2 . 3 . 3 HCPLOT PROGRAM 126 6 . 2 . 3 . 4 L A B E L SUBROUTINE 126 6 . 2 . 3 . 5 LINREG SUBROUTINE 126 6 . 2 . 3 . 6 XX PROGRAM 127 7 . APPENDIX C - PHOTOGRAPHS 128 8 . APPENDIX D - FIGURES 152 9 . APPENDIX E - GRAPHICAL PRESENTATION OF RESULTS 162 9 . 1 NOMENCLATURE FOR GRAPHS 163 1 0 . APPENDIX F - BESSEL FUNCTIONS AND RELATED FORMULAE 206 - v i -L I S T OF FIGURES FIGURE 7 . 1 - GULF CANADA'S ' K U L L U K ' 129 FIGURE 7 . 2 - EXTERIOR VIEW OF OCEAN ENGINEERING CENTRE 130 FIGURE 7 . 3 - INTERIOR VIEW OF OCEAN ENGINEERING CENTRE 131 FIGURE 7 . 4 - ENERGY ABSORBING BEACH MATERIAL 132 FIGURE 7 . 5 - WAVE MAKER 133 FIGURE 7 . 6 - OVERHEAD HOIST 134 FIGURE 7 . 7 - OVERVIEW OF EXPERIMENTAL SETUP 135 FIGURE 7 . 8 - OVERVIEW OF TOWING CARRIAGE SETUP 136 FIGURE 7 . 9 - MOTION GENERATOR SINGLE CYLINDER TEST SETUP 137 FIGURE 7 . 1 0 - MOTION GENERATOR T R I P L E CYLINDER TEST SETUP 138 F IGURE 7 . 1 1 - OVERVIEW OF WAVE PATTERN PRODUCED BY HYDRODYNAMIC T E S T . . . 1 3 9 FIGURE 7 . 1 2 - S INGLE CYLINDER AND MOTION GENERATOR CONNECTIONS 140 FIGURE 7 . 1 3 - DOUBLE CYLINDER MODEL. . 141 FIGURE 7 . 1 4 - T R I P L E CYLINDER MODEL 142 FIGURE 7 . 1 5 - DYNAMOMETER, ADAPTER BLOCK DETAIL 143 FIGURE 7 . 1 6 - HYDRAULIC POWER UNIT 144 FIGURE 7 . 1 7 - DATA ACQUISIT ION HARDWARE 145 FIGURE 7 . 1 8 - ST41B™ SIGNAL CONDITIONER 146 FIGURE 7 . 1 9 - TWO WIRE WAVE PROBE 147 FIGURE 7 . 2 0 - YO-YO POSITION TRANSDUCER 148 FIGURE 7 . 2 1 - DYNAMOMETER STATIC CALIBRATION SETUP 149 FIGURE 7 . 2 2 - DYNAMOMETER WITHOUT PROTECTIVE PLATES 150 FIGURE 8 . 1 - COORDINATE SYSTEM USED AND DEFINIT ION OF MOTIONS 152 FIGURE 8 . 2 - GEOMETRY OF SINGLE CYLINDER MODEL 153 FIGURE 8 . 3 - GEOMETRY OF DOUBLE CYLINDER MODEL 154 - v i i -FIGURE 8.4 - GEOMETRY OF TRIPLE CYLINDER MODEL 155 FIGURE 8.5 - LOCATION OF PRESSURE TRANSDUCERS 156 FIGURE 8.6 - SUBDIVISION OF TRIPLE CYLINDER FLUID DOMAIN FOR MATCHING TECHNIQUE THEORY 157 FIGURE 8.7 - DISCRETIZED CONTROL SURFACE USED IN BOUNDARY ELEMENT METHOD FORMULATION 158 FIGURE 8.8 - ENGINEERING DRAWING OF DYNAMOMETER 159 FIGURE 8.9 - FLOW CHART OF PROGRAM 'DS' 160 FIGURE 8.10 - FLOW CHART OF PROGRAM 'DSEXF' 161 FIGURE 9.1 - CYLINDER DISPLACEMENT TRACE - SAMPLE PLOT ..164 FIGURE 9.2 - CYLINDER DISPLACEMENT SPECTRUM - SAMPLE PLOT 165 FIGURE 9.3 - UNFILTERED SURGE FORCE TRACE - SAMPLE PLOT 166 FIGURE 9.4 - FILTERED SURGE FORCE TRACE - SAMPLE PLOT 167 FIGURE 9.5 - SURGE FORCE SPECTRUM - SAMPLE PLOT 168 FIGURE 9.6 - ORIGINAL 'UNSMOOTHED' ADDED MASS RESULTS FROM BEM AND MT THEORIES 169 FIGURE 9.7 - ORIGINAL 'UNSMOOTHED' DAMPING COEF. RESULTS FROM BEM AND MT THEORIES 170 FIGURE 9.8 - SINGLE CYLINDER - ADDED MASS - DRAFT = 211mm - 171 FIGURE 9.9 - SINGLE CYLINDER - ADDED MASS - DRAFT = 218mm 172 FIGURE 9.10 - DOUBLE CYLINDER - ADDED MASS - OVERALL DRAFT = 625mm 173 FIGURE 9.11 - DOUBLE CYLINDER - ADDED MASS - OVERALL DRAFT = 690-701mm 174 FIGURE 9.12 - TRIPLE CYLINDER - ADDED MASS - OVERALL DRAFT = 867mm 175 FIGURE 9.13 - TRIPLE CYLINDER - ADDED MASS - OVERALL DRAFT - 938mm 176 FIGURE 9.14 - TRIPLE CYLINDER - ADDED MASS - OVERALL DRAFT = 985mm 177 FIGURE 9.15 - TRIPLE CYLINDER - ADDED MASS - OVERALL DRAFT = 1089mm 178 FIGURE 9.16 - TRIPLE CYLINDER - ADDED MASS - OVERALL DRAFT = 1177mm 179 FIGURE 9.17 - TRIPLE CYLINDER - ADDED MASS - OVERALL DRAFT = 1187mm 180 - v i i i -FIGURE 9 . 1 8 - S INGLE CYLINDER - DAMPING COEF. - DRAFT = 211mm 181 FIGURE 9 . 1 9 - S INGLE CYLINDER - DAMPING COEF. - DRAFT = 218mm 182 FIGURE 9 . 2 0 - DOUBLE CYLINDER - DAMPING COEF. - OVERALL DRAFT = 625mm..183 FIGURE 9 . 2 1 - DOUBLE CYLINDER - DAMPING COEF. - OVERALL DRAFT = 690-701mm 184 FIGURE 9 . 2 2 - T R I P L E CYLINDER - DAMPING COEF. - OVERALL DRAFT = 867mm..185 FIGURE 9 . 2 3 - T R I P L E CYLINDER - DAMPING COEF. - OVERALL DRAFT = 938mm..186 FIGURE 9 . 2 4 - T R I P L E CYLINDER - DAMPING COEF. - OVERALL DRAFT = 985mm..187 FIGURE 9 . 2 5 - T R I P L E CYLINDER - DAMPING COEF. - OVERALL DRAFT = 1089mm 188 FIGURE 9 . 2 6 - T R I P L E CYLINDER - DAMPING COEF. - OVERALL DRAFT = 1177mm 189 FIGURE 9 . 2 7 - T R I P L E CYLINDER - DAMPING COEF. - OVERALL DRAFT = 1187mm 190 FIGURE 9 . 2 8 - S INGLE CYLINDER - EXCIT ING FORCE - DRAFT = 211-218mm 191 FIGURE 9 . 2 9 - DOUBLE CYLINDER - EXCIT ING FORCE - OVERALL DRAFT = 625mm 192 FIGURE 9 . 3 0 - DOUBLE CYLINDER - EXCIT ING FORCE - OVERALL DRAFT = 690mm 193 FIGURE 9 . 3 1 - DOUBLE CYLINDER - EXCIT ING FORCE - OVERALL DRAFT = 701mm 194 FIGURE 9 .32 - T R I P L E CYLINDER - EXCIT ING FORCE - OVERALL DRAFT = 867mm 195 FIGURE 9 . 3 3 - T R I P L E CYLINDER - EXCIT ING FORCE - OVERALL DRAFT = 936mm 196 FIGURE 9 . 3 4 - T R I P L E CYLINDER - EXCIT ING FORCE - OVERALL DRAFT = 1177mm 197 FIGURE 9 . 3 5 - S INGLE CYLINDER - WEHAUSEN CALCULATED - DRAFT = 211mm 198 FIGURE 9 . 3 6 - SINGLE CYLINDER - WEHAUSEN CALCULATED COEF. -DRAFT = 218mm 199 FIGURE 9 . 3 7 - DOUBLE CYLINDER - WEHAUSEN CALCULATED - OVERALL DRAFT = 625mm 200 FIGURE 9 .38 - DOUBLE CYLINDER - WEHAUSEN CALCULATED DAMPING COEF. -OVERALL DRAFT = 690mm 201 - i x -FIGURE 9 . 3 9 - DOUBLE CYLINDER - WEHAUSEN CALCULATED DAMPING COEF. -OVERALL DRAFT = 701mm 202 FIGURE 9 . 4 0 - T R I P L E CYLINDER - WEHAUSEN CALCULATED DAMPING COEF. -OVERALL DRAFT = 867mm 203 FIGURE 9 . 4 1 - T R I P L E CYLINDER - WEHAUSEN CALCULATED DAMPING COEF. -OVERALL DRAFT = 936mm 204 FIGURE 9 . 4 2 - T R I P L E CYLINDER - WEHAUSEN CALCULATED DAMPING COEF. -OVERALL DRAFT = 1177mm 205 - x -L I S T OF TABLES TABLE 2 . 2 - 1 - SUMMARY OF TESTS CONDUCTED 12 TABLE 3 -1 - RANGE OF NON-DIMENSIONAL QUANTITIES ENCOUNTERED 16 TABLE 4 . 5 - 1 - RANGE OF ERRORS FOR PLOTTED PARAMETERS 74 TABLE 5 . 2 . 2 . 1 - 1 - A M P L I F I E R SETUP CONFIGURATION 88 TABLE 5 . 2 . 4 . 1 - 1 - DYNAMOMETER SPECIF ICATIONS 95 TABLE 5 . 2 . 4 . 1 . 1 - 1 - RESULTS OF DYNAMOMETER STATIC CALIBRATION 97 TABLE 5 . 2 . 4 . 2 - 1 - YO-YO POTENTIOMETER SPECIF ICATIONS 100 TABLE 5 . 2 . 4 . 3 - 1 - PRESSURE TRANSDUCER SPECIF ICATIONS 101 TABLE 6 . 2 . 1 - 1 - TRANSDUCER CHANNEL ASSIGNMENTS AND USER UNITS ASSUMED BY " D S " 109 TABLE 6 . 2 . 1 - 2 - F I L E NAME EXTENSIONS USED BY " D S " 110 TABLE 6 . 2 . 2 - 1 - F I L E NAME EXTENSIONS USED BY " D S E X F " 124 - x i ACKNOWLEDGEMENTS In the course of conducting t h i s research I have drawn upon the knowledge and resources of many i n d i v i d u a l s , too numerous to mention here. I wish to thank each of them for t h e i r input and assistance. I must, however, sing l e out a few people whose contributions have been p a r t i c u l a r s i g n i f i c a n t to t h i s work. In p a r t i c u l a r , I wish to express my gratitude to Dr. Sander M. C a l i s a l , my supervisor, f o r h i s guidance and support throughout the course of t h i s project. His calm approach to the problems which i n e v i t a b l y arose was a valuable lesson i n both the technological and psychological aspects of research. I also wish to thank the National Sciences and Engineering Research Council (NSERC) of Canada for funding t h i s p r o j e c t . I am deeply indebted to BC Research and the s t a f f at the Ocean Engineering Centre f o r the use of the f a c i l i t y . Thanks to, Gerry Stensgaard, who made s p e c i a l arrangements for me to use the f a c i l i t y a f t e r hours, George Roddan, for h i s valuable advice on instrumentation and Gary Novlesky for h i s always p r a c t i c a l solutions to mechanical problems. I also wish to thank Bruce Hanson i n the Mechanical Engineering Machine Shop f o r h i s advice and the superb q u a l i t y workmanship which was put into every component he fa b r i c a t e d f o r t h i s project. I wish to express my gratitude to Tom N i c o l , i n the Computer Science Department at UBC, for h i s frequent and patient help with software problems. Thanks to Johnson Chan, f o r the use of h i s Boundary Element Method program. I - x i i -also wish to thank the other students i n the Naval Architecture group for t h e i r assistance and support. Thanks, p a r t i c u l a r l y , to those who gave up weekends to help with the experiments. Thanks to Dan McGreer, Gerry Rohling, Gireesh Sadisavan, Jon Mikkelsen and Farshid Namarian. I wish to thank Dean Ross Peters and Associate Dean T. R. Chari i n the Faculty of Engineering at Memorial U n i v e r s i t y of Newfoundland f o r granting me v i s i t i n g student status between J u l y and October of 1986 allowing me to f i n i s h t h i s research i n Newfoundland. P a r t i c u l a r thanks to Dr. Derek Muggeridge f o r permitting me to use h i s l a s e r p r i n t e r from which the f i n a l copy of t h i s thesis was printed. F i n a l l y I wish to express my gratitude to my employer, Mobil O i l Canada Ltd. , f o r the support they have given me toward the completion of t h i s document. x i i i 1. INTRODUCTION Man's quest to explore and e x p l o i t the oceans predates h i s t o r y , but perhaps i n t h i s past decade man has most stretched the l i m i t s of technology by constructing i n c r e a s i n g l y complex structures intended to operate i n i n c r e a s i n g l y harsh environments. The energy c r i s i s of the e a r l y 1970's placed new demands on the petroleum industry. I t was these demands that have been responsible f o r many of the new developments within the offshore industry. I t was 1897 when man f i r s t ventured offshore to explore f o r o i l and gas. Here i n C a l i f o r n i a ' s Santa Barbara channel wells were d r i l l e d from wooden stages. Over the next few years the wharves were extended, some as f a r as 360 metres from shore. Upgraded versions of these stages can be seen today along the co a s t a l highway i n C a l i f o r n i a . Gradually the industry evolved and a c t i v i t y moved to the Gulf of Mexico. Here inland lake technology was implemented offshore. Barges were towed out to l o c a t i o n and made stable by d r i v i n g wooden p i l e s into the seabed. Later barges were u t i l i z e d which could be f l o a t e d out to l o c a t i o n and flooded to r e s t on shallow bottoms. These submersible barges were designed with s u f f i c i e n t freeboard to provide a dry stable work area. Many of these same submersibles are s t i l l used today i n the Gulf of Mexico and more modern examples of t h i s technology are being used today i n the Beaufort Sea. The disadvantage of using a f i x e d g r a v i t y type structure i s that i t becomes incr e a s i n g l y less cost e f f e c t i v e and t e c h n i c a l l y more d i f f i c u l t with increasing depths. Through the 1940's industry experimented with f l o a t i n g barges capable of d r i l l i n g i n water depths of up to 12 metres. D r i l l ships were introduced i n the mid 1950's following a United States Navy experimental program. The - 1 -undesirable motion c h a r a c t e r i s t i c s of the d r i l l ship prevented d r i l l i n g operations from continuing i n apparently moderate sea states. While improvements have been made to the design of today's modern d r i l l ships there arose a need to design f l o a t i n g structures which exhibited more desirable motion response c h a r a c t e r i s t i c s . This l e d to the development of the semisubmersible platform which was f i r s t introduced i n the e a r l y 1960's. The s t r u c t u r a l arrangement of a semisubmersible consists of a deck, supported by a number of v e r t i c a l columns, cross braces and pontoons, which have s u f f i c i e n t buoyancy to f l o a t the e n t i r e structure. In September 1985 there were 184 semisubmersibles working worldwide with about 60 of those working i n the North Sea. The fundamental components which make up the below deck p o r t i o n of a semisubmersible d r i l l i n g u n i t are c y l i n d r i c a l i n shape. Today newer technologies are i n place. Over the l a s t three years Gulf Canada has been using the Kulluk, an e n t i r e l y axisymmetric f l o a t i n g d r i l l i n g u n i t , i n the Beaufort Sea. This d r i l l i n g u n i t i s shown i n Figure 7.1. These complex offshore structures represent s i g n i f i c a n t investments and are u s u a l l y expected to have design l i v e s of 100 years or more. While much e f f o r t has gone into t r y i n g to better understand the theory to p r e d i c t the hydrodynamic loads on these structures, i t i s s t i l l at a r e l a t i v e l y rudimentary l e v e l and designers are forced to r e l y more h e a v i l y on scale model t e s t r e s u l t s or on other trusted empirical data. The a n a l y t i c a l methods which do e x i s t require numerous s i m p l i f i c a t i o n s to model the actual flow around a f u l l scale offshore structure. Some examples of these theories are presented i n Section 3. of t h i s t h e s i s . - 2 -Consider a s i x degree of freedom system moving i n the Cartesian coordinate system shown i n Figure 8.1. Possible motions include t r a n s l a t i o n along and r o t a t i o n about each axis. The naval a r c h i t e c t has assigned s p e c i f i c terms to each degree of freedom. Translations i n the x, y and z di r e c t i o n s are known as surge, sway and heave, r e s p e c t i v e l y . Rotations about these same axis are known as r o l l , p i t c h and yaw, re s p e c t i v e l y . The loading a f l u i d imparts upon a structure i s c a l l e d hydrodynamic loading. The concepts of hydrodynamic forces and the r e s u l t i n g motions are best understood by introducing hydrodynamic c o e f f i c i e n t s . The most general equation describing any dynamic s i x degree of freedom system may be expressed as follows: F ] i M m i j i J (1-D Where, F = Force vector (3x1); M = Moment vector (3x1); X, X, X = Translatory motion vectors (3x1); 6, 9, 6 = Rotary motion vectors (3x1); m = Mass c o e f f i c i e n t matrix (6x6); b = Damping force c o e f f i c i e n t matrix (6x6); c = Restoring force c o e f f i c i e n t matrix (6x6). I f one i s to consider hydrodynamic loading only then the terms which - 3 -comprise the three c o e f f i c i e n t matrices here are known as hydrodynamic c o e f f i c i e n t s . The terms i n phase with acc e l e r a t i o n , v e l o c i t y , and displacement are known as added mass, damping and r e s t o r i n g force c o e f f i c i e n t s , r e s p e c t i v e l y . The added mass terms are designated as a^ where the f i r s t subscript r e f e r s to the d i r e c t i o n of body motion and the second subscript r e f e r s to the d i r e c t i o n of the hydrodynamic force. For a s i x degree of freedom system there are t h i r t y - s i x c o e f f i c i e n t s possible f o r each matrix, but i t can be shown that a •= a . Hence the matrix reduces to twenty-one 1J J i J independent c o e f f i c i e n t s . With other geometric and p h y s i c a l r e l a t i o n s h i p s the number of c o e f f i c i e n t values can be reduced s i g n i f i c a n t l y . In the case of axisymmetric shapes the t h i r t y - s i x possible values are reduced to eight unique-constants i n the added mass matrix, eight unique values i n the damping c o e f f i c i e n t matrix and four unique values i n the r e s t o r i n g force matrix. A l l other values are equal to zero. In the work conducted i n t h i s thesis the problem i s further s i m p l i f i e d by the f a c t that only surge motion or f o r - a f t motion i s considered, thereby, allowing one to consider only a sin g l e term i n each matrix conventionally designated a^, b ^ and c for added mass, damping and r e s t o r i n g force c o e f f i c i e n t s r e s p e c t i v e l y . The r e s t o r i n g force, which i s u s u a l l y r e l a t e d to buoyancy i n a f l u i d medium, does not e x i s t i n surge motion. Hence, c i s always equal to zero. P h y s i c a l l y hydrodynamic c o e f f i c i e n t s a r i s e due to f l u c t u a t i o n s i n the pressure f i e l d around the body. Consider a body moving i n a f l u i d medium, the f l u i d p a r t i c l e s i n some neighbourhood of the body w i l l accelerate at varying rates depending on t h e i r proximity to the body. The added mass, therefore, may be thought of as a measure of the quantity of f l u i d which i s accelerated with the body. Since i n p r i n c i p l e every f l u i d p a r t i c l e i s accelerated to some degree, the added mass i s expressed as an equivalent mass which i s - 4 -accelerated at the same rate as the body. The damping c o e f f i c i e n t i s a measure of the outward f l u x of energy. Most theories presently used assume the flow to be p o t e n t i a l , that i s , the f l u i d i s assumed to be incompressible, i n v i s c i d and the flow i r r o t a t i o n a l . Hence, viscous and s t r u c t u r a l damping e f f e c t s are neglected. For an i d e a l f l u i d the damping energy manifests i t s e l f as wave energy. A d i r e c t r e l a t i o n s h i p may therefore be found between wave height and the damping c o e f f i c i e n t . A body moving p e r i o d i c a l l y i n a f l u i d medium at the free surface w i l l experience hydrodynamic forces which are in-phase and out of phase with the a c c e l e r a t i o n . The in-phase component contributes to the added mass and the component in-phase with the v e l o c i t y contributes to the damping c o e f f i c i e n t . This thesis describes the determination of hydrodynamic c o e f f i c i e n t s for three compound c i r c u l a r c y l i n d e r models, a s i n g l e , a double and a t r i p l e c y l i n d e r . Each c y l i n d e r model underwent simple harmonic surge motion at the free surface i n water of f i n i t e depth. The hydrodynamic c o e f f i c i e n t s were determined t h e o r e t i c a l l y using the two techniques known as the Boundary Element Method (BEM) and the Matching Technique (MT). In a d d i t i o n the hydrodynamic c o e f f i c i e n t s were determined experimentally and the r e s u l t s obtained by each method are compared. Experiments were also conducted to determine the surge e x c i t i n g force due to harmonic waves incident on the r i g i d c y l i n d e r model. These r e s u l t s were also compared to 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 o r e t i c a l determination of these c o e f f i c i e n t s has challenged researchers f o r over 200 years. Some of the f i r s t work was conducted by - 5 -Chevalier du Buat and subsequently a host of well known i n d i v i d u a l s including Bessel, Green, Stokes, Lamb and Darwin studied t h i s problem. A complete t h e o r e t i c a l model of f l u i d structure i n t e r a c t i o n must include the e f f e c t s of the v i s c o s i t y of the f l u i d and the time h i s t o r y of the motion. Consider, an object moving i n a viscous f l u i d . There can be separated flow past the object. This wake or c a v i t y further induces added mass which varies with the shape of the c a v i t y . Therefore, the instantaneous value of t h i s added mass depends on the time h i s t o r y of the motion. In addition, for motion i n a viscous f l u i d , there w i l l e x i s t some viscous damping. For these reasons, determination of the hydrodynamic c o e f f i c i e n t s f o r a r b i t r a r y motion of a body i n a viscous f l u i d medium i s a complicated problem. The analysis can be s i m p l i f i e d however, by considering harmonic motion of an object i n a p o t e n t i a l f l u i d . With these s i m p l i f i c a t i o n s a number of t h e o r e t i c a l techniques have been devised to determine the hydrodynamic c o e f f i c i e n t s of a wide range of bodies. Conformal mapping techniques work well but can only be used f o r two dimensional problems. S i n g u l a r i t y methods are s u i t e d to three dimensional bodies and are used extensively i n the design of offshore structures. This method involves considerable computation and the body surface must be d i s c r e t i z e d c a r e f u l l y . In a d d i t i o n s i n g u l a r i t y methods can give erroneous r e s u l t s at c e r t a i n frequencies. F i n i t e element methods do not s u f f e r from t h i s draw back but they require the e n t i r e flow f i e l d to be d i s c r e t i z e d . This involves considerably more computing time. In t h i s thesis the t h e o r e t i c a l models used make use of the axisymmetry - 6 -of the bodies considered. Two separate t h e o r e t i c a l models were explored and the p r e d i c t i o n s from each theory were compared with the experimental r e s u l t s . The t h e o r e t i c a l models used are known as the Matching Technique and the Boundary Element Method. The Matching Technique was f i r s t introduced by Garrett (1971) i n c a l c u l a t i n g wave forces on a c i r c u l a r dock. Subsequently, Sabuncu and C a l i s a l (1980) s u c c e s s f u l l y applied t h i s technique to determine the hydrodynamic c o e f f i c i e n t s of si n g l e v e r t i c a l c i r c u l a r c y linders and more recently (1984) they applied the theory to double 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 . This technique i s a pplied by d i v i d i n g the s o l u t i o n domain into axisymmetric subregions where the value of the p o t e n t i a l i s expressed i n s e r i e s form i n terms of 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 are determined by matching the pressures and normal v e l o c i t i e s on the boundaries between these subregions. With the Boundary Element Method, as applied to t h i s c l a s s of bodies, i t i s necessary to define an axisymmetric co n t r o l surface. This co n t r o l surface encloses the e n t i r e f l u i d domain around the object. The outer bound of t h i s surface i s assumed to be s u f f i c i e n t l y f a r away from the object so that the only wave energy i s assume to cross t h i s boundary according to the r a d i a t i o n boundary condition. The p o t e n t i a l on the control surface i s determined by d i s c r e t i z i n g the c o n t r o l surface into f i n i t e r i n g elements. The p o t e n t i a l within each element i s determined by so l v i n g a system of l i n e a r simultaneous equations which can be set up by applying the appropriate boundary conditions to the governing equation. - 7 -2. EXPERIMENTAL WORK The experimental work was c a r r i e d out i n the Ocean Engineering Centre of BC Research i n Vancouver, BC. Exterior and i n t e r i o r views of t h i s f a c i l i t y are provided i n Figures 7.2 and 7.3, r e s p e c t i v e l y . Two types of tests were conducted on each of the c y l i n d e r models. These include hydrodynamic tests and wave induced e x c i t i n g force t e s t s . Each test was conducted to v e r i f y the predicted r e s u l t s of the Boundary Element Method and Matching Technique. The subsequent sections of t h i s chapter w i l l describe the experimental setup and procedures f o r each t e s t . In the course of t h i s d i s c u s s i o n frequent reference w i l l be made to equipment which i s described i n greater d e t a i l i n Appendix A. 2.1 HYDRODYNAMIC TESTS The hydrodynamic tests were conducted on the s i n g l e , double and t r i p l e c y l i n d e r models. For each of these tests the motion generator imparted small amplitude s i n u s o i d a l surge motion to each of the c y l i n d e r models at a p a r t i c u l a r frequency. The induced surge force and p i t c h moment was measured by the dynamometer and recorded by the data a c q u i s i t i o n computer. The displacement of the cy l i n d e r model was measured by a yo-yo p o s i t i o n transducer. An array of three pressure transducers was located along a submerged surface on the double and t r i p l e c y l i n d e r models. The l o c a t i o n of these transducers i s shown i n Figure 8.5. A wave probe, located 1 to 2 meters - 8 -away from the c y l i n d e r model, measured the induced wave height. From t h i s measurement i t was possible to v e r i f y the so c a l l e d Wehausen formulation, Equation (3.1.6-1), which r e l a t e s the induced wave height and surge force to the damping c o e f f i c i e n t . A s i g n a l conditioner was used to provide the necessary e x c i t a t i o n and a m p l i f i c a t i o n before any s i g n a l from a transducer was recorded. The data was w r i t t e n i n multiplexed form on floppy diskettes by a MINC 11 mini computer which comprised the data a c q u i s i t i o n system. This data was l a t e r processed by data analysis software programs on a Vax 11/750. D e t a i l s on the data a c q u i s i t i o n system and the data analysis software are provided i n Appendix B. From the displacement record i t was possible to determine the magnitude of the v e l o c i t y and a c c e l e r a t i o n at any time. Furthermore, since the motion generated was purely s i n u s o i d a l , i t was possible to consider points where the v e l o c i t y was maximum and the a c c e l e r a t i o n was zero. Conversely, points exi s t e d where the c y l i n d e r model instantaneously had no v e l o c i t y but the a c c e l e r a t i o n was maximum. At these points i t was possible to i s o l a t e the i n d i v i d u a l c o n t r i b u t i o n of added mass and damping terms i n the following dynamic system equation. . . .(2.1-1) Where, F s = surge force; m = body mass; a 11 = added mass c o e f f i c i e n t ; X = body a c c e l e r a t i o n i n surge d i r e c t i o n ; - 9 -b = damping c o e f f i c i e n t ; X = body v e l o c i t y i n surge d i r e c t i o n . From t h i s equation one can see that i t was possible to determine the added mass c o e f f i c i e n t s and damping c o e f f i c i e n t s r e a d i l y when ei t h e r the v e l o c i t y or a c c e l e r a t i o n terms were zero (excepting of course the t r i v i a l case which a r i s e s i f both are equal to zero). In conducting these experiments each c y l i n d e r was harmonically o s c i l l a t e d at frequencies i n the nominal range between .25 Hz and 2.50 Hz i n increments of .25 Hz. For the larger c y l i n d e r models the forces encountered at the higher frequencies were i n some cases so large that the e n t i r e motion generator would shake v i o l e n t l y , c u t t i n g the top l e v e l of the frequency range which could be tested back to 2.0 or 2.25 Hz. In addition, to conducting tests on d i f f e r e n t models at d i f f e r e n t frequencies, two other parameters were independently va r i e d . Tests were conducted at d i f f e r e n t d r a f t s for each c y l i n d e r and t e s t s were conducted at d i f f e r e n t amplitudes of motion. The theories used i n p r e d i c t i n g the experimental r e s u l t s are l i n e a r i z e d and assume small amplitudes of motion with respect to the c y l i n d e r diameter. By varying the amplitude of motion one can b e t t e r r e a l i z e the l i m i t a t i o n s of the l i n e a r i z i n g assumptions on the theory. In a d d i t i o n by varying the c y l i n d e r d r a f t i t was possible to understand the e f f e c t s i f any of the shallow water region which existed above the l a r g e s t c y l i n d e r section of the double and t r i p l e c y l i n d e r models. A complete l i s t i n g of a l l tests conducted i s provided i n Table 2.2-1. - 10 -2 . 2 WAVE INDUCED EXCIT ING FORCE TESTS F o r t h e wave i n d u c e d e x c i t i n g f o r c e t e s t s t h e c y l i n d e r mode l r e m a i n e d s t a t i o n a r y and waves g e n e r a t e d b y a f l a p p e r t y p e wave maker were s e n t p a s t t h e c y l i n d e r m o d e l . F o r t h e s e t e s t s t h e c y l i n d e r d r a f t was v a r i e d . I n a d d i t i o n , t h e a m p l i t u d e and f r e q u e n c y o f t h e waves was c h a n g e d . I n o t h e r r e s p e c t s t h e s e t up was much t h e same as f o r t h e h y d r o d y n a m i c t e s t s , t he TM m a j o r e x c e p t i o n b e i n g w i t h t h e u s e o f an IBM P e r s o n a l Compute r t o c o n t r o l t h e wave m a k e r . D e t a i l s on t h i s s y s t e m a r e p r o v i d e d i n S e c t i o n 5 . 1 . 2 . T h e s e t e s t s were g e n e r a l l y c o n d u c t e d a t f r e q u e n c i e s i n a r a n g e b e t w e e n .3 and 1 .5 Hz w i t h i n c r e m e n t s o f .2 H z . T h i s r e p r e s e n t s t h e p r a c t i c a l o p e r a t i n g r a n g e o f t h e wave m a k e r . O u t s i d e o f t h i s r a n g e t h e wave f o rms were o f i n s u f f i c i e n t q u a l i t y t o be c o n s i d e r e d r e g u l a r o r s i n u s o i d a l . T e s t s were c o n d u c t e d f o r t h r e e d i f f e r e n t wave a m p l i t u d e s a t e a c h f r e q u e n c y . The s i n g l e , d o u b l e and t r i p l e c y l i n d e r m o d e l s were e a c h t e s t e d b y t h i s m e t h o d . A l i s t i n g o f t h e e x c i t i n g f o r c e t e s t s c o n d u c t e d i s i n c l u d e d i n T a b l e 2 . 2 - 1 . The a m p l i t u d e o f t h e i n d u c e d e x c i t a t i o n f o r c e was compared t o t h a t p r e d i c t e d b y t h e B o u n d a r y E l e m e n t M e t h o d . I n a d d i t i o n t h e Wehausen f o r m u l a t i o n , E q u a t i o n ( 3 . 1 . 6 - 1 ) , c o u l d be u s e d t o p r e d i c t t h e damping c o e f f i c i e n t f r o m t h e m e a s u r e d wave h e i g h t and e x c i t a t i o n f o r c e . The r e s u l t s o b t a i n e d u s i n g t h i s f o r m u l a t i o n were compared w i t h t h e p r e d i c t i o n s o f t he B o u n d a r y E l e m e n t M e t h o d and t h e M a t c h i n g T e c h n i q u e . 11 -TABLE 2 . 2 - 1 SUMMARY OF TESTS CONDUCTED DATE TEST NUMBERS CYL IN MODEL TYPE S / D / T TEST TYPE &NO. D/W AMPL OF MOTION (mm) STEP DRFT (mm) MASS (kg ) DIST WP-CYL (m) 06OCT85 S 1 1 1 1 - S 1 1 1 1 0 S D-7 10 218 52 1 .30 06OCT85 S 1 1 2 1 - S 1 1 2 1 0 S D-8 25 218 52 1 .30 1 .27 06OCT85 S1131 -S11310 S D-9 35 218 52 1 .27 06OCT85 S 2 1 1 1 - S 2 1 5 3 S W-3 - 218 52 1.26 18JAN86 SDA1-SDA10 S D-6 35 218 2 6 . 9 1 .55 18JAN86 SDB1-SDB10 S D-5 25 218 2 6 . 9 1 .55 18JAN86 SDC1-SDC10 S D-4 10 218 2 6 . 9 1 .55 18JAN86 SWA1-SWG3 S W-2 218 2 6 . 9 1 .55 19JAN86 DDA2-DDA10 V D 1 D-15 10 247 6 5 . 4 1 .55 19JAN86 DWA1-DWG3 2 3 D 1 W-6 - 247 6 5 . 4 1 .55 19JAN86 DDB1-DDB9 2 3 D 1 D-16 25 247 6 5 . 4 1 .55 01FEB86 DDC1-DDC9 3 D 2 1 D-14 25 236 6 5 . 4 1 .58 01FEB86 DWH1-DWN3 3 D 2 1 W-5 - 236 6 5 . 4 1 .58 02FEB86 DYN01-DYN03 S D 55 0 2 6 . 9 -02FEB86 SWH1-SWN3 S W - l 211 2 6 . 9 1 .58 02FEB86 SDD1-SDD10 S D - l 10 211 2 6 . 9 1 .58 1 .58 02FEB86 SDE1A-SDE10 S D-2 25 211 2 6 . 9 1 .28 02FEB86 SDF1-SDF10 S D-3 35 211 2 6 . 9 1 .28 03FEB86 DDD1-DDD10 3 D 2 1 D-13 15 236 6 5 . 4 1.28 f Numbers indicate orientation of pressure transducers. The upstream direction (toward the wave maker) is taken to be toward the top of the page. - 12 -TABLE 2.2-1 (CONTINUED) SUMMARY OF TESTS CONDUCTED DATE TEST NUMBERS CYLIN MODEL TYPE S/D/T TEST TYPE &NO. D/W AMPL OF MOTION (nun) STEP DRFT (nun) MASS (kg) DIST WP-CYL (m) 04FEB86 D2WA1-D2WG3 3 D 2 1 W-4 - 171 60.1 1.58 04FEB86 D2DA1-D2DA8 23 D 1 D-10 15 171 60.1 1.28 04FEB86 D2DB1-D2WB10 23 D 1 D - l l 15 171 60.1 1.28 05FEB86 D2DC1-D2WC10 23 D 1 D-12 25 171 60.1 1.28 05FEB86 TDA1-TDA9 23 T 1 D-20 25 171 83.1 1.28 05FEB86 TDB1-TDB9 23 T 1 D-19 10 171 83.1 1.28 15FEB86 TWlAl—TW1G3 31 T 2 W-8 - 169 80.3 1.71 15FEB86 TXA1-TXA9 31 T 2 D&W 25 169 80.3 1.71 15FEB86 TW2A1—TW2G3 31 T 2 W-7 - 100 80.3 1.71 16FEB86 TD2A1—TD2A9 31 T 2 D-18 25 100 80.3 1.54 16FEB86 TD2B1—TD2B9 31 T 2 D-17 10 100 80.3 1.54 16FEB86 TD3A1-TD3A9 31 T 2 D-25 10 410 80.3 1.54 16FEB86 TD3B1—TD3B9 31 T 2 D-27 25 410 80.3 1.54 16FEB86 TD3C1—TD3C9 31 T 2 D-26 15 410 80.3 1.54 16FEB86 TW3A1—TW3G3 31 T 2 W-9 - 410 80.3 1.54 20APR86 TD4A1-TD4A9 31 T 2 D-28 10 420 83.3 1.29 20APR86 TD4B1—TD4B9 31 T 2 D-29 15 420 83.3 1.29 20APR86 TD4C1—TD4C9 31 T 2 D-24 15 322 83.3 1.29 - 13 -TABLE 2.2-1 (CONTINUED) SUMMARY OF TESTS CONDUCTED DATE TEST NUMBERS CYLIN MODEL TYPE S/D/T TEST TYPE &NO. D/W AMPL OF MOTION (mm) STEP DRFT (mm) MASS (kg) DIST WP-CYL (m) 20APR86 TD4D1-TD4D9 31 T 2 D-23 10 322 83.3 1.29 21APR86 TD4E1—TD4E9 31 T 2 D-21 10 218 83.3 1.29 21APR86 TD4F1-TD4F9 31 T 2 D-22 15 218 83.3 1.29 - 14 -3. THEORETICAL MODELS Before discussing the theories at length i t w i l l be b e n e f i c i a l to carry out a dimensional analysis of t h i s problem. By so doing one can determine the r e l a t i v e importance of flow separation and d i f f r a c t i o n e f f e c t s . The force on a f i x e d structure due to an incident wave may be defined as: PgHD = f H V D V T . . . ( 3 - 1 ) Where, L H L D L V D v V T D P g d L H D V = dis p e r s i o n term; = wave steepness term; = d i f f r a c t i o n term; = Reynolds number; = Keulegan-Carpenter number •= density of water; = G r a v i t a t i o n a l constant; = water depth; = wavelength; = wave height; = representative diameter of the body; = representative r e l a t i v e v e l o c i t y of f l u i d with respect to the cy l i n d e r ; = kinematic v i s c o s i t y ; => period of motion. - 15 -A s s t a t e d i n t h e p r e v i o u s s e c t i o n t e s t s were c o n d u c t e d i n t h e r a n g e o f f r e q u e n c i e s b e t w e e n 0 . 2 5 and 2 . 5 0 H z . W i t h t h i s k n o w l e d g e i t i s p o s s i b l e t o d e t e r m i n e a r a n g e f o r some o f t h e s e n o n - d i m e n s i o n a l q u a n t i t i e s . O f p a r t i c u l a r i n t e r e s t i n t h i s p r o b l e m i s t h e K e u l e g a n - C a r p e n t e r number and t h e d i f f r a c t i o n t e r m D / L . T a b l e 3-1 shows t h e v a l u e o f t h e t e rms f o r 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 r e p r e s e n t a t i v e d i a m e t e r , D, o f t h e body i s t a k e n t o be t he - 3 2 maximum d i a m e t e r (386mm) and t h e v i s c o s i t y , v, i s t a k e n t o be 1 . 1 x 1 0 m / s . T a b l e 3 -1 RANGE OF NON-DIMENSIONAL QUANTITIES ENCOUNTERED F r e q (Hz) A m p l i t u d e o f M o t i o n (mm) K e u l e g a n - C a r p e n t e r Number R e y n o l d s Number D i f f r a c t i o n Term 0 . 2 5 10 0 . 1 6 6 . l x l O 3 0 . 0 2 35 0 . 5 7 2 . 1 x 1 0 * 0 . 0 2 1 .00 10 0 . 1 6 2 . 4 x 1 0 * 0 . 2 6 35 0 . 5 7 8 . 5 x 1 0 * 0 . 2 6 2 . 5 0 10 0 . 1 6 6 . 1 x 1 0 * 1 .61 35 0 . 5 7 2 . 1 x l 0 5 1.61 The R e y n o l d s Number i s a l w a y s l e s s t h a n 2 . 1x10 . L a m i n a r f l o w i s c o n s i d e r e d t o e x i s t f o r a R e y n o l d s Number l e s s t h a n 2 . 0 x 1 0 ^ . T h e r e f o r e t e s t i n g i s b e i n g c o n d u c t e d t o 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 r e g i m e and a p o t e n t i a l s o l u t i o n c a n be a p p l i e d t o t h e o u t e r f l o w f i e l d . F o r t h i s p r o b l e m t h e K e u l e g a n - C a r p e n t e r Number, K , i s n e v e r g r e a t e r t h a n 0 . 5 7 . I s a a c s o n c e t . a l . ( 1 9 8 1 ) , p . 3 8 1 , 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 n o m i n a l l y g r e a t e r t h a n 2 . 0 and t h a t d i f f r a c t i o n e f f e c t s a r e c c o n s i d e r e d i m p o r t a n t f o r D / L n o m i n a l l y g r e a t e r t h a n 0 . 2 . T h i s l a t t e r c r i t e r i a - 16 -i s s a t i s f i e d f o r frequencies greater than 0.9 Hz. Tests conducted above t h i s frequency are c l e a r l y within the d i f f r a c t i o n regime and can be solved n e g l e c t i n g viscous e f f e c t s with a p o t e n t i a l f l u i d domain throughout. In tests conducted below 0.9 Hz neither viscous e f f e c t s nor d i f f r a c t i o n may be considered dominant. Isaacson (1981) suggests the force which a r i s e s i n t h i s region i s i n e r t i a l and can be determined using Morison's equation which i s based on empirical data. Nevertheless, a p o t e n t i a l model w i l l also be applied at these low frequencies to these problems to investigate the v a l i d i t y of t h i s model f o r t h i s combination of K and D/L. 3.1 MATCHING TECHNIQUE The matching technique formulation was or i g i n a t e d by Garrett i n 1971 where i t was used to study the s c a t t e r i n g of waves i n the presence of a f l o a t i n g c i r c u l a r dock. I t was l a t e r applied by Sabuncu and C a l i s a l i n 1981 fo r s o l v i n g motion of a sing l e v e r t i c a l c i r c u l a r c y l i n d e r problem and i n 1984 fo r s o l v i n g a double v e r t i c a l c i r c u l a r c y l i n d e r problem 1. This technique i s applied by d i v i d i n g the s o l u t i o n domain into subregions and representing the a n a l y t i c a l s o l u t i o n f o r each of these subregions i n se r i e s form with 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 are obtained by matching the pressures and the normal v e l o c i t i e s on the boundaries between these subregions. This s e c t i o n w i l l describe the Matching Technique i n d e t a i l as i t i s applied to a t r i p l e v e r t i c a l c i r c u l a r c y l i n d e r model. For a d e s c r i p t i o n of 1 E a r l y i n 1987 Sabuncu and C a l i s a l w i l l publish a paper e n t i t l e d "A Generalized Method for the Calculation of Hydrodynamic Forces on Discontinuous Vertical Cylinders" - 17 -t h i s theory as applied to a sing l e and a double c y l i n d e r model the reader i s r e f e r r e d to the referenced papers by Sabuncu and C a l i s a l . Consider a c i r c u l a r c y l i n d e r model with geometry as shown i n Figure 8.4. The s o l u t i o n domain i s divided into four subregions as indicated i n Figure 8.6. They may be mathematically described as follows: REGION 1 2 3 4 0<r<a a <r<a 1 2 a <r<a 3 2 r>a 0<z<d 0<z<d d <z<d 3 0<z<d The shape of these subregions i s convieniently chosen so that the po t e n t i a l s can be most e a s i l y determined and the boundary conditions most e a s i l y applied. 3.1.1 GOVERNING EQUATION 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 Laplace's equation since the flow i s assumed to be i r r o t a t i o n a l and incompressible. Laplace's equation may be expressed as: 7*4 - 0 (3.1.1-1) Since the shapes being studied are axisymmetric i t i s convenient to express Laplace's equation i n c y l i n d r i c a l coordinates. - 18 -«!• + ± d - * + ± £!• + *!* _ 0 . ...(3.1.1.2) a r 2 r 3r r 2 d 6Z 8 z 2 To solve t h i s problem the separation of v a r i a b l e s technique i s applied using the appropriate boundary conditions. I t i s assumed that the p o t e n t i a l function can be defined as the product of four independent functions, each function containing a single v a r i a b l e . The p o t e n t i a l function may therefore be expressed as follows: * ( r , 0 , z ; t ) - R(r)°e(c5)°Z(z)°T(t) ...(3.1.1-3) Since t h i s problem i s concerned with p e r i o d i c motion one can immediately obtain an expression f o r T ( t ) . T(t) = e " i w t ...(3.1.1-4) The p o t e n t i a l can therefore be written as: $(r,c9,z;t) - *e{ <j>{r,6,z) e'iut ) ...(3.1.1-5) Now s u b s t i t u t i n g Equation (3.1.1-5) into the o r i g i n a l p a r t i a l d i f f e r e n t i a l equation, Equation (3.1.1-2), one obtains the following three ordinary d i f f e r e n t i a l equations: *** + A 2 9 = 0 ...(3.1.1-6) d ez - 19 -d * Z j. u2 + B z d z 2 d 2 R 1 dR d r 2 r d r B 2 - A 2 R = 0 . . ( 3 . 1 . 1 - 7 ) ( 3 . 1 . 1 - 8 ) 3 . 1 . 2 BOUNDARY CONDITIONS T h i s h y d r o d y n a m i c p r o b l e m i n v o l v i n g a f l o a t i n g body a t t h e f r e e s u r f a c e h a s i n g e n e r a l f i v e b o u n d a r y c o n d i t i o n s . The c o o r d i n a t e s y s t e m shown i n F i g u r e 8 . 1 w i l l be u s e d t h r o u g h o u t t h i s d i s c u s s i o n . The X - Y p l a n e c o i n c i d e s w i t h t h e s u r f a c e o f t h e w a t e r . The o r i g i n i s l o c a t e d i n t h i s p l a n e a l o n g t he c e n t e r a x i s o f t h e c y l i n d e r m o d e l . The p o s i t i v e Z d i r e c t i o n i s t a k e n t o be v e r t i c a l l y u p w a r d s . The p o s i t i v e X d i r e c t i o n i s t a k e n t o be a l o n g t h e t a n k i n a d i r e c t i o n away f r o m t h e wave m a k e r . The p o s i t i v e Y d i r e c t i o n i s t h e r e f o r e assumed t o be t o w a r d t h e c a r r i a g e s i d e o f t h e t o w i n g t a n k . i ) On b o t t o m s u r f a c e a t z=0 t h e r e e x i s t s an 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 where 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 e q u a l t o z e r o . I n t h i s a n a l y s i s t h e b o t t o m i s assumed t o be f l a t o r s l o p i n g a t a s u c h a s l o w r a t e t h a t i t may be c o n s i d e r e d as f l a t i n some l a r g e n e i g h b o u r h o o d o f t h e o b j e c t . £ 5 - 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 an impe rmeab le - 20 -boundary cond i t i o n which states that the f l u i d v e l o c i t y i n a normal d i r e c t i o n to the body surface must be equal to normal v e l o c i t y component of the body on the surface. In general t h i s may be expressed as: 3$ 3n - - - - - (31 2-2) = V ° n + n ° ( r x n ) ' ' " ^ •• L- z L> S Where V i s the v e l o c i t y vector, fi i s the angular v e l o c i t y vector of the f l o a t i n g body and n i s a un i t vector normal to the body surface. For pure surge motion of the model being studied t h i s expression may be s i m p l i f i e d to the following expressions. 3$ ...(3.1.2-3) •= 0 at z=d r<a az i i z=d a <r<a 2 1 2 z=d a <r<a 3 3 2 |5 = v cos 9 at r=a d < z<d ...(3.1.2-4) 3r s 1 1 2 r=a d <z<d 2 2 3 r=a d <z<d 3 3 V i s the v e l o c i t y of the body surface i n the surge d i r e c t i o n . i i i ) For a wave incident on a submerged surface, S , the normal d e r i v a t i v e of b the d i f f r a c t e d wave p o t e n t i a l , $ d, i s equal to the negative of the normal d e r i v a t i v e of the incident wave p o t e n t i a l , $ . as> ...(3.1.2-5) an 3n 21 -Where the incident wave p o t e n t i a l , 9 , i s defined by Isaacson et. a l . (1981) as: ^ iHg cosh k(z,+d) ^iCkx-ut) ...(3.1.2-6) i 2u cosh(kd) H i s the wave height, k the wave number, d the water depth and w the angular wave frequency. The wave number k can be obtained by s o l v i n g the dispersion r e l a t i o n . w2 . _ , .... ...(3.1.2-7) — = k tanh (kh) iv) At a boundary defined by a con t r o l surface, S , some 'long' distance away from the body surface there e x i s t s a r a d i a t i o n boundary condition. This c o n d i t i o n ensures that waves radiate away from the body. For an axisymmetric body the r a d i a t i o n boundary condition has been determined by Bai (1972) to be of the following form. 5n 2 " S + i k (3.1.2-8) k i s the wave number, and R i s the distance from the body axis to the control surface, S . v) The l a s t boundary condition i s the free surface boundary condition. In f a c t there are two separate boundary conditions which must be s a t i s f i e d . They include a kinematic condition and a dynamic condition. The kinematic boundary - 22 -condition states that f l u i d p a r t i c l e s at the free surface, rj, must remain at the free surface. This can be mathematically expressed as: 3T7 3$ 3n 3$ dn 3$ 3t 3x ox 3y ay 3z (3.1.2-9) The dynamic boundary condition i s the B e r n o u l l i equation which states the pressure, p, i s uniform over the free surface. + 1 / a t 2 |_ "a$ 2 1 "a$" 2 1 "a$ a x T a y T a z } + J + grj = 0 . . (3.1.2-10) In t h i s case the pressure p i s atmospheric and i s taken as constant throughout. These boundary c o n d i t i o r s are non-linear and considerbly d i f f i c u l t to u t i l i z e i n t h e i r present form. Since i t i s desirable to obtain a l i n e a r s o l u t i o n , one can combine Equations (3.1.2-9) and (3.1.2-10) and l i n e a r i z e by neglecting terms of order greater than unity. The r e s u l t of t h i s procedure y i e l d s a s i n g l e l i n e a r i z e d free surface boundary condition. 3$ _ w 3z ~ g $ at z=d (3.1.2-11) 3.1.3 DEFINITION OF POTENTIALS Using these boundary conditions one can now solve Laplace's equation for the p o t e n t i a l within each region. These p o t e n t i a l s w i l l be defined i n terms of unknown c o e f f i c i e n t s and by matching conditions which e x i s t along adjacent boundaries one can determine the value of the p o t e n t i a l everywhere i n the f l u i d domain. The general s o l u t i o n consists of a p a r t i c u l a r s o l u t i o n and a homogeneous s o l u t i o n . Sabuncu and C a l i s a l (1981) describe the p o t e n t i a l by the general expression given as: *( r , z , 0 ; t ) - V d [ <f> (r,z) + <f> (r,z) ] cos 9 e" i w t ...(3.1.3-1) s p h This same reference states that f o r pure surge the p a r t i c u l a r solution, <f> ( r , z ) , i s equal to zero. Therefore one can now write a general expression p f o r the p o t e n t i a l function i n each region around a compound c y l i n d e r as follows: 3>(r,z,0;t) = V d cos 9 e" i w t t>(r,z) ...(3.1.3-2) The homogeneous term, <6(r,z), i n Equation (3.1.3-2) i s defined i n the subsequent sections of t h i s chapter. Each p o t e n t i a l function contains terms i n v o l v i n g Bessel type functions. To understand the notation to be used a l i s t of each Bessel function with i t s corresponding symbol i s provided here. The reader i s r e f e r r e d to Appendix F for various i d e n t i t i e s which are us e f u l i n the a n a l y s i s . J (x) Bessel function of the f i r s t kind of order n n I (x) Modified Bessel function of the f i r s t kind of order n n K (x) Modified Bessel function of the second kind of order n n H (x) Hankel function of the f i r s t kind of order n n Any other terms, which appear i n the equations to follow w i l l be defined as - 24 -they appear. 3.1.3.1 REGION 1 The homogeneous s o l u t i o n f o r the p o t e n t i a l function i n region 1 i s : r a <» I (n7rr/d ) Z i I A T — — cos ( n 7 r z/d ) n I (nTra /d ) ' r n - l l r 1 (3.1.2.1-1) The c o e f f i c i e n t s A and A are unknown and w i l l be solved f o r l a t e r . O n . 3.1.3.2 REGION 2 Solving f o r the homogeneous term i n region 2 one obtains: * 2 ( r , z ) 2 < 2^ 2 a +a 1 2 2 i a a .. 2 1 2 1 r + 2 2 r a +a 1 2 + C a a ., 1 2 1 2 2 a +a 1 2 . 2 2 r a +a 1 2 + V [B V (r) + C W (r) ] cos (n7rz/d ) . . . (3 .1. 3 . 2-1) LA n n n n 2 n - l Where, for n = 1,2,3... V (r) I (n7rr/d ) K (n?ra /d ) 1 2 1 V 2 I (mra /d ) K (nwr/d ) 1 1 2 1 ' 2 I (n?ra /d ) K (n?ra /d ) l 2 ' 2 1 1 2 I (n?ra /d ) K (mra /d ) 1 v 2 1 2 ' 2 ...(3.1.3.2-2) - 25 -f o r n = 1 , 2 , 3 . W ( r ) I (rwrr/d ) K Crura / d ) - I (rwra /d ) K (n7rr /d ) 1 V ' 2 1 2 ^ 2 1 2 2 1 ' 2 l'(ri7ra / d ) K (riTra / d ) - I (riTra / d ) K'(riTra / d ) l v l / 2 1 2 ' 2 1 2 ' 2 1 r 2 . . ( 3 . 1 . 3 . 2 - 3 ) The unknown c o e f f i c i e n t s B , B , C and C w i l l be d e t e r m i n e d l a t e r . 0 n 0 n 3 . 1 . 3 . 3 REGION 3 W i t h i n r e g i o n 3 t h e homogeneous s o l u t i o n t a k e s t h e f o r m : <t> ( r , z ) - f [D X ( r ) + D Y ( r ) ] Z ( r ) 3 Li n n n n n n=0 . . . ( 3 . 1 . 3 . 3 - 1 ) Where , N " 1 / Z s i n h (m ( d - d ) ) 0 0 3 d ( d - d ) m 2 3 ( 3 . 1 . 3 . 3 - 2 ) f o r n = 1 , 2 , 3 . N ~ 1 / 2 s i n (m ( d - d ) ) n n 3 d ( d - d ) m 2 3 n ( 3 . 1 . 3 . 3 - 3 ) o 2. 1 + s i n h (m ( d - d ) ) v o 3 2m ( d - d ) 0 3 ( 3 . 1 . 3 . 3 - 4 ) f o r n = 1 , 2 , 3 . 1 + s i n h (2m ( d - d ) ) n 3 2m ( d - d ) n 3 ( 3 . 1 . 3 . 3 - 5 ) - 26 -mQ and m are the solutions to the equations: w2 - g mo tanh (m (d-d )) - 0 ...(3.1.3.3-6) fo r n = 1,2 , 3 . . . o>2 + g m tan (m (d-d )) = 0 ...(3.1.3.3-7) n n 3 J (m r) H (m a ) - J (m a ) H (m r) X o(r) - - L - 2 L_1_J L_JLJ ...(3.1.3.3-8) J ( m a ) H ( m a ) - J ( m a ) H ( m a ) 1 0 2 1 0 3 1 0 3 1 0 2 f o r n = 1,2,3... I (m r) K (m a ) - I (m a ) K (m r) I n l n 3 l n 3 I n X (r) =— - — — = — — - — ...(3.1.3.3-9) I (m a ) K'(m a ) - l'(m a ) K (m a ) l n 2 l n 3 l n 3 l n 2 J (m r) H (m a ) - J (m a ) H (m r) Y Q ( r ) = — 1 — 2 L _ L J ^ _ £ _ ^ L _ J ...(3.1.3.3-10) J ( m a ) H ( m a ) - J ( m a ) H ( m a ) 1 0 3 1 0 2 1 0 2 1 0 3 for n = 1,2,3.. I ( m r ) K ( a a ) • I, (m a,) K (m r) I n l n 2 l n 2 I n Y (r) = = —= = — = ...(3.1.3.3-11) l'(m a ) K (m a ) - I ( m a ) K (m a ) l n 3 l n 2 l n 2 l n 3 Z = N _ 1 / 2cosh m (z-d ) ...(3.1.3.3-12) 0 0 0 3 f o r n = 1,2,3. Z = N" 1 / 2cos m (z-d ) ...(3.1.3.3-13) n n n 3 The c o e f f i c i e n t D i s to be determined. - 27 -3.1.3.4 REGION 4 In the e x t e r i o r region the homogeneous p o t e n t i a l s o l u t i o n i s : H (k r) oo K (k r) <j> (r,z) = E n - Z (z) + V E — ± — 2 Z ( z ) ...(3.1.3.4-1) H (k a ) n - i K (k r) 1 0 2 I n Where, Z (z) = JV" 1 / 2cosh (k z) 0 0 0 ...(3.1.3.4-2) fo r n = 1,2,3. . . Z (z) = JV"1/2cos (k z) n n n (3.1.3.4-3) o 2 1 + sinh (2k d) o 2k d o (3.1.3.4-4) for n = 1,2,3. . . 1 N 2 1 + s i n (2k d) n 2k d (3.1.3.4-5) k Q and k are the solutions to the equations: w - g k tanh (k d) = 0 ° o o (3.1.3.4-7) fo r n = 1,2,3... w2 + g k tan (k d) - 0 ...(3.1.3.4-8) n n E and E are to be determined, o n - 28 -3 . 1 . 4 SOLVING FOR THE UNKNOWN COEFFICIENTS The unknown c o e f f i c i e n t s e r i e s A , B , C , D , and E a r e d e t e r m i n e d by n n n n n m a t c h i n g t h e p r e s s u r e s and t h e 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 . T h i s l e a v e s 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 . C o n s i d e r two a d j a c e n t r e g i o n s w i t h p o t e n t i a l s $ and $ . From E q u a t i o n ( 3 . 1 . 3 . 2 ) one c a n w r i t e t he a b p o t e n t i a l f u n c t i o n s a s : $ = v d e " i W t c o s 8 <j> . . . ( 3 . 1 . 4 - 1 ) a s a $ - V d e " i W t c o s 8 4> . . . ( 3 . 1 . 4 - 2 ) b s b A l o n g t h e common b o u n d a r y b e t w e e n r e g i o n s a and b t h e r e must be c o n t i n u i t y o f p r e s s u r e . T h e r e f o r e : p = p . . . ( 3 . 1 . 4 - 3 ) a D 3$ a $ •P — = -P — . . . . ( 3 . 1 . 4 - 4 ) a t a t iwp V d e <f> = iwp V d e <f> . . . ( 3 . 1 . 4 - 5 ) s a s b 4> - 6 . . . ( 3 . 1 . 4 - 6 ) a b T h i s p r o v e s t h a t t o s a t i s f y t h e c o n t i n u i t y o f p r e s s u r e a l o n g t h e s h a r e d b o u n d a r y b e t w e e n two r e g i o n s one n e e d o n l y e q u a t e t h e homogeneous s o l u t i o n o f t h e p o t e n t i a l f u n c t i o n s a t t h i s b o u n d a r y . - 29 -For c o n t i n u i t y of r a d i a l v e l o c i t y along the same boundary one may write the f o l l o w i n g r e l a t i o n s : 3$ 3$ — ...(3.1.4-7) 3r 3r 8<f> 8<f> V d e " " * — i - V d e " i W t - b ...(3.1.4-8) 8x 8 8r 8<f> 8<p — - ...(3.1.4-9) 8x 8x By equating the v e l o c i t y and pressure along the common boundary between two regions one now has a l g e b r a i c a l l y defined the required conditions i n terms of the homogeneous s o l u t i o n of the p o t e n t i a l function. However, since each unknown c o e f f i c i e n t i s i n f a c t an i n f i n i t e s e r i e s of terms i t i s convenient to make use of the orthogonal properties of the p o t e n t i a l function to solve f o r each term i n the s e r i e s . This i s done by equating the i n t e g r a l over the depth. From the c o n t i n u i t y of pressure equation, Equation (3.1.4-6), one can write: " <j> Q(z) dz ... (3.1.4-10) b 4> Q(z) dz a d d l l Here Q(z) i s the orthogonal function for d^<z<d . S i m i l a r l y , one can derive a r e l a t i o n from the c o n t i n u i t y of v e l o c i t y equation, Equation - 30 -(3.1.4-9). In t h i s case the normal d e r i v a t i v e of the p o t e n t i a l replaces the p o t e n t i a l f unction of Equation (3.1.4-10). By formulating the problem i n t h i s way one has equated the weighted i n t e g r a l of the pressure and v e l o c i t y along the common boundary. A l t e r n a t i v e l y , one could have equated the pressures and v e l o c i t i e s i n the adjoining regions at a number points. The number of points being equal to the number of terms to be determined i n each s e r i e s . The i n t e g r a l method of s o l u t i o n involves no d i s c r e t i z a t i o n of the boundary and u t i l i z e s the orthogonal properties of the p o t e n t i a l function. The subsequent sections w i l l describe how t h i s formulation i s applied to the t r i p l e c y l i n d e r model. 3.1.4.1 CONTINUITY OF PRESSURE BETWEEN REGIONS 1 AND 2 Applying Equation (3.1.4-10) to t h i s region and taking the orthogonal function over the i n t e r v a l z=0 to z=d as: l Q x(z) = 2 c o s (kwz/d ) I (3.1.4.1-1) one obtains at r=a 2 d i J t£ cos(k7rz/d i) dz 2 d l . <f>2 cos(k7rz/d i) dz (3.1.4.1-2) Using the properties of orthogonal functions, Equation (3.1.4.1-2) may be w r i t t e n as : - 31 -f o r n = 0 , 1 , 2 , 3 . . . 00 00 A - V a B - J 8 C - 0 . . . ( 3 . 1 . 4 . 1 - 3 ) n U t n t Li t n t W h e r e , t = 0 . t = 0 2a a a — — . . . ( 3 . 1 . 4 . 1 - 4 ) 00 2 2 a + a 1 2 f o r n = 1 , 2 , 3 . . On ( 3 . 1 . 4 . 1 - 5 ) f o r t = 1 , 2 , 3 . a d s i n ( t ; rd / d ) V ( a ) . . . ( 3 . 1 . 4 . 1 - 6 ) tO 7T d _ t 1 1 t f o r n = 1 , 2 , 3 . . . and t = l , 2 , 3 . . . 2d d t s i n (t?rd / d ) a = ( - 1 ) " — — — V ( a ) . . . ( 3 . 1 . 4 . 1 - 7 ) t n 7T -,2 2 ,2 2 t 1 d t - d n 1 2 2 2 a - a 8 - — ^ . . . ( 3 . 1 . 4 . 1 - 8 ) ^ 0 0 2 2 a + a 1 2 f o r n = 1 , 2 , 3 . B - 0 . . . ( 3 . 1 . 4 . 1 - 9 ) On f o r t = 1 , 2 , 3 . „ d s i n (t?rd / d ) a --4- — W (a ) . . . ( 3 . 1 . 4 . 1 - 1 0 ) K t 0 7T d _ t 1 1 t - 32 -f o r n - 1 , 2 , 3 . . . and t = 1 , 2 , 3 . tn (-D 2d d n 1 2 t s i n (d / d ) v r 2 d V 1 A2 2 - d n 2 W ( a ) t I . . ( 3 . 1 . 4 . 1 - 1 1 ) 3 . 1 . 4 . 2 CONTINUITY OF VELOCITY BETWEEN REGIONS 1 AND 2 Now, u s i n g E q u a t i o n ( 3 . 1 . 4 - 1 0 ) , one m u l t i p l i e s b y t h e f u n c t i o n c o s ( k7 rz /d 2 ) and i n t e g r a t e s a l o n g t h e d e p t h a t r = a 1 > a s ^ n E q u a t i o n ( 3 . 1 . 4 . 2 - 1 ) . d i 3 * i - r — c o s ( k 7 r z / d ) dz d r ' l 0 -=— c o s ( k 7 r z / d ) dz dx ' l 0 ( 3 . 1 . 4 . 2 - 1 ) A g a i n u s i n g t h e p r i n c i p l e s o f o r t h o g o n a l f u n c t i o n s one c a n w r i t e a s e c o n d s y s t e m o f e q u a t i o n s a s : f o r n = 0 , 1 , 2 , 3 . C - f £ A = b n Li tn t n t = 0 ( 3 . 1 . 4 . 2 - 2 ) Where , o d ( 3 . 1 . 4 . 2 - 3 ) f o r n = 1 , 2 , 3 . 2 d 2 — : — s in (n7 rd / d ) (nn) d ( 3 . 1 . 4 . 2 - 4 ) "oo d . . . ( 3 . 1 . 4 . 2 - 5 ) - 33 -f o r n = 1 , 2 , 3 . . . 1 d f o n " 7 ~ T IT s i n ( n * V d a ) . . . ( 3 . 1 . 4 . 2 - 6 ) (rur) i f o r n = 1 , 2 , 3 . . . and t = 0 , 1 , 2 , 3 . . . 2d d l'(tira / d ) ( - l ) f c s i n (mrd / d ) 12 1 1 1 1 2 cj = — — — — — . . . ( 3 . 1 . 4 . 2 - 7 ) t n ff T t*. IA \ A 2 2 j 2 2 I (tna / d ) d n - d t 1 1 1 1 2 3 . 1 . 4 . 3 CONTINUITY OF PRESSURE BETWEEN REGIONS 2 AND 4 U s i n g a s i m i l a r p r o c e d u r e t o t h a t d e s c r i b e d i n S e c t i o n 3 . 1 . 4 . 1 one o b t a i n s a t h i r d s y s t e m o f e q u a t i o n s a s : f o r n - 0 , 1 , 2 , 3 . . . CO B - V e E - 0 . . . ( 3 . 1 . 4 . 3 - 1 ) n Ld t i t t t = 0 Where , f o r n - 0 , 1 , 2 , 3 . . . H (k a ) e = — - — ° — ^ - Z . . . ( 3 . 1 . 4 . 3 - 2 ) n0 - . nO H (k a ) 1 0 2 f o r n = 0 , 1 , 2 , 3 . . . and t = 1 , 2 , 3 . . . K (k a ) e 5 _ J _ t . . . ( 3 . 1 . 4 . 3 - 3 ) n t K'(k a ) n t 1 0 2 - 34 -oo d A Z dz o ( 3 . 1 . 4 . 3 - 4 ) 2 J V ~ 1 / 2 s i n h (k d ) o v o 2' k d 0 2 ( 3 . 1 . 4 . 3 - 5 ) f o r n = 1 , 2 , 3 . . . t = ±r-nO d 2 Z cos(n7rz/d ) dz o ' z ( 3 . 1 . 4 . 3 - 6 ) 2 ( - l ) n J V " 1 / 2 (k d ) sinh (k d ) 0 0 2 0 2 (k d ) 2 + (nTr)2 0 2 ( 3 . 1 . 4 . 3 - 7 ) f o r t = 1 , 2 , 3 . . . A z = t r -ot d 2 Z dz t . ( 3 . 1 . 4 . 3 - 8 ) 2 / 1 / 2 s i n (k d ) t t 2 k d t 2 ( 3 . 1 . 4 . 3 - 9 ) f o r n - 1 , 2 , 3 . . . and t = 1 , 2 , 3 . . A Z = T i -nt d 2 Z cos(n?rz/d2) dz ( 3 . 1 . 4 . 3 - 1 0 ) 2 ( - l ) n J V ; 1 / 2 (k d ) s i n (k d ) ( k f c d 2 ) 2 + (nTr)2 ( 3 . 1 . 4 . 3 - 1 1 ) 3 . 1 . 4 . 4 CONTINUITY OF PRESSURE BETWEEN REGIONS 3 AND 4 The f o u r t h s y s t e m o f e q u a t i o n s i s f o u n d b y e q u a t i n g t h e p r e s s u r e s be tween r e g i o n s 3 and 4 . U s i n g t he o r t h o g o n a l f u n c t i o n : - 35 -Q ( z ) = n Z ( z ) n d - d 3 . ( 3 . 1 . 4 . 4 - 1 ) t h e f o u r t h s y s t e m o f e q u a t i o n s i s f o u n d t o b e : f o r n = 0 , 1 , 2 , 3 . D - Y" A E = 0 n L i n t t t = 0 ( 3 . 1 . 4 . 4 - 2 ) Where , f o r n = 0 , 1 , 2 , 3 . n0 H (k a ) 1 0 2 H ' ( k a ) 1 0 2 nO ( 3 . 1 . 4 . 4 - 3 ) f o r n = 0 , 1 , 2 , 3 . . . and t - 1 , 2 , 3 . . K (k a ) A ~ 1 ° 2 L n t K ' ( k a ) n t 1 0 2 ( 3 . 1 . 4 . 4 - 4 ) oo d - d Z Z dz 0 0 ( 3 . 1 . 4 . 4 - 5 ) (N N ) o o 1 / 2 k s i n h (k d ) 0 0 3 (d - d ) ( k 2 - m 2 ) 3 0 0 ( 3 . 1 . 4 . 4 - 6 ) f o r n = 1 , 2 , 3 . . no d - d Z Z dz 0 0 ( 3 . 1 . 4 . 4 - 7 ) (N N ) " 1 / 2 k s i n h (k d ) n o o 0 3 (d - d ) ( k 2 - m 2 ) 3 0 n ( 3 . 1 . 4 . 4 - 8 ) - 36 -for t = 1 , 2 , 3 L ot d - d Z Z dz t o . . . ( 3 . 1 . 4 . 4 - 9 ) (N JV ) o t 1 / 2 k sinh (k d ) t t 3 (d - d ) (k 2 - m2) 3 t 0 . . . ( 3 . 1 . 4 . 4 - 1 0 ) for n = 1 , 2 , 3 . and t - 1 , 2 , 3 . nt d - d Z Z dz t 0 ( 3 . 1 . 4 . 4 - 1 1 ) (N N ) n t 1 / 2 k sinh (k d ) t t 3 (d - d ) (k 2 - m2) 3 t n ( 3 . 1 . 4 . 4 - 1 2 ) Z and Z are defined as before i n Equations ( 3 . 1 . 3 . 3 - 1 2 ) , ( 3 . 1 . 3 . 3 - 1 3 ) , ( 3 . 1 . 3 . 4 - 2 ) and ( 3 . 1 . 3 . 4 - 3 ) . 3 . 1 . 4 . 5 CONTINUITY OF VELOCITY BETWEEN REGIONS 2 , 3 AND 4 The f i f t h and f i n a l system of equations i s formed by equating the r a d i a l v e l o c i t i e s from regions 2 and 3 to the r a d i a l v e l o c i t y from region 4 . In th i s case, using Equation ( 3 . 1 . 4 - 9 ) , the following equations can be written. a$ 3$ — - . . . ( 3 . 1 . 4 . 5 - 1 ) 3 r 3 r a$ a$ — . . . ( 3 . 1 . 4 . 5 - 2 ) 3 r a r Using the following as the orthogonal functions - 37 -f o r t = 0,1,2,3... Q ( z ) — Z (z) ...(3.1.4.5-3) t dk t t By m u l t i p l y i n g Equations (3.1.4.5-1) and (3.1.4.5-2) by the orthogonal function, i n t e g r a t i n g along the common boundary and adding the two r e s u l t i n g equations one obtains the f i f t h set of equations. CO CO CO CO E - V C T B - y f C - V 7 D -Y 6 V = e ...(3.1.4.5-4) t i-i nt n L i n t n L nt n L i n t n t n=0 n=0 t=0 t=0 Where, N ' 1 1 2 [sinh (k d ) - sinh(k d )] e ...(3.1.4.5-5) o 2 ( d k o ) 2 for t = 1,2,3... JV" 1 / 2 [sinh (k d ) - sinh(k d )] e — — ...(3.1.4.5-6) ' ( d k t ) 2 for t = 0,1,2,3... , 2 2 d a - a a Z ...(3.1.4.5-7) ot , ,, , 2 2 . ot 4dk a (a + a ) t 2 1 f o r n - 1,2,3... and t = 0,1,2,3... a - •.—-— V ' ( a ) n £ ...(3.1.4.5-8) nt n 2 „t t f o r t = 0,1,2,3... d a C = — Z ...(3.1.4.5-9) ot . 2 2 . ot 2dk (a + a ) t 2 1 - 38 -for n = 1,2,3..: and t = 0,1,2,3... 7T n t 2dk W (a ) n t n 2 n t (3.1.4.5-10) fo r n = 0,1,2,3... and t = 0,1,2,3... d - d 7 . = n t dk X (a ) m t n 2 n n t (3.1.4.5-11) fo r n = 0,1,2,3... and t = 0,1,2,3. d - d n t dk Y (a ) m Z n 2 n n t (3.1.4.5-12) for n = 0,1,2,3. V (a,) n 2 I (nsra /d ) K (n?ra /d ) - I (nrca /d ) K (n?ra /d ) 1 2 / 2 1 V z' l x l ' 2 1 Z1 2 I (nrca /d ) K'(njra /d ) - l'(njra /d ) K (nwa /d ) 1 Z1 2 1 V 2 1 l ' 2 1 Z' Z1 (3.1.4.5-13) fo r n = 0,1,2,3. W n(a 2) I (n?ra /d ) K (mra /d ) - I (n?ra /d ) K (n*a /d ) 1 2 / 2 1 2 / 2 1 Z' 2 1 2 / 2 l ' ( n 7 r a /d ) K (n?ra /d ) - I (mra /d ) K' (mra /d ) 1 l ' 2 1 2' 2 1 Z1 2 1 l ' 1 (3.1.4.5-14) X (a ) = 0 2 J (m a ) H (m a ) - J (m a ) H (m a ) 1 0 2 1 0 3 1 0 3 1 0 2 J ( m a ) H ( m a ) - J (m a ) - H ( m a ) 1 0 2 1 0 3 1 0 3 1 0 2 (3.1.4.5-15) for n = 1,2,3... X „ ( a 2 ) = I (m a ) K (m a ) - I (m a ) K (m a ) 1 n 2 1 n 3 1 n 3 1 n 2 I (m a ) K' (m a ) - l'(m a ) K (m a ) 1 V n z' 1 V n Z' 1 n 3 7 1 V n z' (3.1.4.5-16) - 39 -Y ( a ) 0 2 J ( m a ) H ( m a ) - J ( m a ) H ( m a ) 1 0 2 1 ^ 0 2 1 V 0 2 1 0 2 J (m a ) H (m a ) - J (m a ) H (m a ) 1 0 3 1 0 2 1 0 2 1 0 3 . ( 3 . 1 . 4 . 5 - 1 7 ) f o r n = 1 , 2 , 3 . I (m a ) K (m a ) - I (m a ) K (m a ) 1 V n z' I n z' 1 V n z' I n z' l ' ( m a ) K (m a ) - I (m a ) K ' ( m a ) 1 n 3 I n 2 l v n 2 7 I n 3 y ( 3 . 1 . 4 . 5 - 1 8 ) 3 . 1 . 4.6 SOLVING FOR THE COEFFICIENTS OF THE SERIES A t t h i s p o i n t f i v e s y s t e m s o f e q u a t i o n s r e m a i n t o be s o l v e d f o r t he unknown c o e f f i c i e n t s . The e q u a t i o n s a l r e a d y d e t e r m i n e d a r e r e w r i t t e n h e r e f o r c o n v e n i e n c e . CO CO A - [« B - [ ( 3 C = 0 . . . ( 3 . 1 . 4 . 6 - 1 ) n L i t n t L. t n t t = 0 t = 0 CO C - V £ A = b . . . ( 3 . 1 . 4 . 6 - 2 ) n L i t n t n ' t = 0 CO B - V e E = 0 . . . ( 3 . 1 . 4 . 6 - 3 ) n L i n t t t = 0 CO D - y A E = 0 . . . ( 3 . 1 . 4 . 6 - 4 ) n L n t t x ' t = 0 CO CO CO CO E - y c7 B - y f c - y 7 D - y« » = e ...0 . 1.4.6-5) t L i n t n L i n t n L i n t n L n t n t n = 0 n = 0 t = 0 t = 0 I n t h e s e e q u a t i o n s a l l t e rms a r e known, e x c e p t i n g t h e c o e f f i c i e n t s o f t h e s e r i e s A , B , C , D and E , where n = 0 , l , 2 . . . I n s o l v i n g t h i s s y s t e m o f n n n n n e q u a t i o n s i t i s o b v i o u s l y i m p r a c t i c a l t o t a k e t o o many t e rms i n t h e s e r i e s . S a b u n c u a n d C a l i s a l (1984) i n s o l v i n g a d o u b l e c y l i n d e r p r o b l e m t r u n c a t e d t he s e r i e s t o 20 t e r m s and assumed t h e s o l u t i o n t o be c o n v e r g e n t . Howeve r , t h e s e a u t h o r s s t a t e t h a t , i n g e n e r a l , 10 te rms s h o u l d g i v e " s u f f i c i e n t l y g o o d " - 40 -values f o r the hydrodynamic c o e f f i c i e n t s . For t h i s problem a t o t a l of at le a s t 15 terms was considered to ensure a convergent s o l u t i o n . To determine 15 terms f o r each unknown c o e f f i c i e n t series i t i s necessary to solve a system of 75 simultaneous equations. One can write the system of equation i n matrix form as: [A][x] - [B] ...(3.1.4.6-6) Where f o r t h i s problem [A] i s a complex square matrix of order 75, [x] i s a complex vector containing the unknown c o e f f i c i e n t s and [B] i s a vector representing the terms on the r i g h t hand side of the system of equations. 3.1.5 CALCULATION OF THE HYDRODYNAMIC COEFFICIENTS Having now determined the p o t e n t i a l at a l l points i n the f l u i d domain i t i s now poss i b l e to determine the hydrodynamic c o e f f i c i e n t s of the t r i p l e c y l i n d e r i n surge motion. The expression r e l a t i n g added mass c o e f f i c i e n t , damping c o e f f i c i e n t and the f l u i d p o t e n t i a l i s of the following form: F a b l wpW^ pV wpV s^ Where, F = t o t a l hydrodynamic surge force; a ^ = surge added mass c o e f f i c i e n t ; b = surge damping c o e f f i c i e n t ; - 41 -— + i 1 1 - \ JJ *(r,tf,z) ni dS ...(3.1.5-1) p = density of f l u i d medium; V = displaced volume of the cylinder; u> = frequency of motion; V »= v e l o c i t y i n surge d i r e c t i o n . n^ = u n i t normal vector i n surge d i r e c t i o n ; ds = i n t e g r a l over the body surface; s b $(r,9,z) = p o t e n t i a l of f l u i d medium; To use t h i s formulation one must make use of the p o t e n t i a l i n a l l regions. For t h i s problem, therefore, the surface i n t e g r a l i n Equation (3.1.5-1) can be wr i t t e n as: JJs <Pcr,0,z)(-n_)dS = u$ (a ,0,z;t) (-n ) a dz o d dB V d e • i W t 1 \y o "'d (a ,z) (-n )cos(f?) a dz dd i j r j V d Tr e i W t 4> (a ,z) a dz i j J ...(3.1.5-2) ...(3.1.5-3) ...(-3.1.5-4) When wri t t e n without v a r i a b l e subscripts, the i n t e g r a l i n Equation (3.1.5-4) i s of the following form, f or the t r i p l e c y l i n d e r geometry. - 42 -rdu fd2 U<A (a ,z) a dz = d> (a ,z) a dz J d i j j J d 2 1 1 a dz + d , . 2 0 <a .z) a 3 dz ... (3.1.5-5) d 3 uc6 (a ,z) a dz = IN + IN + IN ...(3.1.5-6) J d i j J 1 2 3 ' l Where the f i r s t second and t h i r d i n t e g r a l s on the r i g h t hand side of Equation (3.1.5-5) are designated IN^ , IN 2 a n d IN 3,respectively. Performing the i n t e g r a t i o n from the expressions f or <j>^, 4>2 and <f>^ i n Equations (3.1.3.1-1), (3.1.3.2-1) and (3.1.3.3-1), one obtains the following: 2 2 2 a a 1 a - a IN = B — (d - d ) + i C — — a (d - d ) 1 0 2 2 2 1 2 0 2 2 1 2 1 a + a a + a 2 1 2 1 =0 a d + V — 1 — * - [ B V (a ) + C W (a ) ] [ - sin (nnd /d ) ] ... (3.1.5-7) L Tin 1 n n 1 n n l J L V 2 1 n = 1 H (k a ) N'112 IN = E — - — — a [ sinh (k d ) - sinh (k d ) 1 2 o - . . 2 1 o 3 o 3 J H (k a ) k 1 0 2 0 K (k a ) N'1/Z + y E — - — — a [ sin (k d ) - sin (k d ) ] ] ... (3.1.5-8) i-i n ' 2 n 3 n l n = l K (k a ) k 1 n 2 n IN - a D X (a ) + D Y (a ) — sinh {m (d-d ) 3 3 l 0 0 3 0 0 3 J m 0 3 0 1 co 1 + a y [ D X (a ) + D Y (a ) ] — s i n {m (d-d )} ...(3.1.5-9) 3 Li n n 3 n n 3 m n 3 n " 1 n One can now rewrite Equation (3.1.5-1) i n terms of IN , IN 2 and IN 3 as: - 43 -F a b -— + i — — = - J v d i r e" l W t [ IN + IN + IN ] . . . ( 3 . 1 . 5 - 1 0 ) 3 . 1 . 6 CALCULATION OF SURGE EXCIT ING FORCE F o r v e r t i c a l a x i s y m m e t r i c s h a p e s t h e e x c i t i n g f o r c e c a n be c a l c u l a t e d f r o m t h e damp ing c o e f f i c i e n t u s i n g a s i m p l e a l g e b r a i c e q u a t i o n . T h i s r e l a t i o n s h i p was d e r i v e d b y Wehausen (1971) and i s g i v e n i n e q u a t i o n ( 3 . 1 . 6 - 1 ) . Pg w H Where , F = s u r g e e x c i t i n g f o r c e ; b = s u r g e damp ing 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 medium; g = a c c e l e r a t i o n due t o g r a v i t y ; H = i n c i d e n t wave h e i g h t ; d = w a t e r d e p t h ; w = f r e q u e n c y o f m o t i o n ; k = wave number . 1 + 2 k d s i n h (2kd) ( 3 . 1 . 6 - 1 ) 3 . 1 . 7 CYL3 PROGRAM A c o m p u t e r p r o g r a m named CYL3 was w r i t t e n t o s o l v e t h i s p r o b l e m u s i n g - 44 -t h e m a t c h i n g t e c h n i q u e . T h i s p r o g r a m t a k e s f o r i n p u t i n f o r m a t i o n s u c h as the c y l i n d e r g e o m e t r y , t h e t a n k d e p t h , t h e f r e q u e n c y and t h e number o f t e rms t o c o n s i d e r e d i n t h e i n f i n i t e s e r i e s . T h i s i n f o r m a t i o n i s u s e d t o d e t e r m i n e the p o t e n t i a l s i n r e g i o n s 1 t o 4 . The p r o g r a m u s e s t h e MTS ( M i c h i g a n T e r m i n a l S y s t e m s ) G a u s s i a n E l i m i n a t i o n r o u t i n e CDSOLN t o d e t e r m i n e t h e unknown c o e f f i c i e n t s and t h e n computes t he 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 t h e c y l i n d e r i n s u r g e m o t i o n . 3 . 2 BOUNDARY ELEMENT METHOD Th i ' s s e c t i o n w i l l d e s c r i b e how one c a n a p p l y t h e o r y o f t h e B o u n d a r y E l e m e n t M e t h o d (BEM) t o t h e same p r o b l e m d e s c r i b e d i n S e c t i o n 3 . 1 , t h a t i s , d e t e r m i n i n g t h e 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 f o r a t r i p l e c y l i n d e r a t t he f r e e s u r f a c e . I n a d d i t i o n , t h i s s o l u t i o n method w i l l be a p p l i e d t o d e t e r m i n e t h e e x c i t i n g f o r c e i n d u c e d by i n c i d e n t waves on t h e c y l i n d e r m o d e l s . F o r t h i s t h e o r y t h e same c o o r d i n a t e s y s t e m , i n t r o d u c e d i n S e c t i o n 3 . 1 . 2 and shown i n F i g u r e 8 . 1 w i l l be u s e d . I n a p p l y i n g t h e B o u n d a r y E l e m e n t M e t h o d , i t i s n e c e s s a r y t o d e f i n e a c o n t r o l v o l u m e . F o r a x i s y m m e t r i c s h a p e s i t i s d e s i r a b l e t o d e f i n e t he o u t e r s u r f a c e o f t h e c o n t r o l vo lume t o be c y l i n d r i c a l i n s h a p e . I t i s a t t h i s o u t e r s u r f a c e t h a t 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 w i l l be a p p l i e d . F o r t he p u r p o s e o f t h i s d i s c u s s i o n an o b s e r v a t i o n p o i n t , P , w i l l be d e f i n e d . P may be l o c a t e d e i t h e r i n s i d e o r o u t s i d e t h e c o n t r o l s u r f a c e . I t i s a l s o n e c e s s a r y t o d e f i n e a c o n t r o l p o i n t , Q, l o c a t e d on t he c o n t r o l s u r f a c e . As one w o u l d e x p e c t , 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 w h i c h were i n t r o d u c e d i n S e c t i o n s 3 . 1 . 1 and 3 . 1 . 2 a p p l y t o t h i s t h e o r y as w e l l . - 45 -The p o t e n t i a l on the control surface can be determined by d i s c r e t i z i n g the surface into f i n i t e elements as shown i n Figure 8.7. In the case of axisymmetric shapes these elements are r i n g shaped. The p o t e n t i a l within each element i s determined by so l v i n g a system of simultaneous equations which can be set up by applying the appropriate boundary conditions to the governing equation. D e t a i l s of t h i s procedure are presented i n the following sections. 3.2.1 SOLVING FOR THE POTENTIAL FUNCTION Wehausen (1971) expressed the v e l o c i t y p o t e n t i a l f o r the flow f i e l d i n the neighbourhood of the c y l i n d e r as the sum of three terms. $ = ... (3.2.1-1) I D F 7 Where, $ = the t o t a l p o t e n t i a l of the f l u i d medium; $ i = the p o t e n t i a l of the incoming waves; $ = the p o t e n t i a l of the r e f l e c t e d waves; D $ f = the p o t e n t i a l induced by the motion of the body. The i n t e r a c t i o n between each of the terms i s assumed to be n e g l i g i b l e . The $ and $ p terms are u s e f u l i n c a l c u l a t i n g the e x c i t i n g forces and moments on the body. The d i s t r i b u t i o n of $ f on the body i s used to determine the added mass and damping c o e f f i c i e n t s . The $ term, introduced i n the next section, i s equal to the $ ? term since i n t h i s case the $ j and quantities are assumed to be zero for pure s i n u s o i d a l motion of a body. - 46 -3.2.2 DETERMINATION OF THE MOTION INDUCED POTENTIAL Applying the l i n e a r i z e d boundary conditions to the governing equation one obtains the s o l u t i o n suggested by Brebbia (1978) and Chan (1984). $ ( P ) + rf . 3 G ( P , Q ) $(Q> — ^ dS 3n 3$(Q) 3n G(P,Q) dS . (3.2.2-1) Where, S R n = the t o t a l surface area bounding the con t r o l volume; = an a r b i t r a r y point with p o t e n t i a l , $ ( P ) , located on the con t r o l surface at (x ,y ,z ); p p p = an a r b i t r a r y point with a p o t e n t i a l , $(Q), located on the con t r o l surface (x,y,z); = Green's function; 1 4TTR for 3 dimensional problems ; the distance between P and Q; -/ (x - x ) 2 + (y - y ) 2 + (z - z ) 2 p p p = u n i t normal vector on con t r o l surface p o i n t i n g out of the c o n t r o l volume. For axisymmetric shapes the p o t e n t i a l , $ i s a function of r and z only. The c o n t r o l surface can be d i s c r e t i z e d into r i n g type elements as shown i n Figure 8.7. The p o t e n t i a l value on each of these elements i s assumed to be constant and equal to value c a l c u l a t e d at the center of the element. The t o t a l c o n t r o l surface i s made up of four sections where the d i f f e r e n t boundary conditions are applied. Therefore, the following - 47 -f o r m u l a t i o n a p p l i e s . S = S + S + S + S t o b £ i ( 3 . 2 . 2 - 2 ) Where , S = b o t t o m s u r f a c e ; = body s u r f a c e ; = f r e e s u r f a c e ; = o u t e r s u r f a c e where r a d i a t i o n c o n d i t i o n i s assumed t o e x i s t . A p p l y i n g E q u a t i o n ( 3 . 2 . 2 - 1 ) t o t h e c o n t r o l s u r f a c e s i n d i v i d u a l l y Chan (1984) d e v e l o p e d t h e f o l l o w i n g f o r m u l a t i o n : II 3n r dd dl = $(P) + II. + 11 $ s 3G an- r de < 3G 85 r d9 " 8G CO g ' dG ; an 1 r d6 d r r d * dz . . . ( 3 . 2 . 2 - 3 ) Chan (1984) i n t e g r a t e d E q u a t i o n ( 3 . 2 . 2 - 3 ) i n t e r m s o f 6 and m u l t i p l i e d b y 47T t o o b t a i n t h e f o l l o w i n g : - 48 -dl = s 47T $(P) - 4 s r n + r ( z - z )n r p z ((a - b) /a + b r n + r ( z - z )n r p z 2n E ( 5 , ? ) + ^ -/ a + b 2n ((a - b) /a + b -4 (z-z ) ^ — E<«,f) • (a - b) /a + b •2 r 2 - r 2 - (z-z ) 2 p p / a + b y dr , 2 4w r ((a - b) /a + b g / a + b 4 i k r / a + b > dz W . f ) • dz ...(3.2.2-4) Where, E(J,f) n n complete e l l i p t i c a l function of f i r s t kind, as per Equation (10-9), Appendix F; complete e l l i p t i c a l function of second kind, as per Equation (10-11), Appendix F; 2b «| a+b 2 2 . .2 = r + r + (z - z ) ; p p = 2rr ; p = r component of u n i t normal vector; — z component of u n i t normal vector. Equation (3.2.2-4) i s an i n t e g r a l equation f o r axisymmetric shapes. The surface i n t e g r a t i o n of Equation (3.2.2-4) i s replaced by a l i n e or contour i n t e g r a t i o n . I f one l e t s point P be on element i and point Q on element j one can write an equation for the unknown $^  i n terms of 4^ using Equation - 49 -( 3 . 2 . 2 - 4 ) . S i n c e t h i s i s a c l o s e d s u r f a c e w i t h N r i n g e l e m e n t s one c a n w r i t e N e q u a t i o n s w i t h N unknowns and t h e r e f o r e s o l v e f o r t h e p o t e n t i a l e v e r y w h e r e on t h e c o n t r o l s u r f a c e . 3 . 2 . 3 DETERMINATION OF THE HYDRODYNAMIC COEFFICIENTS H a v i n g s o l v e d f o r t he p o t e n t i a l i n t h e f l u i d medium one c a n now e v a l u 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 . Chan (1984) g i v e s t h e f o l l o w i n g r e l a t i o n f o r t h e 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 : F a b . . $ l I I . . 11 1 f l + i — - i f i m r dl . . . ( 3 . 2 . 3 - 1 ) wpV V J r „ . w o W pV . „ .. „ l s V c o s b 1 Where , F = h y d r o d y n a m i c f o r c e i n s u r g e d i r e c t i o n ; a = added mass c o e f f i c i e n t ; n b = damp ing c o e f f i c i e n t ; u> = f r e q u e n c y o f m o t i o n ; p = d e n s i t y o f f l u i d medium; V = v o l u m e o f f l u i d d i s p l a c e d b y t h e o b j e c t ; V = v e l o c i t y o f o b j e c t i n s u r g e d i r e c t i o n . 3 . 2 . 4 DETERMINATION OF THE SURGE EXCITING FORCE The s u r g e i n d u c e d e x c i t i n g f o r c e i s g i v e n b y Chan (1984) a s : - 50 -F = -2mpe J n A g c o s h k ( z+d ) c o s h k d J ( kp ) + i $ w 1 D 1 d t ( 3 . 2 . 4 - 1 ) Where , F l k J l d = s u r g e i n d u c e d e x c i t i n g f o r c e ; = wave number ; = B e s s e l f u n c t i o n , as p e r E q u a t i o n ( 1 0 - 1 ) , A p p e n d i x F ; = d e p t h o f f l u i d medium; = r e f l e c t e d wave p o t e n t i a l as g i v e n d e t e r m i n e d b y E q u a t i o n ( 3 . 2 . 4 - 2 ) ; wave amp1i t u d e . The r e f l e c t e d wave p o t e n t i a l i s g i v e n b y Chan (1984) a s : 2TT $ (P) D 1 + 1 % < 0 1 r n + r ( z - z ) r r p b / a + b / a + b E ( 5 , - ) y T T b 2 / a + b ( a - b ) E ( 5 , f ) ^ d£ D 0 1 - 4 r n + r ( z - z ) r r P b / a + b a - b 2 2 / a + b E ( 6 , - ) ^ — F ( o , f ) + / a + b / a + b ( a - b ) y d t 7- r $ « b D l / a + b E ( S ^ ) + g 2 z - z -au> P g F ( « , ^ ) ( z - z ) a 2 5 E ( 5 , £ ) / a +b / a +b (a - b ) V dR - 51 -2( 2 r - a ) b / a + b 4 i k r / a + b E ( 5 , - ) / a +b * — F ( 5 , f ) dz J _ w c o s h k d 4 r J ( k r ) n c o s h k ( z+h ) + J ( k r ) n s i n h k ( z + d ) 1 r 1 z / a +b F < 0 - / a + b E ( f i , £ ) d t . . . ( 3 . 2 . 4 - 2 ) By a s i m i l a r p r o c e d u r e t o t h a t i n t r o d u c e d i n S e c t i o n 3 . 2 . 2 one o b t a i n s N e q u a t i o n s w i t h N unknowns w h i c h c a n be s o l v e d s i m u l t a n e o u s l y t o g i v e t h e r e f l e c t e d wave p o t e n t i a l e v e r y w h e r e on t h e c o n t r o l s u r f a c e . 3 . 2 . 5 A l l P r o g r a m The c o m p u t e r p r o g r a m , A l l , o r i g i n a l l y w r i t t e n b y Chan (1984) was r u n on t h e Vax™ 1 1 / 7 5 0 c o m p u t e r i n t h e Depa r tmen t o f M e c h a n i c a l E n g i n e e r i n g a t t he 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 . Some s l i g h t m o d i f i c a t i o n s were made t o t h i s p r o g r a m b y t h i s t h i s a u t h o r t o a l l o w i t t o r u n f o r a l l c l a s s e s o f waves i n c l u d i n g deep w a t e r waves and t h e f o r m a t t i n g o f i n p u t and o u t p u t was changed t o s u i t t h i s a u t h o r . To r u n t h i s p r o g r a m , w h i c h w i l l wo rk f o r any a r b i t r a r y a x i s y m m e t r i c s h a p e , one must c r e a t e an i n p u t f i l e as s p e c i f i e d i n t h e p rog ram d o c u m e n t a t i o n . T h i s f i l e i s t o c o n t a i n t h e c o o r d i n a t e s o f t h e d i s c r e t i z e d c o n t r o l s u r f a c e and t h e f r e q u e n c i e s a t w h i c h one w i s h e s t o d e t e r m i n e t he 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 p r o g r a m , SETUP, was w r i t t e n b y t h i s a u t h o r t o a u t o m a t i c a l l y c r e a t e t h e i n p u t f i l e b y k n o w i n g t h e d r a f t and t h e c y l i n d e r t y p e . - 52 -4. PRESENTATION AND ANALYSIS OF RESULTS The experimental r e s u l t s are compared with the Boundary Element Method and the Matching Technique Formulation. These comparisons are gr a p h i c a l l y i l l u s t r a t e d i n Appendix E. The r e s u l t s presented may be di v i d e d into f i v e separate groups as follows: 1) Sample data p l o t s ; 2) Added mass c o e f f i c i e n t p l o t s from hydrodynamic t e s t s ; 3) Damping c o e f f i c i e n t p l o t s from hydrodynamic t e s t s ; 4) Induced surge force p l o t s from incident wave t e s t s ; 5) Wehausen c a l c u l a t e d damping c o e f f i c i e n t p l o t s from incident wave te s t s ; Within categories 2 through 5 the r e s u l t s may be further categorized into s i n g l e , double and t r i p l e c y l i n d e r t e s t r e s u l t s . A l l experimental r e s u l t s , i n c l u d i n g determination of the hydrodynamic c o e f f i c i e n t s and the incident wave forces, were compared with the r e s u l t s of the Boundary Element Method. The Matching Technique was applied to determine only the hydrodynamic c o e f f i c i e n t s of the double and t r i p l e c y l i n d e r models. C a l i s a l and Sabuncu (1981) have applied t h i s theory to a single c y l i n d e r and complete solutions are g r a p h i c a l l y presented i n the c i t e d reference. Chan (1984) compared the r e s u l t s of the Matching Technique with those of the Boundary Element Method fo r a s i n g l e c y l i n d e r i n f i n i t e water depth and found agreements to be within 10% f o r the peak damping c o e f f i c i e n t and within 6% f o r peak added mass c o e f f i c i e n t . Over the range of frequencies considered, the Matching technique was found to over-predict the hydrodynamic c o e f f i c i e n t s when compared with the r e s u l t s of the Boundary Element Method. A c o m p l e t e l i s t i n g o f a l l t e s t s c o n d u c t e d i s p r o v i d e d i n T a b l e 2 . 2 - 1 . The o n l y e x p e r i m e n t a l l y o b t a i n e d d a t a w h i c h i s n o t p r e s e n t e d i n A p p e n d i x E was f r o m t e s t s w i t h known equ ipmen 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 . One s u c h c a s e o c c u r r e d d u r i n g a s e r i e s o f h y d r o d y n a m i c t e s t s c o n d u c t e d i n O c t o b e r o f 1985 where t h e c y l i n d e r mode l was f o u n d t o have l e a k e d c r e a t i n g a f r e e s u r f a c e e f f e c t i n s i d e t h e c y l i n d e r . A n o t h e r t e s t s e r i e s was n o t a n a l y z e d b e c a u s e t h e d i s p l a c e m e n t 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 . O f t he f o r t y t e s t s e r i e s c o n d u c t e d o n l y f o u r were s c r u b b e d due t o p r o c e d u r a l e r r o r s o r e q u i p m e n t b r e a k d o w n . To a l l o w t h e r e s u l t s t o 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 i v a l e n t p r o p o r t i o n s 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 . The samp le p l o t s o f t h e s i g n a l t r a c e s and s i g n a l s p e c t r u m s a r e , h o w e v e r , p r e s e n t e d w i t h t h e a c t u a l m e a s u r e d u n i t s . The f r e q u e n c y , to, i s n o n - d i m e n s i o n a l i z 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 o f t h e c y l i n d e r , R , and t h e a c c e l e r a t i o n due max 2 t o g r a v i t y , g , as ——IlLif.. xhe added mass c o e f f i c i e n t , a , i s ^ a n o n - d i m e n s i o n a l i z e d as b e f o r e where i t was e x p r e s s e d as — — — . H e r e , p i s t he pv d e n s i t y o f t h e f l u i d medium and V i s t h e vo lume o f t h e d i s p l a c e d f l u i d . The b damp ing c o e f f i c i e n t , b ^ , i s n o n - d i m e n s i o n a l i z e d as ^ , w h i l e t h e s u r g e Fs f o r c e , Fs , i s n o n - d i m e n s i o n a l i z e d as — „ . , where A i s t h e a m p l i t u d e o f t he pVgA i n c i d e n t w a v e . R e s u l t s o b t a i n e d f r om t h e B o u n d a r y E l e m e n t M e t h o d were f o u n d t o o s c i l l a t e , as i l l u s t r a t e d i n F i g u r e s 9 .6 and 9 . 7 . Chan (1984) o b s e r v e d t h i s b e h a v i o u r and c o n c l u d e d t h a t t h i s was r e l a t e d t o t h e s i z e o f t h e s u r f a c e e l e m e n t s . F u r t h e r , C h a n , o b s e r v e d t h a t d e c r e a s i n g t h e s i z e o f t h e s u r f a c e e l e m e n t s s u p p r e s s e d t h i s o s c i l l a t o r y b e h a v i o u r . I n c o n d u c t i n g t h e a n a l y s i s - 54 -t h e c o n t r o l s u r f a c e was d i s c r e t i z e d i n t o e l e m e n t s w h i c h were as s m a l l as p r a c t i c a l l y p o s s i b l e . The r e s u l t i n g s o l u t i o n was t h e n f i l t e r e d t o remove the o s c i l l a t o r y componen t . T h i s new s o l u t i o n i s a n i m p r o v e d i n t e r p r e t a t i o n o f the v a l u e s p r e d i c t e d b y t h e B o u n d a r y E l e m e n t M e t h o d . A l l s u b s e q u e n t p l o t s p o r t r a y t h e smoo thed s o l u t i o n f r o m t h e B o u n d a r y E l e m e n t Me thod p r o g r a m , A l l . 4 . 1 SAMPLE DATA PLOTS F i g u r e s 9 .1 t h r o u g h t o 9 . 4 c o n t a i n samp le p l o t s f r o m t h e t e s t SDF4 . The i n d e p e n d e n t v a r i a b l e s s e l e c t e d f o r t h i s t e s t a r e as f o l l o w s : C y l i n d e r t y p e : S i n g l e D r a f t : 211 mm A m p l i t u d e o f M o t i o n : 35 mm M a s s : 2 6 . 9 Kg N o m i n a l F r e q u e n c y S e t t i n g : 1 H z . F i g u r e 9 . 1 shows a f i l t e r e d t r a c e o f t h e f i r s t 10 s e c o n d s o f d a t a s a m p l e d f r o m 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 . S a m p l i n g was c o n d u c t e d a t t he n o m i n a l r a t e o f 25 s a m p l e s p e r c y c l e and t h e d u r a t i o n o f a s i n g l e t e s t was a p p r o x i m a t e l y 25 c y c l e s . T h i s s i g n a l , w i t h o u t e x c e p t i o n , c o n t a i n e d t h e l e a s t amount o f n o i s e and was u s e d as t h e c h a n n e 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 was d e t e r m i n e d . The f i l t e r i n g p r o c e s s h a d t h e e f f e c t o f s l i g h t l y r e d u c i n g t h e a m p l i t u d e s o f t h e p e a k s . F i g u r e 9 . 2 c o n t a i n s a p l o t o f t he s p e c t r u m as d e t e r m i n e d b y t he s u b r o u t i n e , F F T . One c a n s e e 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 . T h i s p l o t i n d i c a t e s a d i s p l a c e m e n t a m p l i t u d e o f 3 4 . 1 mm a t a f r e q u e n c y o f .969 H z . The - 55 -a c t u a l 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 i s known t o be 35 mm ± . 1 % . The d i s c r e p a n c y b e t w e e n t h e d i s p l a c e m e n t a m p l i t u d e s i s c a u s e d m a i n l y b y a m p l i t u d e r e d u c t i o n d u r i n g f i l t e r i n g b u t o t h e r c o n t r i b u t i n g e f f e c t s i n c l u d e c a l i b r a t i o n a c c u r a c y and t h e i n a b i l i t y t o measure t h e a b s o l u t e peak o f a s i g n a l when s a m p l e d p e r i o d i c a l l y . F o r s u b s e q u e n t p r o c e s s 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 was c o r r e c t e d t o i t s known v a l u e . T h i s c o r r e c t i o n was r a r e l y g r e a t e r t h a n 1 mm. F i g u r e s 9 . 3 and 9 . 4 r e s p e c t i v e l y show u n f i l t e r e d and f i l t e r e d t r a c e s f r o m t h e s u r g e c h a n n e l o f t h e dynamometer f o r t e s t SDF4 . The u n f i l t e r e d t r a c e i s t h e i n p u t d a t a f o r t he s u b r o u t i n e , F I L T E R . The o u t p u t f r o m F I L T E R a p p e a r s as t h e d a t a p l o t t e d i n F i g u r e 9 . 4 . The u n f i l t e r e d d a t a , shown i n F i g u r e 9 . 3 , c o n t a i n s a c o n s i d e r a b l e amount o f n o i s e . T h i s i s n o t s u r p r i s i n g s i n c e a 2200 N dynamometer i s b e i n g u s e d t o measure a 60 N f o r c e . F o r t e s t s where l a r g e r s u r g e f o r c e s were m e a s u r e d t h e amount o f n o i s e i n t h e s i g n a l was o b s e r v e d t o be g r e a t l y r e d u c e d . C o n v e r s e l y , f o r t e s t s c o n d u c t e d a t l ow f r e q u e n c i e s ( i . e . 0 . 2 5 Hz and l e s s ) t h e n o i s e t o s i g n a l r a t i o was so h i g h t h a t t h e d a t a o b t a i n e d d u r i n g t h e s e t e s t s s h o u l d be c o n s i d e r e d u n r e l i a b l e . F o r t h e samp le p l o t o f F i g u r e 9 . 3 , t h e s u b r o u t i n e , F I L T E R , s u c c e s s f u l l y removed t h e n o i s e f r o m t h e raw d a t a and t h e r e s u l t i n g s p e c t r u m , shown i n F i g u r e 9 . 5 , a p p e a r s as e x p e c t e d w i t h v i r t u a l l y a l l t h e e n e r g y c o n c e n t r a t e d a t one f r e q u e n c y . The p l o t o f F i g u r e 9 . 5 shows a 5 7 . 1 N s u r g e f o r c e o c c u r r i n g a t a f r e q u e n c y o f 965 H z . F i g u r e s 9 . 6 and 9 .7 show t h e unsmoo thed o u t p u t f r o m t h e BEM and MT t h e o r i e s as d i s c u s s e d p r e v i o u s l y . - 56 -4 . 2 HYDRODYNAMIC TEST RESULTS 4 . 2 . 1 ADDED MASS COEFFICIENTS 4 . 2 . 1 . 1 SINGLE CYLINDER The s i n g l e c y l i n d e r e x p e r i m e n t a l r e s u l t s f o r added mass c o e f f i c i e n t s , shown i n F i g u r e s 9 . 8 and 9 . 9 , a g r e e c l o s e l y w i t h t h e r e s u l t s o b t a i n e d by t he BEM. E x p e r i m e n t s were p e r f o r m e d f o r two s i m i l a r d r a f t s : 211 mm and 218 mm, a t t h r e e d i f f e r e n t a m p l i t u d e s : 10 mm, 25 mm and 35 mm. The 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 211 mm d r a f t g e n e r a l l y u n d e r e s t i m a t e t h e t h e o r y e x c e p t i n t h e v i c i n i t y o f peak v a l u e where t h e added mass was f o u n d t o be a b o u t 7% g r e a t e r t h a n t h a t p r e d i c t e d b y t h e t h e o r y . The e x p e r i m e n t a l r e s u l t s f r o m t h e 218 mm d r a f t f o l l o w a s i m i l a r t r e n d t o t h e r e s u l t s a t t h e 211 mm d r a f t . A t h i g h e r f r e q u e n c i e s t h e r e s u l t s f r o m the 218 mm d r a f t t e s t more c l o s e l y ma tch t h o s e d e t e r m i n e d b y t h e BEM. The peak v a l u e s were o b s e r v e d t o be g r e a t e r t h a n t h e t h o s e f o u n d b y t h e t h e o r y and were a l s o o b s e r v e d t o be g r e a t e r t h a n t h o s e f o u n d b y 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 218 mm d r a f t . A c o n s i s t e n t t r e n d o b s e r v e d f r o m b o t h p l o t s was a r e d u c t i o n i n t h e added mass c o e f f i c i e n t f o r i n c r e a s i n g a m p l i t u d e s o f m o t i o n . On t h e b a s i s o f t h e s e p l o t s i t may be c o n c l u d e d t h a t t h e l i n e a r BEM t h e o r y a p p l i e s w e l l t o t h e s i n g l e c y l i n d e r geome t r y f o r a m p l i t u d e s o f m o t i o n up t o 9% o f t h e d i a m e t e r . - 57 -4 . 2 . 1 . 2 DOUBLE CYLINDER The D o u b l e c y l i n d e r t e s t r e s u l t s a r e c o n t a i n e d i n F i g u r e s 9 . 1 0 and 9 . 1 1 . T e s t s were p e r f o r m e d a t t h r e e d r a f t s : 171 mm, 236 mm and 247 mm, b u t b e c a u s e m u t u a l l y e x c l u s i v e a m p l i t u d e s o f m o t i o n were t e s t e d a t t h e n e a r e q u a l d r a f t s o f 236 mm and 247 mm, t h e s e t e s t r e s u l t s were p l o t t e d t o g e t h e r . I n g e n e r a l , t h e r e s u l t s f r o m b o t h p l o t s i n d i c a t e a good c o r r e l a t i o n b e t w e e n t h e BEM and MT f o r m u l a t i o n . When compared w i t h t h e MT t h e o r y a t t he 171 mm d r a f t , h o w e v e r , t h e BEM t h e o r y was o b s e r v e d t o h a v e a n a r r o w e r peak w h i c h s l i g h t l y o v e r e s t i m a t e d t h e peak v a l u e o f t h e a d d e d m a s s . T h i s c a n be a t t r i b u t e d t o t h e e r r o r i n s m o o t h i n g t h e BEM t h e o r y r e s u l t s s i n c e t h i s p a r t i c u l a r l i n e was smoothed u s i n g f r o m f e w e r a b s c i s s a p o i n t s . F o r t h i s r e a s o n , g r e a t e r c o n f i d e n c e s h o u l d be p l a c e d i n t h e MT t h e o r y f o r t h i s p l o t . F o r t h e s e c o n d g r a p h t h e ag reemen t was o b s e r v e d t o be much g r e a t e r . The e x p e r i m e n t a l r e s u l t s do n o t c l o s e l y f o l l o w t h e d a t a f o r e i t h e r p l o t . A t h i g h e r f r e q u e n c i e s t he added mass c o e f f i c i e n t was c o n t i n u a l l y i n c r e a s i n g , r a t h e r t h a n t e n d i n g t o a s t e a d y v a l u e as b o t h t h e o r i e s p r e d i c t . A t l o w e r f r e q u e n c i e s f o r t h e 171 mm d r a f t t h e d a t a u n d e r e s t i m a t e s t h e t h e o r i e s w i t h a p e a k v a l u e 30% l e s s t h a n t h a t p r e d i c t e d b y t h e MT. The d a t a d o e s , h o w e v e r , f o l l o w t h e c o r r e c t t r e n 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 up t o 2 . 0 . F o r the 236 mm-247 mm s t e p d r a f t , t h e d a t a i s s c a t t e r e d on b o t h s i d e s o f t he t h e o r e t i c a l p r e d i c t i o n s w i t h no c o n s i s t e n t t r e n d . - 58 -4 . 2 . 1 . 3 T R I P L E CYLINDER The t r i p l e c y l i n d e r was t e s t e d most e x t e n s i v e l y , s i n c e i t was t h e most c o m p l e x o f t h e m o d e l s s t u d i e d i t was o f t h e g r e a t e s t i n t e r e s t . T e s t s were c o n d u c t e d a t s i x d i f f e r e n t s t e p d r a f t s , r a n g i n g b e t w e e n 100 mm and 420 mm, a n d a t l e a s t two a m p l i t u d e s o f m o t i o n b e t w e e n 10 mm and 25 mm. T h e s e r e s u l t s a r e c o n t a i n e d i n F i g u r e s 9 . 1 2 t o 9 . 1 7 . A g a i n f o r e a c h o f t h e s e p l o t s good ag reemen t was o b s e r v e d b e t w e e n t he two t h e o r i e s p a r t i c u l a r l y f o r t h e d e e p e r d r a f t s . D i s c r e p a n c i e s a t t he s h a l l o w e r d r a f t s may be a t t r i b u t e d t o t h e i n a b i l i t y t o smooth t h e d a t a where t h e f r e q u e n c y o f a b s c i s s a p o i n t s i s n e a r t h e f r e q u e n c y o s c i l l a t i o n . O t h e r p l o t s c o n t a i n e d more c l o s e l y s p a c e d a b s c i s s a p o i n t s f o r t h e BEM. F o r t he s h a l l o w e r d r a f t g r e a t e r r e l i a b i l i t y s h o u l d be a s s o c i a t e d w i t h t h e MT r a t h e r t h a n t h e BEM r e s u l t s . A g a i n f o r e a c h o f t h e s e t e s t s p o o r c o r r e l a t i o n was o b s e r v e d b e t w e e n t he 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 i e s . Above a n o n - d i m e n s i o n a l f r e q u e n c y o f 1 .0 t h e e x p e r i m e n t a l l y d e t e r m i n e d added mass c o e f f i c i e n t s t e n d e d t o i n c r e a s i n g l y g r e a t e r v a l u e s w h i l e t h e v a l u e s d e t e r m i n e d t h e o r e t i c a l l y were o b s e r v e d t o r e a c h a s t e a d y s t a t e v a l u e i n n o n - d i m e n s i o n a l u n i t s o f b e t w e e n 0 . 6 and 0 . 7 . F o r t h e l o w e r r a n g e o f f r e q u e n c i e s s t u d i e d t h e d a t a f o l l o w e d t h e same t r e n d as t h e t h e o r y b u t was o b s e r v e d t o be u n d e r e s t i m a t e t h e p e a k v a l u e s o f t h e t h e o r e t i c a l r e s u l t s b y amounts up t o 30%. I n a l l c a s e s , e x c l u d i n g t h e r e s u l t s o b t a i n e d f r o m t e s t s c o n d u c t e d a t t h e 410 mm s t e p d r a f t , t h e p e a k c o e f f i c i e n t p r e d i c t e d b y t h e t h e o r y c o r r e s p o n d e d t o a l o c a l p e a k i n t h e e x p e r i m e n t a l l y d e t e r m i n e d c o e f f i c i e n t s . A l s o t h e a m p l i t u d e o f t he p e a k , m e a s u r e d w i t h r e s p e c t t o t h e a b s o l u t e l o w e s t a d d e d mass c o e f f i c i e n t - 59 -e n c o u n t e r e d , was o b s e r v e d t o c o r r e l a t e q u i t e w e l l 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 . T h a t i s , f o r s h a l l o w d r a f t s a l a r g e a m p l i t u d e p e a k was o b s e r v e d . As t h e d r a f t i n c r e a s e s t h e peak was o b s e r v e d t o f l a t t e n o u t somewhat . T h i s was o b s e r v e d b y b o t h t h e e x p e r i m e n t a l and t h e t h e o r e t i c a l r e s u l t s . No c o n s i s t e n t t r e n d was n o t i c e d f o r t he 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 . I t was g e n e r a l l y o b s e r v e d t h a t i n c r e a s e d a m p l i t u d e s o f m o t i o n p r o d u c e d m a r g i n a l l y g r e a t e r v a l u e s f o r t h e added mass c o e f f i c i e n t s a t a g i v e n f r e q u e n c y . T h e r e were e x c e p t i o n s h o w e v e r , most n o t a b l y f o r t h e t e s t done a t t h e s h a l l o w e s t s t e p o f 100 mm. The- t e s t s c o n d u c t e d a t t h e 410 mm d r a f t p r o d u c e d some r a t h e r u n t y p i c a l r e s u l t s . T h e s e r e s u l t s do n o t f o l l o w t h e c o n s i s t e n t t r e n d s y i e l d e d f r o m t e s t s a t o t h e r d r a f t s . A g e n e r a l d i s c u s s i o n o f d i s c r e p a n c i e s b e t w e e n t he e x p e r i m e n t s and t h e r e s u l t i s p r o v i d e d i n S e c t i o n 4 . 4 . H o w e v e r , a t t h i s p o i n t i t i s p e r t i n e n t t o make t h e f o l l o w i n g comments. I n t h e c o u r s e o f c o n d u c t i n g t h e s e e x p e r i m e n t s i t was o b s e r v e d t h a t f o r t h e l a r g e r c y l i n d e r mode ls ( i . e . t h e d o u b l e and t r i p l e m o d e l s ) 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 1 .5 H z , c o r r e s p o n d i n g t o a n o n - d i m e n s i o n a l f r e q u e n c y o f 1 . 7 5 , t h e c y l i n d e r was o b s e r v e d t o s w i n g w i t h r e s p e c t t o t h e m o t i o n g e n e r a t o r . T h i s was most e v i d e n t when t h e c y l i n d e r was a t d e e p e r d r a f t s . The c y l i n d e r mode l was b a l l a s t e d t o p r o v i d e h y d r o s t a t i c s t a b i l i t y d u r i n g i n s t a l l a t i o n . T h i s b a l l a s t c o n c e n t r a t e d t h e m a j o r i t y o f t h e mass a t t h e b a s e o f t h e c y l i n d e r , w h i l e t h e i n d u c e d s u r g e m o t i o n was a p p l i e d t h r o u g h t h e t o p o f t h e c y l i n d e r m o d e l . When t h e s e f a c t o r s were c o m b i n e d w i t h t he e l a s t i c i t y o f t he s y s t e m a pendu lum e f f e c t was o b s e r v e d . T h i s 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 o r was u n d e s i r a b l e . I n some c a s e s t h i s e f f e c t was g r e a t enough t o p e r m a n e n t l y d e f o r m t h e a d a p t e r b l o c k p o r t i o n o f t h e f a s t e n i n g . Once t h i s was i d e n t i f i e d - 60 -as a p o s s i b l e p r o b l e m e f f o r t s were made t o s t r e n g t h e n t h e l i n k a g e . R i g i d c o l l a r s r e p l a c e d t h e t h r e a d e d r o d s and a t o t a l o f s i x t e s t s e r i e s were c o n d u c t e d o v e r t h r e e d r a f t s on t h e t r i p l e c y l i n d e r mode l i n l a t e A p r i l o f 1 9 8 6 . T h e s e t e s t s c o r r e s p o n d t o t h e f o l l o w i n g s t e p d r a f t s : 218 mm, 322 mm and 420 mm. A l l t h e s e t e s t s p r o d u c e d s i m i l a r d a t a t r e n d s t o t h o s e o b s e r v e d i n e a r l i e r t e s t s f o r s h a l l o w d r a f t s . The r e s u l t s f o r t h e 410 mm d r a f t a r e assumed t o be e r r o n e o u s b e c a u s e o f t h i s so c a l l e d pendu lum e f f e c t . T h i s e f f e c t may a l s o be r e s p o n s i b l e f o r t h e t e n d e n c y o f t h e d a t a t o ' b l o w u p ' a t h i g h e r f r e q u e n c i e s b u t o t h e r f a c t o r s d i s c u s s e d i n S e c t i o n 4 . 4 w o u l d a l s o c o n t r i b u t e t o t h i s t r e n d . 4 . 2 . 2 DAMPING COEFFICIENTS 4 . 2 . 2 . 1 S INGLE CYLINDER The s i n g l e c y l i n d e r t e s t r e s u l t s f o r damping c o e f f i c i e n t s a r e p r e s e n t e d i n F i g u r e s 9 . 1 8 and 9 . 1 9 . H e r e t h e e x p e r i m e n t a l d a t a was f o u n d t o c l o s e l y f o l l o w t h e p r e d i c t e d r e s u l t s o f t h e BEM. T h i s t h e o r y , i n g e n e r a l s l i g h t l y u n d e r - p r e d i c t e d t h e e x p e r i m e n t a l l y d e t e r m i n e d v a l u e s . The h i g h v a l u e s s e e n a t 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 c a n be a t t r i b u t e d t o t h e i n a b i l i t y o f t h e dynamometer t o r e s o l v e f o r c e s o f t h e l o w m a g n i t u d e b e i n g m e a s u r e d . I n mos t c a s e s t h e f o r c e i n p h a s e w i t h t h e v e l o c i t y was l e s s t h a n t h e f o r c 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 b y a b o u t one o r d e r o f m a g n i t u d e . No n o t i c e a b l e change i n t h e d a t a t r e n d was o b s e r v e d w i t h t h e d a t a when t h e a m p l i t u d e o f m o t i o n was v a r i e d . As was t h e c a s e f o r t h e added mass c o e f f i c i e n t s , t h e r e s u l t s f r o m the - 61 -218 mm d r a f t t e s t s o v e r - p r e d i c t e d t h e t h e o r e t i c a l l y d e t e r m i n e d peak v a l u e c o e f f i c i e n t s . A t h i g h e r f r e q u e n c i e s , h o w e v e r , t h i s t r e n d was r e v e r s e d . 4 . 2 . 2 . 2 DOUBLE CYLINDER The damp ing c o e f f i c i e n t r e s u l t s f r o m t h e d o u b l e c y l i n d e r t e s t s i s p r e s e n t e d i n F i g u r e s 9 . 2 0 and 9 . 2 1 . The MT and BEM t h e o r i e s do c o r r e l a t e v e r y c l o s e l y . F o r t h e 247 mm s t e p d r a f t t h e c u r v e s a p p e a r t o c o i n c i d e . F o r 171 mm s t e p d r a f t t h e MT s l i g h t l y u n d e r - p r e d i c t s t h e p e a k v a l u e o f t h e damping c o e f f i c i e n t when compared w i t h t h e BEM. The e x p e r i m e n t a l r e s u l t s a l s o c l o s e l y f o l l o w t h e t r e n d p r e d i c t e d b y t h e two t h e o r i e s . A t t h e s h a l l o w e s t d r a f t t he d a t a i s s c a t t e r e d on b o t h s i d e s o f 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 h i l e a t t he d e e p e s t d r a f t t h e t h e o r y c l e a r l y u n d e r - p r e d i c t s t h e r e s u l t s o f t he e x p e r i m e n t s . A l s o , t h e peak damping c o e f f i c i e n t o c c u r s a t a s l i g h t l y g r e a t e r f r e q u e n c y t h a n t h e t h e o r y p r e d i c t s . S i n c e t he e x p e r i m e n t s were c o n d u c t e d a t d i s c r e t e f r e q u e n c y s e t t i n g s i t was n o t p o s s i b l e t o p r e c i s e l y d e t e r m i n e t he amount o f t h i s d i s c r e p a n c y . The damping c o e f f i c i e n t was g e n e r a l l y o b s e r v e d t o be l e s s f o r i n c r e a s i n g a m p l i t u d e s o f m o t i o n and f o r d e e p e r d r a f t s . 4 . 2 . 2 . 3 T R I P L E CYLINDER The t r i p l e c y l i n d e r t e s t r e s u l t s f o r damping c o e f f i c i e n t s a r e p r e s e n t e d i n F i g u r e s 9 . 2 2 t o 9 . 2 7 . F o r t h e s e p l o t s good ag reemen t was o b s e r v e d be tween t h e BEM and t h e MT t h e o r i e s . F o r t h e two s h a l l o w e s t d r a f t s t h e MT u n d e r - p r e d i c t e d t h e damping c o e f f i c i e n t s w i t h r e s p e c t t o t h e BEM. - 62 -The e x p e r i m e n t a l d a t a was f o u n d t o f o l l o w t h e t r e n d s p r e d i c t e d by the t h e o r y b u t was s c a t t e r e d on b o t h s i d e s t h e t h e o r e t i c a l c u r v e s . The d a t a o b t a i n e d f r o m t e s t s c o n d u c t e d a t t h e 410 mm s t e p d r a f t was a g a i n o b s e r v e d t o be u n t y p i c a l . I t d i d n o t f o l l o w t h e t r e n d s o f o t h e r d a t a o r t h e t h e o r y . An h y p o t h e s i s e x p l a i n i n g t h i s b e h a v i o u r i s p r o p o s e d i n S e c t i o n 4 . 2 . 1 . 3 . The 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 o f 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 , as d i s c u s s e d p r e v i o u s l y . F o r t h e d e e p e r s t e p d r a f t s , 322 and 420 mm, t h e t h e o r y u n d e r - p r e d i c t s t h e e x p e r i m e n t a l r e s u l t s b y as much as 50%. T h i s i s l i k e l y due t o t he h y d r o e l a s t i c e f f e c t d e s c r i b e d i n S e c t i o n 4 . 2 . 1 . 3 . W i t h some o f t h e e x p e r i m e n t a l r e s u l t s s p o r a d i c a l l y h i g h v a l u e s f o r t he damp ing c o e f f i c i e n t s were o b s e r v e d a t h i g h e r f r e q u e n c i e s . T h i s may i n p a r t be due t o e x p e r i m e n t a l e r r o r b u t i t s c o n s i s t e n c y o f a p p e a r a n c e w o u l d i n d i c a t e t h a t some p h y s i c a l phenomena n o t b e i n g m o d e l e d b y t h e t h e o r y a r e o c c u r r i n g . I n c o n d u c t i n g t h e s e t e s t s 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 o f a b o u t 2 . 2 5 t he w a l l e f f e c t , v i s i b l e as h i g h f r e q u e n c y s t a n d i n g waves b e t w e e n t h e c y l i n d e r and t h e t a n k w a l l , was most e v i d e n t . T h i s b e h a v i o u r i s c l e a r l y v i s i b l e i n F i g u r e 7 . 1 1 . T h i s e f f e c t w o u l d m i g h t e x p l a i n t h e s e a n o m a l i e s . • 4 . 3 INCIDENT WAVE TEST RESULTS One common p r o b l e m w i t h t h e i n c i d e n t wave t e s t s was t h e i n a b i l i t y o f t he dynamometer t o a c c u r a t e l y r e s o l v e t h e v e r y l o w m a g n i t u d e o f t h e i n d u c e d - 63 -e x c i t i n g f o r c e . I n a b s o l u t e te rms t h e maximum e x c i t i n g f o r c e measu red a t any f r e q u e n c y was l e s s t h a n 60 N w h i c h i s l e s s t h a n 3% o f t h e f u l l s c a l e r ange o f t h e dynamomete r . One w o u l d t h e r e f o r e e x p e c t t h e r e s u l t s f r o m t h e s e t e s t s t o be q u i t e s c a t t e r e d . I n r e c o n d u c t i n g s i m i l a r e x p e r i m e n t s i t i s i m p e r a t i v e t h a t a l o w r a n g e dynamometer be u s e d t o o b t a i n more r e l i a b l e r e s u l t s . The s u b s e q u e n t c o n c l u s i o n s and o b s e r v a t i o n s w i l l be made w i t h t h e i m p l i e d u n d e r s t a n d i n g t h a t t h e dynamometer i s o p e r a t i n g n e a r t h e l i m i t s o f i t s r e s o l u t i o n . F o r t u n a t e l y t o o f f s e t t h i s d e t r a c t i o n t h e s i g n a l was r e l a t i v e l y f r e e o f n o i s e b e c a u s e o f t h e smoo thness o f t h e wave a c t i o n . F o r t h e h y d r o d y n a m i c t e s t s , t h e m e c h a n i c a l m o t i o n , i n d u c e d v i b r a t i o n s c r e a t i n g more n o i s e i n t he s i g n a l . D u r i n g t h e s e t e s t s t h e wave a m p l i t u d e was k e p t s m a l l so t h a t t h e l i n e a r wave t h e o r y c o u l d be assumed t o a p p l y . 4 . 3 . 1 WAVE INDUCED EXCITING FORCE I n t h e s u b s e q u e n t l y i n t r o d u c e d t e s t s t h e wave i n d u c e d e x c i t i n g f o r c e i s compared w i t h p r e d i c t i o n s f r o m t h e BEM t h e o r y . 4 . 3 . 1 . 1 S INGLE CYLINDER F i g u r e 9 . 2 8 i s a c o m b i n e d p l o t o f t he a l l i n c i d e n t wave t e s t s f o r t he 211 mm and 218 mm d r a f t s . The e x p e r i m e n t a l r e s u l t s d e s i g n a t e d as 218X c o r r e s p o n d s t o t h e f i r s t t e s t c o n d u c t e d where t h e c y l i n d e r f i l l e d w i t h w a t e r . - 64 -The f r e e s u r f a c e w i t h i n t h e c y l i n d e r w o u l d e f f e c t t h e d a t a o n l y i f t he c y l i n d e r moved . F o r t h e s e t e s t s t h e c y l i n d e r was s t a t i o n a r y . The r e s u l t s i n d i c a t e t h a t t h e e x p e r i m e n t s and t h e o r y f o l l o w s i m i l a r t r e n d s w i t h p e a k v a l u e s o b s e r v e d t o o c c u r a t t h e same f r e q u e n c i e s . The t h e o r y u n d e r - p r e d i c t s t h e peak e x c i t i n g f o r c e b y a b o u t 30%. A t o t h e r f r e q u e n c i e s t he 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 p r e d i c t i o n s more c l o s e l y c o n c u r . 4 . 3 . 2 . 3 DOUBLE CYLINDER The d o u b l e c y l i n d e r t e s t r e s u l t s a r e c o n t a i n e d i n F i g u r e s 9 . 2 9 t o 9 . 3 1 . These r e s u l t s e x h i b i t s i m i l a r b e h a v i o r t o t h e t h o s e f o u n d i n t h e s i n g l e c y l i n d e r t e s t s . They f o l l o w t h e same t r e n d as t h e t h e o r y b u t t h e t h e o r y u n d e r - p r e d i c t s t h e peak v a l u e s b y a b o u t 30%. The p e a k s were o b s e r v e d t o o c c u r a t t h e same f r e q u e n c i e s i n b o t h t h e e x p e r i m e n t s and t h e t h e o r y . 4 . 3 . 2 . 2 T R I P L E CYLINDER The t r i p l e c y l i n d e r e x c i t i n g f o r c e t e s t r e s u l t s a r e p r e s e n t e d i n F i g u r e s 9 . 3 2 t o 9 . 3 4 . These r e s u l t f o l l o w t h e BEM t h e o r y more c l o s e l y t h a n t he r e s u l t s f r o m t h e d o u b l e and s i n g l e c y l i n d e r t e s t . T h i s may be due i n p a r t t o t h e i n c r e a s e d a b i l i t y o f t h e dynamometer t o d i s c e r n t h e l a r g e r f o r c e s a s s o c i a t e d w i t h t h e l a r g e r c y l i n d e r . The f r e q u e n c y a t w h i c h t h e peak v a l u e s o c c u r do n o t c o n c u r e x a c t l y w i t h t h e t h e o r y . A t t h e s h a l l o w e s t d r a f t t he t h e o r y p r e d i c t s t h e peak t o o c c u r a t f r e q u e n c y g r e a t e r t h a n t h a t o b s e r v e d by e x p e r i m e n t . A t t h e d e e p e s t d r a f t t h e t r e n d i s r e v e r s e d and t h e t h e o r y - 65 -p r e d i c t s a l o w e r peak f r e q u e n c y t h a n t h a t d e t e r m i n e d e x p e r i m e n t a l l y . A t t h e s h a l l o w e s t d r a f t t h e t h e o r y u n d e r - p r e d i c t s t h e e x p e r i m e n t a l l y d e t e r m i n e d p e a k e x c i t i n g f o r c e b y 15%. F o r i n c r e a s i n g d r a f t s t h i s d i s c r e p a n c y i n c r e a s e s . 4 . 3 . 2 INDIRECTLY DETERMINED DAMPING COEFFICIENT U s i n g t h e Wehausen f o r m u l a t i o n d e s c r i b e d i n S e c t i o n 3 . 1 . 6 , i t was p o s s i b l e f r o m a measurement o f t h e i n c i d e n t wave h e i g h t and t h e wave i n d u c e d e x c i t i n g t o i n d i r e c t l y d e t e r m i n e t h e damping c o e f f i c i e n t o f a b o d y . T h i s i n d i r e c t l y d e t e r m i n e d damping c o e f f i c i e n t w i l l be s u b s e q u e n t l y r e f e r r e d t o as t h e Wehausen damp ing c o e f f i c i e n t . The r e s u l t s d i s c u s s e d i n t h i s s e c t i o n compare t h e damp ing 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 BEM a n d MT t h e o r i e s w i t h t h e v a l u e s c a l c u l a t e d b y t h e Wehausen f o r m u l a t i o n f r o m t h e 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 p u t s o f i n c i d e n t wave h e i g h t and wave i n d u c e d s u r g e f o r c e . 4 . 3 . 2 . 1 S INGLE CYLINDER The s i n g l e c y l i n d e r r e s u l t s a r e p r e s e n t e d i n F i g u r e s 9 . 3 5 and 9 . 3 6 . T h e s e r e s u l t compare t h e Wehausen damping c o e f f i c i e n t w i t h t h e BEM t h e o r y . 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 l e s s t h a n 0 . 5 t h e r e s u l t s show l i t t l e s c a t t e r and c l o s e l y c o r r e s p o n d t o t h e t h e o r e t i c a l r e s u l t s . Above t h i s f r e q u e n c y t h e r e s u l t show a h i g h e r d e g r e e o f s c a t t e r and do n o t c o r r e l a t e 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 . A t t he peak v a l u e t h e t h e o r y u n d e r - p r e d i c t s t h e e x p e r i m e n t a l l y d e t e r m i n e d r e s u l t s b y 50%. - 66 -4 . 3 . 2 . 2 DOUBLE CYLINDER The d o u b l e c y l i n d e r t e s t r e s u l t s a r e compared w i t h b o t h t h e MT and BEM t h e o r i e s . The p l o t t e d r e s u l t s a r e f o u n d i n F i g u r e s 9 . 3 7 t o 9 . 3 9 . These r e s u l t s a l s o d e m o n s t r a t e good c o r r e l a t i o n 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 t l o w f r e q u e n c i e s . A t h i g h e r v a l u e s t he d a t a shows a h i g h e r d e g r e e o f s c a t t e r and i t was o b s e r v e d t h a t b o t h t h e o r i e s u n d e r - p r e d i c t t h e e x p e r i m e n t a l l y d e t e r m i n e d peak v a l u e s b y 25 t o 60%. The t r e n d s o b s e r v e d were c o n s i s t e n t w i t h t h e t h e o r i e s . The t e s t s c o n d u c t e d a t t h e s h a l l o w e s t s t e p d r a f t o f 171 mm d e m o n s t r a t e d t h e b e s t c o r r e l a t i o n w i t h t h e o r y . 4 . 3 . 2 . 3 T R I P L E CYLINDER The t r i p l e c y l i n d e r t e s t r e s u l t s , shown i n F i g u r e s 9 . 4 0 t o 9 . 4 2 , show somewhat i m p r o v e d ag reemen t w i t h t h e two t h e o r i e s . F o r t e s t s c o n d u c t e d a t t h e s h a l l o w e s t d r a f t t h e t h e o r e t i c a l l y d e t e r m i n e r e s u l t s show s t r o n g c o r r e l a t i 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 . 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 1 .0 t h e r e was o b s e r v e d a g r e a t e r t e n d e n c y f o r t h e e x p e r i m e n t a l d a t a t o f a l l b e l o w t h e p r e d i c t i o n s o f t h e t h e o r i e s . F o r t h e deep d r a f t t e s t s t h e t h e o r y c o n s i s t e n t l y u n d e r e s t i m a t e d t h e e x p e r i m e n t a l r e s u l t s . 4 . 4 ANALYSIS OF RESULTS H a v i n g p r e s e n t e d and d i s c u s s e d a t l e n g t h t h e r e s u l t s o b t a i n e d i n t h i s r e s e a r c h i t i s now p o s s i b l e t o make some f u r t h e r g e n e r a l comments on - 67 -d i s c r e p a n c i e s b e t w e e n t he t h e o r e t i c a l r e s u l t s and t h e e x p e r i m e n t a l r e s u l t s . When t h e t h e o r i e s were d i s c u s s e d i n C h a p t e r 3 . o f t h i s t h e s i s t h e y were i n t r o d u c e d w i t h some s i m p l i f y i n g a s s u m p t i o n s . Some o f t h e s e a s s u m p t i o n a r e r e s t a t e d h e r e as f o l l o w s : . - The f l u i d was assumed i n v i s c i d w h i c h p r e c l u d e s t h e e x i s t e n c e o f v o r t i c e s . - The t h e o r y i s l i n e a r i z e d so t h a t t h e a m p l i t u d e s o f m o t i o n and wave a m p l i t u d e s a r e assumed t o be s m a l l . - S h a l l o w w a t e r e f f e c t s a r e n o t a p p l i e d t o 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 . - The w a l l o f t h e t a n k i s assumed t o be i n t h e f a r f i e l d so as n o t t o a f f e c t t h e r e s u l t s . A d i s c u s s i o n o f e a c h o f t h e s e e f f e c t s f o l l o w s . The p r e s e n c e o f v o r t i c e s h a s n o t b e e n m o d e l e d b y t h e t h e o r y . To 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 i s e f f e c t i t i s n e c e s s a r y t o i n c o r p o r a t e a more s o p h i s t i c a t e d v o r t e x r i n g m o d e l . A s was m e n t i o n e d i n S e c t i o n 3 . , h o w e v e r , 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 l e n g t h ( D / L ) r a t i o s . V i s c o u s e f f e c t s w o u l d , t h e r e f o r e , be e x p e c t e d t o be l o w e r f o r i n c r e a s i n g f r e q u e n c i e s . Howeve r , 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 l y d e t e r m i n e d r e s u l t s and t h e t h e o r i e s were o b s e r v e d t o be g r e a t e r as t h e f r e q u e n c y i n c r e a s e d . I t i s t h e r e f o r e c o n c l u d e d t h a t the e f f e c t s o f v i s c o s i t y a r e v e r y s m a l l w i t h r e s p e c t t o d i f f r a c t i o n e f f e c t s f o r a l l f r e q u e n c i e s t e s t e d . One l i n e a r i z i n g a s s u m p t i o n s t a t e s t h a t t h e a m p l i t u d e o f m o t i o n must be s m a l l w i t h r e s p e c t t o c y l i n d e r d i a m e t e r . W i t h t h i s a s s u m p t i o n s m a l l r a t i o s - 68 -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 may be assumed t o be n e g l i g i b l e . The r e s u l t s showed no s t r o n g s e n s i t i v i t y t o t h e a m p l i t u d e o f m o t i o n . I t was o b s e r v e d on a few o c c a s i o n s t h a t t h e h i g h e r a m p l i t u d e s o f m o t i o n r e d u c e d the m a g n i t u d e 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 , b u t t h i s e f f e c t was m i n i m a l and n o t c o n s i s t e n t . I t may t h e r e f o r e be c o n c l u d e d t h a t t h e l i n e a r i t y a s s u m p t i o n as a p p l i e d t o t h e a m p l i t u d e o f m o t i o n i s n o t s e n s i t i v e t o v a r i a t i o n s i n the a m p l i t u d e / d i a m e t e r r a t i o b e t w e e n 0 . 0 2 6 and 0 . 0 9 1 . Whe the r t h e l i n e a r t h e o r y m i g h t c o r r e l a t e b e t t e r 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 l e s s t h a n .026 h a s n o t b e e n t e s t e d . The l i n e a r wave t h e o r y assumes t h a t t h e wave a m p l i t u d e i s s m a l l w i t h r e s p e c t t o t h e wave l e n g t h . F o r waves g e n e r a t e d b y t h e wave maker t h i s was t y p i c a l l y t h e c a s e s i n c e s m a l l a m p l i t u d e l o n g waves we re g e n e r a t e d . F o r t he h y d r o d y n a m i c t e s t s , h o w e v e r , h i g h e r f r e q u e n c y s t e e p e r waves were g e n e r a t e d by t h e m o t i o n g e n e r a t o r . F i g u r e 4 . 1 7 c o n t a i n e d i n t h e c i t e d r e f e r e n c e b y I s a a c s o n (1981) shows t h a t a h i g h e r o r d e r wave t h e o r y i s b e t t e r s u i t e d f o r 2 deep w a t e r waves h a v i n g H/gT r a t i o s g r e a t e r t h a n . 0 0 1 . T h i s c o r r e s p o n d s t o a 4 . 4 mm wave h e i g h t a t a f r e q u e n c y o f 1 .5 H z . T h i s l i m i t was o f t e n e x c e e d e d . One o f t h e o r i g i n a l b o u n d a r y c o n d i t i o n s a p p l i e d t o t h e f r e e s u r f a c e s t a t e s t h a t 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 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 k i n e m a t i c 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 s i n t r o d u c e d i n S e c t i o n 3 . 1 . 2 . T h i s b o u n d a r y c o n d i t i o n was c l e a r l y v i o l a t e d a t h i g h e r f r e q u e n c i e s 2 (w R / g > 3) where a s p l a s h i n g e f f e c t was o b s e r v e d . The s h a l l o w w a t e r e f f e c t s were n o t o b s e r v e d t o a d v e r s e l y e f f e c t t he e x p e r i m e n t a l r e s u l t s . I t was i n f a c t o b s e r v e d t h a t t h e e x p e r i m e n t a l r e s u l t s more c l o s e l y c o r r e s p o n d e d w i t h t h e t h e o r y f o r s h a l l o w e r d r a f t s . - 69 -D e s p i t e t h e f a c t t h a t t h e c y l i n d e r m o t i o n was p a r a l l e l t o t he w a l l s o f t h e t a n k , a w a l l e f f e c t was o b s e r v e d i n c o n d u c t i n g t h e e x p e r i m e n t s . The p r e s e n c e o f t h e w a l l was o b s e r v e d t o change t he r a d i a t i o n o f waves o u t w a r d f r o m t h e c y l i n d e r . T h i s e f f e c t i s c l e a r l y v i s i b l e i n t h e F i g u r e 7 . 1 1 . S t a n d i n g waves c o u l d be o b s e r v e d v i s u a l l y i n t h e r e g 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 w a l l . B e c a u s e o f t h i s e f f e c t t h e p o t e n t i a l o f t h e f a r f i e l d i s n o t m o d e l e d c o r r e c t l y b y t h e t h e o r y . The p r e s s u r e f i e l d i n t h e n e i g h b o u r h o o d o f t h e c y l i n d e r i s c h a n g e d . T h i s e f f e c t c o u l d be m o d e l e d u s i n g i m a g i n g m e t h o d s . I t i s t h e r e f o r e c o n c l u d e d , s i n c e good ag reemen t was a c h i e v e d b e t w e e n t he e x p e r i m e n t and t h e o r y d u r i n g t h e s i n g l e c y l i n d e r h y d r o d y n a m i c t e s t s , t h a t t he w a l l e f f e c t does n o t a c c o u n t f o r a l a r g e p o r t i o n o f t h e a d d e d mass c o e f f i c i e n t d i s c r e p a n c i e s o b s e r v e d a t h i g h f r e q u e n c i e s f o r t h e d o u b l e and t r i p l e c y l i n d e r m o d e l s . I t h a s b e e n p r e v i o u s l y d i s c u s s e d t h a t t h e 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 d 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 he 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 i s t h o u g h t t o be r e s p o n s i b l e f o r t he t e n d e n c y f o r t h e added mass c o e f f i c i e n t s t o ' b l o w u p ' a t h i g h e r f r e q u e n c i e s . I n a d d i t i o n t o 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 f u r t h e r d i s c r e p a n c i e s were i n t r o d u c e d b e c a u s e o f l i m i t a t i o n s w i t h t h e e x p e r i m e n t a l f a c i l i t i e s and p r o c e d u r e s . The wave maker a t t he OEC f a c i l i t y i s l i m i t e d i n i t s a b i l i t y t o . g e n e r a t e r e g u l a r waves a t a l l f r e q u e n c i e s . As d i s c u s s e d i n S e c t i o n 5 . 1 . 2 e n e r g y l e a k a g e i n t o t h e m a n o e u v e r i n g b a s i n c a u s e s t h e waves t o skew and l o o s e - 70 -r e g u l a r i t y . A n o t h e r l i m i t a t i o n i s t h e i n a b i l i t y o f t h e dynamometer t o r e s o l v e the s m a l l a m p l i t u d e f o r c e s , I t was n o t p o s s i b l e t o f i n d a s i n g l e dynamometer w e l l s u i t e d f o r u s e w i t h a l l t e s t s b e c a u s e o f t h e w i d e r a n g e o f f o r c e s b e i n g m e a s u r e d . I t i s t h e r e f o r e recommended t h a t two dynamomete rs , a h i g h range u n i t and a l o w r a n g e u n i t , be u s e d i n s u b s e q u e n t t e s t s . F o r t h e s e t e s t s a p p r o p r i a t e s i z e d dynamometers s h o u l d h a v e f u l l s c a l e r a t i n g s o f 1000 N and 100 N f o r t h e h i g h and l o w r a n g e u n i t s , r e s p e c t i v e l y . A c c u r a t e d e t e r m i n a t i o n o f t he p h a s e r e l a t i o n s h i p s b e t w e e n c h a n n e l s i s c r i t i c a l l y n e c e s s a r y 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 . E x t e n s i v e e f f o r t s were made t o c o n s e r v e p h a s e r e l a t i o n s h i p s and much t i m e was s p e n t d e v e l o p i n g t h i s a s p e c t o f t h e d a t a a n a l y s i s s o f t w a r e . S i n c e t h e a d d e d 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 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 . 4 . 5 ANALYSIS OF ERROR I t i s p o s s i b l e t o q u a n t i f y t h e m a g n i t u d e o f t h e e r r o r b y 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 . F o r e x a m p l e , i f one w i s h e s t o d e t e r m i n e t h e e r r o r i n a q u a n t i t y , W, w h i c h i s a f u n c t i o n o f t h r e e m e a s u r e d q u a n t i t i e s , x , y and z , 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 . - 71 -I f , W - / [ x . y . z ] . . . ( 4 . 5 - 1 ) T h e n , d W = a W d x + | W d y + | W d z . . . ( 4 . 5 - 2 ) dx dy dz 2 F o u r p a r a m e t e r s have b e e n p l o t t e d as a f u n c t i o n o f w R / g . The 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 be f r o m f o r c e measurements u s i n g t h e dynamometer w i t h s e c o n d a r y e f f e c t s f r o m wave a m p l i t u d e measurements u s i n g t h e wave p r o b e . T h i s s e c t i o n w i l l d e r i v e an 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 o f t h e e r r o r a s s o c i a t e d w i t h e a c h p l o t t e d p a r a m e t e r . 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 and f r e q u e n c i e s a r e i n s i g n i f i c a n t l y s m a l l s i n c e t i m e i s measu red t o a t l e a s t t h e n e a r e s t m i l l i s e c o n d and d i s p l a c e m e n t i s known t o t h e n e a r e s t .025 mm. I t i s s t a t e d i n A p p e n d i x A t h a t t h e RMS e r r o r i n dynamometer measuremen ts i s ± 6 . 9 N and t h e wave p r o b e i s assumed t o have an a c c u r a c y o f ± 1 mm. The f o l l o w i n g f u n c t i o n a l r e l a t i o n s h i p s have b e e n u s e d i n d e t e r m i n i n g t h e p l o t t e d p a r a m e t e r s . Added Mass a F m a ^ - . . . ( 4 . 5 - 3 ) pV w2XpV pV Damping C o e f f i c i e n t b F b — - . . . ( 4 . 5 - 4 ) 2 wpV u> XpV - 72 -Wave I n d u c e d E x c i t i n g F o r c e F F . . . ( 4 . 5 - 5 ) pVgA I n d i r e c t l y D e t e r m i n e d Damping C o e f f i c i e n t 2 b * 11 D = 2 2 copV 16np gW g F A ( 4 . 5 - 6 ) Where , a — n o n - d i m e n s i o n a l s u r g e added mass c o e f f i c i e n t ; b = n o n - d i m e n s i o n a l s u r g e damping c o e f f i c i e n t ; * n o n - d i m e n s i o n a l wave i n d u c e d e x c i t a t i o n f o r c e ; a ^ •= s u r g e added mass c o e f f i c i e n t ; b = s u r g e damp ing c o e f f i c i e n t ; w = f r e q u e n c y ; F = s u r g e f o r c e ; p = d e n s i t y o f f l u i d medium; V = d i s p l a c e d v o l u m e ; A = wave a m p l i t u d e ; V = wave g roup v e l o c i t y ; s X = a m p l i t u d e o f m o t i o n ; g = a c c e l e r a t i o n due t o g r a v i t y . 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 v a l u e s were u s e d d u r i n g t h e t e s t s , p = 1000 K g / m 3 g - 9 . 8 0 6 6 5 m / s 2 ?r = 3 .1415927 - 73 -V = 2 4 . 4 x 1 0 " 3m3 to 7 4 . 3 x l O " V H y d r o d y n a m i c T e s t s X = 10 mm t o 35 mm w - 1 .57 s " 1 t o 1 5 . 7 1 s " 1 I n c i d e n t wave t e s t s A = 2 mm t o 60 mm F = 1 N t o 60 N V - 1 .24 m/s t o 3 . 12 m/s 8 <a = 1 .24 s " 1 t o 1 0 . 9 s " 1 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 p a r a m e t e r v a l u e s g i v e n t he 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 : TABLE 4 . 5 - 1 ANALYSIS OF ERROR FOR PLOTTED PARAMETERS P a r a m e t e r Minimum 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 * b 0 . 0 1 1 0 . 0 4 5 1 1 . 5 * F 0 . 1 5 8 0 . 3 6 9 1 4 . 4 * (Wehausen) b 0 . 0 0 0 1 0 . 0 0 3 1 1 1 . 2 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 . F o r 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 t h e i n d i r e c t l y d e t e r m i n e d damping 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 l i g h t e s t c y l i n d e r w i t h t h e s m a l l e s t a m p l i t u d e wave . T h i s c o m b i n a t i o n o f e v e n t s p h y s i c a l l y d i d - 74 -n o t o c c u r . To do a e x h a u s t i v e e r r o r a n a l y s i s 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 t h e o r d e r o f m a g n i t u d e e s t i m a t e d e s i r e d h e r e t he ' 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 e r r o r . 4 . 6 U T I L I T Y OF RESULTS H a v i n g p r e s e n t e d a l l t h i s d a t a , one m i g h t a s k how c a n i t be u s e d t o s o l v e a n e n g i n e e r i n g p r o b l e m . C o n s i d e r a n a x i s y m m e t r i c f l o a t i n g s t r u c t u r e o f known g e o m e t r y w h i c h i s t o be d e s i g n e d t o e x i s t i n a n o c e a n e n v i r o n m e n t w i t h a n assumed s e a s t a t e . Assume , i t i s n e c e s s a r y t o d e t e r m i n e t h e wave i n d u c e d s u r g e f o r c e on t h e s t r u c t u r e and t h e dynamic s u r g e r e s p o n s e o f t h e s t r u c t u r e t o t h i s f o r c e . I t i s p o s s i b l e t o e x p e r i m e n t a l l y d e t e r m i n e , as h a s b e e n d e m o n s t r a t e d i n t h i s t h e s i s , 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 c a l e d down mode l o f t h e f l o a t i n g b o d y . I n a d d i t i o n e s t i m a t e s o f t h e s e c o e f f i c i e n t s c a n be n u m e r i c a l l y c a l c u l a t e d u s i n g one o f two methods d i s c u s s e d i n S e c t i o n 3 . As a f i r s t o r d e r a p p r o x i m a t i o n W e h a u s e n ' s f o r m u l a t i o n ( E q u a t i o n 3 . 1 . 6 - 1 ) may t h e n be u s e d t o p r e d i c t t h e i n c i d e n t wave i n d u c e d s u r g e e x c i t i n g f o r c e on t he b o d y . O t h e r w i s e s c a l e mode l e x p e r i m e n t s o r t h e BEM c a n be u s e d t o p r e d i c t t h i s f o r c e u s i n g t h e m e t h o d o l o g y p r e s e n t e d i n t h i s t h e s i s . T h i s f o r c e i s t h e n a p p l i e d u s i n g t h 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 d i s c r i b i n g t h e m o t i o n o f a dynamic s y s t e m ( E q u a t i o n 2 . 1 - 1 ) and t h e m o t i o n s c a n be s o l v e d f o r . I n r e a l i t y i t m i g h t be d e s i r a b l e t o know t h e e n t i r e m o t i o n o f t h e body f o r 6 d e g r e e s o f f r e e d o m , i n w h i c h c a s e e x p e r i m e n t s a n d / o r t h e o r y w o u l d be u s e d t o s o l v e f o r a l l o f t h e 8 u n i q u e added mass and damp ing c o e f f i c i e n t t e r m s . These t e rms a r e t h e n a p p l i e d as a m a t r i x t o E q u a t i o n 1-1 where t h e m o t i o n f o r a l l s i x d e g r e e s o f f r e e d o m c a n be s o l v e d f o r . 75 -CONCLUSIONS 1. The p r e d i c t i o n s o f two t h e o r e t i c a l m o d e l s , t h e B o u n d a r y E l e m e n t Me thod (BEM) a n d t h e M a t c h i n g T e c h n i q u e (MT) , were i n v e s t i g a t e d i n t h i s t h e s i s . The MT f o r m u l a t i o n r e q u i r e s d e t a i l e d m a t h e m a t i c a l p r e p a r a t i o n b e f o r e i t c a n be a p p l i e d t o a p a r t i c u l a r c l a s s o f a x i s y m m e t r i c s h a p e s . The BEM f o r m u l a t i o n i s d i r e c t l y a p p l i c a b l e t o any t y p e o f a x i s y m m e t r i c shape a n d t h e c o n t r o l s u r f a c e n e e d o n l y be d i s c r e t i z e d f o r i t s i m p l e m e n t a t i o n . The d i s a d v a n t a g e o f t he BEM f o r m u l a t i o n i s t h a t i t r e q u i r e s a much g r e a t e r c o m p u t a t i o n a l e f f o r t t h a n t he MT f o r m u l a t i o n t o s o l v e f o r t h e p o t e n t i a l i n t h e f l u i d d o m a i n . T h i s g r e a t e r e f f o r t i s due i n l a r g e p a r t t o t h e r e l a t i v e m a t r i x s i z e w h i c h must be s o l v e d b y e a c h p r o g r a m . A n o t h e r a d v a n t a g e o f t h e MT f o r m u l a t i o n i s i t s a p p l i c a b i l i t y t o a w i d e r a r r a n g e m e n t o f b o d i e s . The BEM t h e o r y as p r e s e n t 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 t o a s i n g l e a x i s y m m e t r i c shape w h i l e t h e MT f o r m u l a t i o n c a n be r e a d i l y a d a p t e d t o a r r a y s o f s h a p e s . One s t r o n g d i s a d v a n t a g e o b s e r v e d w i t h t h e BEM f o r m u l a t i o n was t h e t e n d e n c y f o r t h e r e s u l t s t o o s c i l l a t e a b o u t a mean t r e n d a t h i g h e r f r e q u e n c i e s . To e l i m i n a t e t h i s e f f e c t w o u l d r e q u i r e d i s c r e t i z i n g t h e c o n t r o l t o s u r f a c e t o s u c h a d e g r e e t h a t i t w o u l d r e q u i r e p r o h i b i t i v e q u a n t i t i e s o f c o m p u t i n g t i m e . By f i l t e r i n g o u t t h e o s c i l l a t i n g component v e r y s t r o n g ag reemen t was o b s e r v e d i n t h e p r e d i c t i o n o f 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 . 2 . 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 a n d 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 due t o t h e f o l l o w i n g i n a p p r o x i m a t e o r d e r o f i m p o r t a n c e : 1) A h y d r o e l a s t i c • e f f e c t w h i c h c r e a t e s r e l a t i v e m o t i o n be tween t h e c y l i n d e r mode l and t h e m o t i o n g e n e r a t o r . T h i s e f f e c t c a n be m i n i m i z e d b y u s i n g a more r i g i d m o t i o n g e n e r a t o r a s s e m b l y . - 76 -2) The i n a b i l i t y o f t he dynamometers t o r e s o l v e l o w a m p l i t u d e f o r c e s . T h i s e f f e c t c o u l d be r e m e d i e d b y h a v i n g a v a i l a b l e two o r more dynamometers w i t h d i f f e r e n t f u l l s c a l e r a n g e s . The u n i t w h i c h g i v e s t h e g r e a t e s t r e s o l u t i o n b u t h a s a f u l l s c a l e r a n g e w i t h i n t h e maximum a n t i c i p a t e d f o r c e c o u l d t h e n be u s e d f o r e a c h t e s t . 3) R e s u l t s o f l i n e a r i z i n g a s s u m p t i o n s o f t h e o r y , p a r t i c u l a r l y as a p p l i e d t o 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 s . H i g h e r o r d e r t h e o r i e s a r e b e t t e r s u i t e d t o mode l t h e h i g h e r f r e q u e n c y waves a t t h e f r e e s u r f a c e . 4 ) E f f e c t s o f w a l l on t h e f l o w f i e l d . A i m a g i n g t e c h n i q u e c o u l d be a p p l i e d t o t h e M a t c h i n g T e c h n i q u e f o r m u l a t i o n t o mode l t h i s e f f e c t . 3 . By c o m p a r i n g t h e e x p e r i m e n t a l r e s u l t s o f t h e h y d r o d y n a m i c t e s t s t he t h e o r e t i c a l r e s u l t s t h e f o l l o w i n g o b s e r v a t i o n a r e n o t e d . 1) E x c e l l e n t 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 i n 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 f o r t he s i n g l e c y l i n d e r . 2) The t h e o r i e s more c l o s e l y mode l 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 s h a l l o w e r d r a f t s o f t h e d o u b l e and t r i p l e c y l i n d e r s . 3) F o r t h e d o u b l e and t r i p l e c y l i n d e r m o d e l s a t 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 up t o a b o u t 1 .0 t h e t r e n d s o f t h e a d d e d mass c o e f f i c i e n t s we re i d e n t i c a l f o r b o t h t h e e x p e r i m e n t s a n d t h e t h e o r i e s b u t t he e x p e r i m e n t a l r e s u l t s were o b s e r v e d t o be g e n e r a l l y a b o u t 30% l e s s t h a n t h e v a l u e s p r e d i c t e d b y e i t h e r o f t h e two t h e o r i e s . A t f r e q u e n c i e s g r e a t e r t h a n 1 .0 p o o r ag reemen t was a c h i e v e d b e t w e e n the e x p e r i m e n t s and t h e t h e o r i e s . T h i s i s b e l i e v e d t o be due i n l a r g e p a r t t o a h y d r o e l a s t i c e f f e c t . - 77 -4) F o r t h e d o u b l e and t r i p l e c y l i n d e r m o d e l s g e n e r a l l y good agreement was o b s e r v e d 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 f o r damping c o e f f i c i e n t s and t h o s e v a l u e s d e t e r m i n e d b y t h e t h e o r i e s . The e x p e r i m e n t a l r e s u l t s were s c a t t e r e d on b o t h s i d e s o f 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 . 5) V a r i a t i o n s i n t h e added mass and damp ing c o e f f i c i e n t s f o r d i f f e r e n t a m p l i t u d e s o f m o t i o n were o b s e r v e d b u t no c o n s i s t e n t t r e n d was e v i d e n t . I t i s t h e r e f o r e c o n c l u d e d t h a t v a r y i n g t he a m p l i t u d e / d i a m e t e r r a t i o b e t w e e n .026 and .091 h a s no s i g n i f i c a n t e f f e c t on t h e l i n e a r i t y a s s u m p t i o n o f t h e t h e o r y . 6) A r e d u c t i o n i n 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 was o b s e r v e d f o r a l l c y l i n d e r m o d e l s when t e s t e d a n d e e p e r d r a f t s . 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 i n c i d e n t waves t e s t 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 o b s e r v a t i o n s a r e n o t e d . 1) The same t r e n d s were f o u n d t o e x i s t f o r b o t h t h e e x p e r i m e n t a l r e s u l t 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 as a p p l i e d t o b o t h t h e wave i n d u c e d e x c i t i n g f o r c e and t h e i n d i r e c t l y d e t e r m i n e d damp ing c o e f f i c i e n t s . 2) B e t t e r 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 a c h i e v e d a t 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 0 . 5 . 3) The i n d i r e c t l y d e t e r m i n e d damp ing c o e f f i c i e n t was o b s e r v e d t o most c l o s e l y f o l l o w t h e o r y f o r t h e t r i p l e c y l i n d e r m o d e l . F o r o t h e r mode l s t h e t h e o r i e s were o b s e r v e d t o u n d e r - p r e d i c t t h e e x p e r i m e n t a l r e s u l t s a t 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 0 . 5 . B e l o w t h i s f r e q u e n c y good c o r r e l a t i o n was a c h i e v e d . 4 ) The c o n s i d e r a b l e s c a t t e r i n t h i s d a t a i s a t t r i b u t e d t o t h e i n a b i l i t y o f t h e dynamometer t o r e s o l v e l o w a m p l i t u d e f o r c e s . - 78 -RECOMMENDATIONS 1. I f f u t u r e work i s t o be c o n d u c t e d i n t h i s a r e a e f f o r t s h o u l d be d e v o t e d t o i m p l e m e n t i n g some o f t h e s u g g e s t i o n s i n C o n c l u s i o n 2 . t o m i n i m i z e d i s c r e p a n c i e s b e t w e e n t h e t h e o r e t i c a l and e x p e r i m e n t a l r e s u l t s . A s a minimum i t i s recommended t h a t a t l e a s t t h e two f o l l o w i n g r e c o m m e n d a t i o n s be i m p l e m e n t e d : 1) A s y s t e m o f two dynamometers be e m p l o y e d , whe reby t h e f u l l s c a l e o f t h e h i g h r a n g e u n i t w o u l d be an o r d e r o f m a g n i t u d e g r e a t e r t h a n t h e f u l l s c a l e r a n g e o f t he s m a l l e r u n i t . The two u n i t s w o u l d t h e n be swapped o u t d u r i n g d i f f e r e n t t e s t s b a s e d on t h e maximum a n t i c i p a t e d f o r c e . 2) The m o t i o n g e n e r a t o r and c y l i n d e r m o d e l s be made more r i g i d . P a r t i c u l a r a t t e n t i o n s h o u l d be g i v e n t o t h e a r e a where t h e c y l i n d e r i s 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 o r . I f t h e c y l i n d e r mode ls t h e m s e l v e s c a n n o t be s t r e n g t h e n e d c o n s i d e r a t i o n s h o u l d be g i v e n t o u s i n g s m a l l e r mode l s f o r t h e h y d r o d y n a m i c t e s t s . 2 . Improvements be made t o t h e BEM p r o g r a m t o remove t h e o s c i l l a t o r y b e h a v i o u r o f i t s r e s u l t s . - 79 -BIBLIOGRAPHY A b r a m o w i t z , M. and I . A . S t e g u n , 1964 , 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 i e 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 . A y a d , A . M . , 1 9 8 3 , Radiation of Short Surface Waves by Oscillating Submerged Smooth Cylinders, J . E n g g . M a t h . , Vo lume 1 7 , p p . 5 5 - 7 2 . B e n d a t and P i e r s o l , 1 9 7 1 , Random D a t a , p p . 2 8 6 - 3 4 3 . B a u m e i s t e r , T h e o d o r e ( e d i t o r ) , 1978 , M a r k s ' S t a n d a r d Handbook f o r E n g i n e e r s , E i g h t h e d i t i o n , New Y o r k : M c G r a w - H i l l . 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 i n g , P e n t e c h P r e s s , p p . 4 6 - 7 2 . 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 8 , p p . 2 5 - 6 3 . C a l i s a l , S . M. and T . S a b u n c u , 1984 , 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. , 1 9 8 5 a , MECH 541 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 . C a l i s a l , S . M. and J . K. L . C h a n , 1985b , 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 ' 8 5 , S h a n g h a i , Dec 2 - 8 , 1 9 8 5 . C h a n , J o h n s o n L . K . , 1984 , 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 . F r y e r , D. K . , and M. W. Thomas, 1 9 7 5 , A Linear Twin Wire Probe for Measuring Water Waves, J o u r n a l o f P h y s i c s E : S c i e n t i f i c I n s t r u m e n t s , Vo lume 8 , p p . 4 0 5 - 4 0 8 . G a r r i s o n , C . J . , 1974 , Hydrodynamics of Large Objects in the Sea - Fart I: Hydrodynamic Analysis, J . H y d r o n a u t i c s , Vo lume 8 , No 1 . , p p . 5 - 1 2 . - 80 -G a r r i s o n , C . J . , 1 9 7 5 , Hydrodynamics of Large Objects in the Sea - Part II: Motion of Free-Floating Bodies, J . H y d r o n a u t i c s , Vo lume 9 , No 2 , p p . 5 8 - 6 3 . H a v e l o c k , T . H . , 1 9 6 3 , C o l l e c t e d P a p e r s , O N R / A C E - 1 0 3 , US Government P r i n t i n g O f f i c e , W a s h i n g t o n D . C . H i c k m a n , The H o n o r a b l e T . A . ( C h a i r m a n ) , 1984 , R e p o r t One: The L o s s o f The S e m i s u b m e r s i b l e D r i l l R i g Ocean R a n g e r and i t s C r e w , R e p o r t o f t h e C a n a d i a n R o y a l C o m m i s s i o n on t h e Ocean R a n g e r M a r i n e D i s a s t e r , O t t a w a : C a n a d i a n Government P u b l i s h i n g C e n t r e . I s a a c s o n , M i c h a e l and T u r g u t 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. K e u l e g a n , G a r b i s H . , and L l o y d H. 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Wehausen , J o h n 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 n a r d , J o h n K . , and R o b e r t L . S t r e e t , 1 9 7 6 , E l e m e n t a r y F l u i d M e c h a n i c s , F i f t h e d i t i o n , T o r o n t o : J o h n W i l e y and Sons I n c . V e n u g o p a l , M a d a n , 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 Of S i n g l e and Doub le 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 a d a n , 1 9 8 4 b , 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 . - 81 -APPENDIX A 5 . EXPERIMENTAL S E T - U P 5 .1 EXPERIMENTAL F A C I L I T I E S A l l 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 a t t h e Ocean E n g i n e e r i n g C e n t e r (OEC) o f BC R e s e a r c h (BCR) i n V a n c o u v e r , B C . 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 . 5 . 1 . 1 TOWING TANK Two t a n k s a r e i n p l a c e a t t h e OEC i n c l u d i n g a s h i p mode l m a n o 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 . 4 4 m and a t o w i n g t a n k , m e a s u r i n g 6 7 . 1 0 m x 3 . 6 6 m x 2 . 4 4 m d e e p . The two t a n k s a r e s e p a r a t e d b y a n a luminum b u l k h e a d w i t h a 1 .22 m d r a f t h e n c e f r e e c o m m u n i c a t i o n e x i s t s b e l o w t h i s d e p t h . F i g u r e 7 . 2 shows t h e two b a s i n s a t t h e OEC. The t o w i n g t a n k i s p r i m a r i l y u s e d f o r 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 t owed t h e l e n g t h o f t h e t a n k by a t o w i n g c a r r i a g e e q u i p p e d w i t h t h e i n s t r u m e n t a t i o n and d a t a c o l l e c 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 r e m o t e l y d r i v e n by a h y d r a u l i c mo to r c a b l e s y s t e m . The c a r r i a g e i s c l e a r l y v i s i b l e i n F i g u r e 7 . 8 . The t o w i n g t a n k i s a l s o e q u i p p e d w i t h t h r e e u n d e r w a t e r v i e w i n g windows u s e d 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 . I n t h e J u l y 1985 t h i s a u t h o r d e s i g n e d and i n s t a l l e d an o v e r h e a d h o i s t i n g mechan ism p o s i t i o n e d above t he w o r k i n g a r e a o f t h e t o w i n g b a s i n u s e d i n t h e s e t e s t s . T h i s c h a i n h o i s t - 82 -(1300 N maximum s a f e l o a d ) , p i c t u r e d i n F i g u r e . 7 . 6 , r e p l a c e d an e x i s t i n g i n s t a l l a t i o n w h i c h was d e c l a r e d u n s a f e b y a gove rnmen t r e g u l a t o r y b o a r d . The m e c h a n i c a l h o i s t 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 . 4 m a l o n g i t s l e n g t h . T h i s was t h e p r i n c i p a l 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 d u r i n g t e s t s . I n a d d i t i o n , t h e t o w i n g t a n k i s e q u i p p e d w i t h a wave m a k e r , w h i c h i s d e s c r i b e d i n more d e t a i l i n n e x t s e c t i o n . A t t h e o p p o s i t e end o f t h e t a n k t h e r e i s a n a r t i f i c i a l b e a c h w h i c h i s d e s i g n e d t o a b s o r b t h e wave e n e r g y t o m i n i m i z e r e f l e c t i o n o f t h e w a v e s . The 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 a p p r o x i m a t e l y m i d - s t a t i o n o f t h e t o w i n g t a n k . The t o w i n g c a r r i a g e w h i c h c o n t a i n e d 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 and a s s o c i a t e d p a r a p h e r n a l i a was p o s i t i o n e d a d j a c e n t t o t h e wo rk a r e a . D u r i n g t h e dynamic t e s t s a t e m p o r a r y a r t i f i c i a l b e a c h , w h i c h c o n s i s t e d 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 , was p l a c e d a l o n g t h e s u r f a c e o f t h e w a t e r a b o u t 8 m e t e r s u p s t r e a m t o w a r d t h e wave m a k e r . T h i s a r r a n g e m e n t i s d e p i c t e d i n F i g u r e 7 . 4 . T h i s i n s t a l l a t i o n i n c o n j u n c t i o n w i t h t h e pe rmanen t b e a c h a t t h e o p p o s i t e end o f t h e t a n k h e l p e d t o dampen t h e waves i n d u c e d by t h e m o t i o n o f t h e c y l i n d e r more q u i c k l y . 5 . 1 . 2 WAVE MAKER The wave maker a t t h e OEC i s a f l a p p e r t y p e u n i t w h i c h p i v o t s a b o u t an a x i s l o c a t e d 1.2 m b e l o w t h e w a t e r s u r f a c e . The f l a p p e r i s o s c i l l a t e d b y a 2 0 . 7 M p a , 5 0 . 8 mm x 9 1 4 . 4 mm ( b o r e x s t r o k e ) d o u b l e a c t i n g R o y a l h y d r a u l i c c y l i n d e r . A p h o t o g r a p h o f t h i s d e v i c e i s p r o v i d e d i n F i g u r e 7 . 5 . - 83 -The wave maker may be c o n t r o l l e d by any number o f i n p u t s i g n a l g e n e r a t o r s . F o r t h e t e s t s d i s c u s s e d i n t h i s r e p o r t 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 an 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 ) , was u s e d t o p r o v i d e t h e i n p u t s i g n a l . A p r o g r a m c a l l e d SWAVE g e n e r a t e d s i n e waves o f a s p e c i f i e d f r e q u e n c y and s i g n a l v o l t a g e a m p l i t u d e . The g e n e r a t e d wave a m p l i t u d e i s d e p e n d a n t on b o t h t h e i n p u t s i g n a l f r e q u e n c y and v o l t a g e a m p l i t u d e b u t a t r a n s f e r f u n c t i o n r e l a t i n g t h e two h a s n o t b e e n d e t e r m i n e d . Some e x p e r i m e n t a t i o n was n e c e s s a r y i n some c a s e s t o d e t e r m i n e t h e i n p u t s i g n a l a m p l i t u d e r e q u i r e d t o c r e a t e t h e d e s i r e d wave o u t p u t . The i n p u t s i g n a l goes t o a wave s y n t h e s i z e r where i t i s compared w i t h t h e 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 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 maker i s a b l e t o f u n c t i o n m e c h a n i c a l l y t h r o u g h f r e q u e n c i e s g r e a t e r t h a n 2 . 5 Hz and w i t h i n p u t v o l t a g e s i g n a l s up t o 300 mV. Howeve r , t he waves g e n e r a t e d a t f r e q u e n c i e s o u t s i d e o f t h e n o m i n a l r a n g e b e t w e e n .3 Hz and 1 .5 Hz a r e n o t w e l l f o r m e d s i n u s o i d a l w a v e s . F u r t h e r m o r e , t h e a m p l i t u d e s . o f waves g e n e r a t e d o u t s i d e t h i s r a n g e were so s m a l l t h a t t h e y c o u l d n o t i n d u c e a f o r c e w h i c h c o u l d be s a t i s f a c t o r i l y m e a s u r e d b y t h e dynamometer . Some i r r e g u l a r i t i e s were o b s e r v e d i n many o f t h e g e n e r a t e d wave f o r m s . T h i s c a n be a t t r i b u t e d t o l e a k a g e and i n t e r f e r e n c e f r o m t h e m a n o e u v e r i n g b a s i n w h i c h i s s e p a r a t e d f r o m t h e t o w i n g b a s i n b y a c o m p l i a n t a l um inum h a l f - w a l l . I n a d d i t i o n o t h e r i r r e g u l a r i t i e s may h a v e a r i s e n f r o m r e f l e c t e d w a v e s , a i r c u r r e n t s and r e s i d u a l e f f e c t s f r o m p r e v i o u s t e s t s . I n t e r f e r e n c e f r o m p r e v i o u s t e s t s was m i n i m i z e d b y a l l o w i n g t h e t a n k t o " c a l m down" a f t e r e a c h t e s t i f t h e s u r f a c e o f t h e w a t e r was n o t i c e a b l y d i s t u r b e d . The s p e c t r a l a n a l y s i s - 84 -p r o c e d u r e u s e d i n t h e d a t a a n a l y s i s a l l o w e d one 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 s i n u s o i d a l wave i n c i d e n t on t h e c y l i n d e r a t t h e f r e q u e n c y o f i n t e r e s t . I n o t h e r w o r d s , i t was p o s s i b l e t o m a t h e m a t i c a l l y remove any i n t e r f e r e n c e w h i c h m i g h t h a v e e x i s t e d a t f r e q u e n c i e s w h i c h were n o t n e a r t h e o p e r a t i n g f r e q u e n c y . 5 . 2 EXPERIMENTAL EQUIPMENT 5 . 2 . 1 MOTION GENERATOR The m o t i o n g e n e r a t o r c o n s i s t e d o f a h y d r a u l i c a l l y p o w e r e d s l i d e r mechan ism s u p p o r t e d b y an a luminum f r a m e . I t i s v i s i b l e i n many o f t he p h o t o g r a p h s i n A p p e n d i x C b u t most c l e a r l y i l l u s t r a t e d i n F i g u r e 7 . 9 . 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 i n t he 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 m a c h i n e shop a t UBC i n 1 9 8 2 . I t was f i r s t u s e d b y Madan V e n u g o p a l i n t h e summer o f 1983 f o r t e s t i n g t h e heave 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 same c y l i n d e r m o d e l s u s e d i n t h e s e s u r g e m o t i o n e x p e r i m e n t s . I n t h e w i n t e r o f 1985 d e s i g n m o d i f i c a t i o n s were made b y t h i s a u t h o r t o t r a n s f o r m t h e h e a v e m o t i o n g e n e r a t o r i n t o a s u r g e m o t i o n g e n e r a t o r . The r e s u l t i n g s t r u c t u r e c o n s i s t e d o f a h o r i z o n t a l s l i d e r p l a t e c a n t i l e v e r e d f r om a v e r t i c a l f r ame w h i c h was mounted on a b r i d g e s p a n n i n g t h e t o w i n g t a n k . V i r t u a l l y a l l s t r u c t u r a l components were m a n u f a c t u r e d f r o m a l u m i n u m . To p r o v i d e a d d i t i o n a l s u p p o r t and r i g i d i t y t o t h e s t r u c t u r e an a u x i l i a r y b r i d g e was l o c a t e d f o r w a r d o f t h e c a n t i l e v e r e d s l i d i n g p l a t e . T h i s p l a t e was t h e n s u p p o r t e d b y two ' t e l e s c o p i n g ' b r a c e s f r o m t h e a u x i l i a r y b r i d g e . The s l i d i n g - 85 -p l a t e was c o n n e c t e d t o a s c o t c h yoke mechan ism and t r a v e l l e d on l i n e a r b e a r i n g s . The f o r c e dynamometer was r i g i d l y b o l t e d t o t h e s l i d e r p l a t e and t h e c y l i n d e r mode l was c o n n e c t e d t o t h e dynamometer b y an a d a p t e r b l o c k . T h i s a r r a n g e m e n t i s shown i n F i g u r e 7 . 1 5 . The s l i d e r p l a t e was d r i v e n b y a s c o t c h yoke mechan ism w h i c h was powered 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 c a p a b l e o f p r o v i d i n g 680 N-m o f o u t p u t t o r q u e . Howeve r , 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 a l power s u p p l y t h e h y d r a u l i c m o t o r c o u l d o n l y d e l i v e r t h i s r a t e d t o r q u e up t o a f r e q u e n c y o f .83 H z , d i m i n i s h i n g t o 225 N-m o f o u t p u t t o r q u e a t 2 . 5 H z . The m o t o r , w i t h a d i s p l a c e m e n t o f .208 1 / r e v o l u t i o n , was d r i v e n b y 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 w h i c h e n s u r e s t i g h t c o n t r o l o v e r a s p e e d r a n g e f r o m 3 t o 150 rpm. The h y d r a u l i c power u n i t c o n s i s t e d o f a 1750 rpm, 3 .7 kW, 440 V e l e c t r i c m o t o r w h i c h d r o v e 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 h a v i n g 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 . M e c h a n i c a l s t o p s l i m i t e d t h e d i s p l a c e m e n t o f t h e pump t o 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 . A p h o t o g r a p h o f t h i s u n i t i s p r o v i d e d i n F i g u r e 7 . 1 6 . 5 . 2 . 2 DATA COLLECTION EQUIPMENT The d a t a c o l l e c t i o n e q u i p m e n t was p r o v i d e d b y BC R e s e a r c h and c o n s i s t e d o f a MINC™ 11 m i n i compu te r and an ST41B™ s i g n a l c o n d i t i o n e r . 86 -5 . 2 . 2 . 1 ST41B SIGNAL CONDITIONER T h i s s i g n a l c o n d i t i o n e r was c u s t o m 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 S y s t e m s L t d . i n V a n c o u v e r , B C . 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 s i g n a l c o n d i t i o n i n g i n mode l b o a t t e s t s . The p r i n c i p a l 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 s y s t e m 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 c h a n n e l (2 t o 10 VDC) - 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 ( ± 1 3 VDC) - 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 1, ^ o r f u l l 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 (50 Hz) on e a c h c h a n n e l . The s i g n a l c o n d i t i o n e r was g e n e r a l l y c o n f i g u r e d as f o l l o w s : - 87 -T a b l e 5 . 2 . 2 . 1 - 1 A m p l i f i e r S e t u p C o n f i g u r a t i o n 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 # V o l t a g e 1 P r e s s u r e t r a n s d u c e r #1 5. .0 500 2 P r e s s u r e t r a n s d u c e r #2 5. .0 500 3 P r e s s u r e t r a n s d u c e r #3 5. .0 200 4 Y o - Y o P o s i t i o n T r a n s d u c e r 10, .0 5 5 Dynamometer S u r g e C h a n n e l 10. .0 1000/500 6 1 Dynamometer Heave C h a n n e l 10. .0 1000/500 7 Dynamometer P i t c h C h a n n e l 10. .0 1000/500 8 Wave P r o b e ±15. .o 2 1 Heave c h a n n e l on dynamometer n o t g e n e r a l l y u s e d . 2 E x c i t a t i o n p r o v i d e d b y s o u r c e e x t e r n a l t o s i g n a l c o n d i t i o n e r . W i t h t h e s i g n a l c o n d i t i o n e r c o n f i g u r e d as s p e c i f i e d e a c h c h a n n e l was t h e n b a l a n c e d o r z e r o e d so t h a t t h e mean s i g n a l w o u l d be a p p r o x i m a t e l y z e r o . I t n e e d n ' t be p r e c i s e l y z e r o s i n c e t h e d a t a a n a l y s i s s o f t w a r e makes f u r t h e r c o r r e c t i o n s t o remove t h e DC o f f s e t f r o m t h e s i g n a l . The e x c i t a t i o n and r e t u r n s i g n a l a l l t r a v e l on 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 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 . E a c h p l u g i s i d e n t i f i e d w i t h i t s c h a n n e l number and t a k e s a B e n d i x PT02A 1 2 - 1 0 S c o n n e c t o r . L o c a t e d j u s t above t h e i n p u t p l u g s i s an 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 c o n n e c t s t h e s e BNC t e r m i n a l s t o f o u r t e r m i n a l s l o c a t e d on t h e f a c e o f t he MINC™ 11 M i n i Computer and f o u r o t h e r i n t e r n a l p l u g s . F u r t h e r d e t a i l s on t he o p e r a t i o n o f t h e ST41B™ s i g n a l c o n d i t i o n e r may be f o u n d i n t h e o p e r a t i o n manua l a v a i l a b l e a t BC R e s e a r c h . - 88 -5 . 2 . 2 . 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 phase a n g l e b e t w e e n e a c h c h a n n e l . The s i g n a l c o n d i t i o n e r and i n p a r t i c u l a r t h e l ow p a s s f i l t e r w i t h i n t h e s i g n a l c o n d i t i o n e r i n t r o d u c e a p h a s e s h i f t t o t he i n c o m i n g s i g n a l . To 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 i s p h a s e s h i f t i t was n e c e s s a r y t o c o n d u c t a s e r i e s o f t e s t s . S i n c e t h e s i g n a l c o n d i t i o n e r c o n s i s t e d o f e i g h t i d e n t i c a l l y c o n f i g u r e d c h a n n e l s i t was d e c i d e d t h a t o n l y two c h a n n e l s w o u l d be t e s t e d and i f t h e i r r e s u l t s were f o u n d t o be s i m i l a r t h e n i t c o u l d be assumed t h a t t h e s e r e s u l t s were r e p r e s e n t a t i v e o f a l l t h e c h a n n e l s . I n c o n d u c t i n g t h e s e t e s t s a N i c o l e t S p e c t r u m A n a l y z e r was u s e d i n c o n j u n c t i o n w i t h a random s i g n a l g e n e r a t o r . O u t p u t f r o m t h e random s i g n a l g e n e r a t o r was s e t t o p r o v i d e u n i f o r m e n e r g y a t a l l f r e q u e n c i e s up t o 10 H z . T h i s was d i f f i c u l t t o a c h i e v e a t f r e q u e n c i e s w h i c h we re n e a r z e r o so t h e a n a l y s i s d i d n o t c o n s i d e r f r e q u e n c i e s i n t h i s r e g i o n . The w h i t e n o i s e s i g n a l was f e d i n t o c h a n n e l A on t h e s p e c t r u m a n a l y z e r as w e l l as t h r o u g h t h e s i g n a l c o n d i t i o n e r . The o u t p u t f r o m t h e s i g n a l c o n d i t i o n e r was c o n n e c t e d t o c h a n n e l B o f t h e s p e c t r u m a n a l y z e r . The t r a n s f e r f u n c t i o n w h i c h t r a n s f o r m e d t h e s i g n a l f r o m c h a n n e l A t o c h a n n e l B was d e t e r m i n e d b y t h e s p e c t r u m a n a l y z e r . The r e s u l t i n g t r a n s f e r f u n c t i o n was d i s p l a y e d on CRT s c r e e n and s e n t t o the Vax™ 1 1 / 7 5 0 as a d i g i t i z e d d a t a f i l e f o r f u r t h e r p r o c e s s i n g . T h i s t e s t was r e p e a t e d f o r a l l t e n p o s s i b l e g a i n s on c h a n n e l s 1 and 5 o f t h e s i g n a l c o n d i t i o n e r . A n a l y s i s was c a r r i e d o u t on e a c h o f t h e t w e n t y t e s t s i n d i v i d u a l l y and a - 89 -l e a s t s q u a r e s a n a l y s i s d e t e r m i n e d a l i n e a r f u n c t i o n r e l a t i n g t h e f r e q u e n c y and p h a s e a n g l e . P e r f o r m i n g a s t a t i s t i c a l a n a l y s i s on t h e s l o p e s o f t h e s e r e l a t i o n s an a v e r a g e s l o p e o f - 2 . 7 9 7 d e g / H z w i t h a s t a n d a r d d e v i a t i o n o f .068 d e g / H z was d e t e r m i n e d . T h i s i n d i c a t e s t h a t t he p h a s e s h i f t i n t r o d u c e d was n o t s i g n i f i c a n t l y r e l a t e d t o t h e c h a n n e l number o r g a i n s e t t i n g . F u r t h e r m o r e , t h e s e r e s u l t s i n d i c a t e a c o n s i s t e n t l i n e a r r e l a t i o n s h i p b e t w e e n p h a s e l a g and f r e q u e n c y . The l e a s t s q u a r e s a n a l y s i s o f a l l t h e d a t a y i e l d e d t h e f o l l o w i n g r e l a t i o n s h i p . 4> = - 2 . 7 6 3 f - .598 . . . ( 5 . 2 . 2 . 1 . 1 - 1 ) W h e r e , <f> = 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 ) ; f = f r e q u e n c y o f s i g n a l , ( H z ) . When u s i n g t h e s i g n a l c o n d i t i o n e r l a t e r i n t h e h y d r o d y n a m i c t e s t s i t was p r i m a r i l y p r o c e s s i n g a s i g n a l w i t h one dom inan t f r e q u e n c y . T h a t w a s , t h e o p e r a t i n g f r e q u e n c y as s e l e c t e d b y t h e e x p e r i m e n t e r . S i n c e a l l c h a n n e l s were r e c o r d i n g s i g n a l s o f s i m i l a r f r e q u e n c y t h e p h a s e l a g i n t r o d u c e d was v i r t u a l l y t h e same f o r e a c h c h a n n e l . The i m p o r t a n t f a c t o r i n t h e h y d r o d y n a m i c e x p e r i m e n t s was f o r 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 c h a n n e l s t o be c o n s e r v e d . S i n c e e a c h c h a n n e l was f o u n d t o l a g b y t h e same amount i t was c o n c l u d e d t h a t 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 . - 90 -5 . 2 . 2 . 2 MINC™ 11 MINI COMPUTER A l l d a t a g a t h e r i n g was c o n d u c t e d u s i n g t he MINC 11 m i n i - c o m p u t e r i n t he Ocean E n g i n e e r i n g C e n t e r . The s y s t e m u s e s d u a l f l o p p y d i s k d r i v e s r e q u i r i n g 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 c o n d u c t i n g t h e h y d r o d y n a m i c t e s t s 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 c o n t a i n e d on one d i s k e t t e and t h e d a t a f i l e s we re s t o r e d on t h e o t h e r . The c o m p u t e r i t s e l f u s e s R T - 1 1 f o r m a t t i n g w h i c h makes i t c o m p a t i b l e w i t h t h e D i g i t a l PDP™ 11 s e r i e s c o m p u t e r s . 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 module c o n t a i n s 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 t e r h a r d w a r e w h i c h t r a n s l a t e s t he v o l t a g e o u t p u t f r o m e a c h t r a n s d u c e r i n t o a d i g i t a l number w h i c h t h e compu te r c a n u n d e r s t a n d and w r i t e t o t h e d i s k e t t e . The A / D c o n v e r t e r i s c a p a b l e o f r e a d 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 . E a c h o f t h e 16 p o r t s w i l l 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. I t c o n v e r t s t h e s e v o l t a g e s t o an i n t e g e r number b e t w e e n 0 and 4096 by t h e r e l a t i o n d e s c r i b e d i n E q u a t i o n ( 5 . 2 . 2 . 2 - 1 ) . I - I N T ( 4 0 0 . * V ) + 2048 . . . ( 5 . 2 . 2 . 2 - 1 ) Where , I = i n t e g e r v a l u e ; V = a n a l o g u e v o l t a g e ; 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 . The c o m p u t e r a l s o u s 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 w h i c h a l l o w s one t o v i s u a l l y d i s p l a y t h e r e c o r d e d s i g n a l s . I n a d d i t i o n t h e r e e x i s t s - 91 -a t t h e OEC a T e k t r o n i x s c r e e n dump p r i n t e r w h i c h d u p l i c a t e s t h e c o n t e n t s o f t h e v i d e o s c r e e n , and a DEC l i n e p r i n t e r , f o r o b t a i n i n g more s t a n d a r d o u t p u t s . T h e s e p r i n t e r s , h o w e v e r , were n o t e x t e n s i v e l y u s e d b y t he e x p e r i m e n t e r d u r i n g t h e c o u r s e o f t e s t i n g . 5 . 2 . 3 CYLINDER MODELS H y d r o d y n a m i c t e s t i n g was c o n d u c t e d on t h r e e c y l i n d e r m o d e l s . These m o d e l s h a v e a n i d e n t i c a l geome t r y t o t h o s e u s e d b y V e n u g o p a l i n 1982 and 1983 f o r s i m i l a r e x p e r i m e n t s w i t h heave m o t i o n . The m o d e l s were c o n s t r u c t e d i n t h e m a c h i n e shop o f t h e Depa r tmen t o f M e c h a n i c a l E n g i n e e r i n g a t UBC. The geomet r y o f e a c h c y l i n d e r mode l i s p r o v i d e d i n F i g u r e s 8 . 2 , 8 . 3 and 8 . 4 and p h o t o g r a p h s o f t h e s i n g l e , d o u b l e and t r i p l e c y l i n d e r m o d e l s may be f o u n d i n F i g u r e s 7 . 1 2 , 7 . 1 3 and 7 . 1 4 . The s i n g l e c y l i n d e r c o n s i s t e d o f a 455 mm s e c t i o n o f 384 mm PVC t u b i n g w i t h a lum inum end p l a t e s . The d o u b l e c y l i n d e r c o n s i s t e d o f e s s e n t i a l l y t h e same c y l i n d e r s e c t i o n w i t h t o p p l a t e b e i n g r e p l a c e d b y a n a luminum c y l i n d r i c a l s e c t i o n 613 mm h i g h and 219 mm i n d i a m e t e r . The t r i p l e c y l i n d e r was made b y r e p l a c i n g t h e b o t t o m p l a t e o f t h e d o u b l e c y l i n d e r w i t h a s i m i l a r a lum inum u n i t t o t h a t on t o p . 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 t r i p l e c y l i n d e r i s 1 .38 m. TM E a c h c y l i n d e r mode l was k e p t w a t e r t i g h t b y B u n a - N O - r i n g m a t e r i a l w h i c h f i t t e d i n t o m a c h i n e d 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 . A l l t h e c y l i n d e r s were h e l d t o g e t h e r b y a c o m p r e s s i v e f o r c e w h i c h was a p p l i e d b y 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 . These r o d s - 92 -were a t t a c h e d t o t h e b o t t o m p l a t e and e x t e n d e d up a l o n g t h e e n t i r e l e n g t h o f t h e c y l i n d e r m o d e l t h r o u g h t h e t o p p l a t e . The t h r e a d e d r o d s a l s o 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 t o the d e s i r e d d r a f t . The r o d s , w h i c h e x t e n d e d a b o u t 160 mm o u t o f t h e t o p p l a t e , we re o r i g i n a l l y t e n s i o n e d b y h e x a g o n a l n u t s a g a i n s t t h e t o p p l a t e . The a d a p t e r b l o c k was t h e n a t t a c h e d w i t h f o u r p a i r s o f h e x a g o n a l n u t s t o t h e s t i c k - u p s e c t i o n o f t h r e a d e d r o d . The a d a p t e r b l o c k was t h e n c o n n e c t e d t o t he dynamometer w h i c h was i n t u r n 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 . The a d a p t e r b l o c k c o u l d be s e t anywhere a l o n g t h e s t i c k - u p s e c t i o n o f t h e t h r e a d e d r o d by s i m p l y t i g h t e n i n g n u t s on b o t h s i d e s o f i t s b a s e p l a t e . F i g u r e s 7 . 1 2 and 7 .16 show t h i s a r r a n g e m e n t . I t was o b s e r v e d , h o w e v e r , w i t h t h e d o u b l e and t r i p l e c y l i n d e r mode l s t h a t t h e r e was n o t i c e a b l e f l e x i b i l i t y i n t h e s y s t e m . T h e r e was c o n c e r n t h a t t h i s e l a s t i c i t y c o u l d e f f e c t t h e r e s u l t s o b t a i n e d . The v i d e o t a p e s were r e p l a y e d and t h e s y s t e m r e s t u d i e d . I t was t h e n d e c i d e d t o wo rk w i t h t h e w e a k e s t l i n k i n t h e s y s t e m . T h a t was b e l i e v e d t o be t h r e a d e d r o d s t i c k - u p . A s t i f f s e c t i o n o f s t e e l p i p e was p l a c e d b e t w e e n t h e a d a p t e r b l o c k and t h e t o p p l a t e o f t h e c y l i n d e r . W i t h t h i s m o d i f i c a t i o n o n l y a s i n g l e n u t p e r t h r e a d e d r o d was r e q u i r e d t o p r o v i d e t h e c o m p r e s s i v e l o a d w h i c h f i x e d t h e c y l i n d e r t o t h e a d a p t e r b l o c k . T h i s more r i g i d a s s e m b l y g r e a t l y r e d u c e d t h e e x i s t e n c e 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 a d a p t e r b l o c k and t h e c y l i n d e r . F o u r s t i f f e n i n g c o l l a r s o f v a r y i n g t h i c k n e s s e s were made up a l l o w i n g t h e d r a f t t o be v a r i e d t o i n t e r m e d i a t e s e t t i n g s . I n t h i s c o n f i g u r a t i o n t h e f i n a l 6 s e r i e s o f 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 t h e t r i p l e c y l i n d e r . - 93 -5 . 2 . 4 INSTRUMENTATION USED 5 . 2 . 4 . 1 LOAD C E L L DYNAMOMETER I n p r e v i o u s h y d r o d y n a m i c t e s t s , V e n u g o p a l (1984b) u s e d one o f two dynamometers d e s i g n e d and c o n s t r u c t e d b y BC R e s e a r c h . T h e r e e x i s t e d a 350 N and a 3500 N u n i t . Upon i n s p e c t i o n i n J u l y o f 1985 damage t o b o t h u n i t s was o b s e r v e d i n c l u d i n g c r a c k s on t h e l o a d b e a r i n g e l e m e n t s . A n a t t e m p t was made t o c a l i b r a t e t h e h e a v i e r u n i t b u t n o n - r e p e a t a b l e r e s u l t s were f o u n d e v e n a f t e r a c a l i b r a t i o n was c o n d u c t e d w i t h t h e s u r g e f o r c e and p i t c h moment c o u p l e d . I t was t h e r e f o r e n e c e s s a r y , t o a c q u i r e a new dynamometer f o r t h e e x p e r i m e n t s . I t was a n t i c i p a t e d t h a t t h e maximum l o a d i n g w o u l d be o f t h e o r d e r o f 1000 N o f s u r g e f o r c e c o m b i n e d w i t h a p i t c h moment o f abou t 2000 N-m. F o l l o w i n g a s u r v e y o f what was c o m m e r c i a l l y a v a i l a b l e i t was e s t a b l i s h e d t h a t t h e r e was no s u i t a b l e p r o d u c t r e a d i l y a v a i l a b l e on t o d a y ' s m a r k e t . 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 t d . o f V a n c o u v e r , BC were t h e r e f o r e c o m m i s s i o n e d t o d e s i g n and w i r e a s u i t a b l e dynamometer . Two s u c h u n i t s we re made a l l o w i n g one t o f u n c t i o n as a b a c k u p f o r t h e o t h e r . The a c t u a l m a c h i n i n g and f a b r i c a t i o n o f t h e components was c o n d u c t e d i n t he m a c h i n e shop o f t h e Depa r tmen t o f M e c h a n i c a l E n g i n e e r i n g a t UBC. By Sep tember 3 0 , 1985 t h i s new dynamometers were d e l i v e r e d and c a l i b r a t e d . 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 v i s i b l e i n F i g u r e s 7 . 1 5 and 7 . 2 2 and 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 8 . 9 . The gauges - 94 -a r e e p o x y s u b s t a t e c o n s t a n t r o s e t t e s epoxy bonded t o a m a c h i n e d 7075-T6 a lum inum b l a n k . The l o a d c e l l s have b a s i c p r o t e c t i o n 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 s a r e as f o l l o w s : TABLE 5 . 2 . 4 . 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 s 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 3 , 3 3 0 N 1 3 , 0 5 0 N-m 1 2 , 4 5 0 N 4 4 , 4 7 0 N 2 7 , 1 1 0 N-m 2 , 0 3 0 N-m c o m b i n e d i n d i v i d u a l i n d i v i d u a l i n d i v i d u a l i n d i v i d u a l S u r g e / P i t c h 2% O 1% F S / 4 C a l l b r i d g e s o 1% p e r 2 2 . 2 C .1% 20 m i n 10 VDC 350 n 5 . 2 . 4 . 1 . 1 DYNAMOMETER CALIBRATION The dynamometer was s t a t i c a l l y c a l i b r a t e d i n t h e s t r e n g t h l a b o r a t o r y o f 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 a t UBC. The c a l i b r a t i o n was c o n d u c t e d on a T u n i s - O l s e n U n i v e r s a l T e s t i n g Machine™ u s i n g p u r p o s e b u i l t a t t a c h m e n t s w h i c h a l l o w e d one t o a p p l y s u r g e , h e a v e and p i t c h l o a d i n g - 95 -i n d e p e n d e n t l y . By r e c o n f i g u r i n g t h e a t t a c h m e n t s i t was p o s s i b l e t o a p p l y s i m u l t a n e o u s s u r g e and p i t c h l o a d i n g o r s i m u l t a n e o u s h e a v e and p i t c h l o a d i n g . The ST41B™ s i g n a l c o n d i t i o n e r was u s e d i n t he c a l i b r a t i o n s e t u p . The e x c i t a t i o n v o l t a g e was s e t t o 1 0 . 0 v o l t s and t he g a i n t o 1 0 0 0 . The o u t p u t f r o m t h e s i g n a l c o n d i t i o n e r was r e c o r d e d and t he d a t a t r a n s f e r r e d t o t he Vax™ 1 1 / 7 5 0 c o m p u t e r a t UBC. A n o v e r v i e w o f t h i s c a l i b r a t i o n s e t - u p i s p r o v i d e d i n F i g u r e 7 . 2 1 . The a n a l y s i s o f t h e c a l i b r a t i o n was c a r r i e d o u t u s i n g t h r e e m e t h o d s . The f i r s t method assumes e a c h c h a n n e l i s e n t i r e l y i n d e p e n d e n t . F o r e x a m p l e , i f a s u r g e f o r c e i s a p p l i e d t o t h e c e n t e r l i n e o f t h e l o a d c e l l i t w i l l have no e f f e c t on t h e o t h e r c h a n n e l s . The s e c o n d l e v e l o f a n a l y s i s assumes t h a t o n l y s u r g e and p i t c h a r e c o u p l e d b u t h e a v e i s i n d e p e n d e n t . T h i r d l y , one i n v e s t i g a t e s t h e c a s e w h i c h assumes a l l c h a n n e l s a r e c o u p l e d . I n e a c h c a s e t h e r e s u l t i n g t r a n s f o r m a t i o n m a t r i x i s s q u a r e w i t h t h e number o f unknown te rms e q u a l t o t h e s q u a r e o f t h e d e g r e e o f c o u p l i n g a s s u m e d . F o r o r d e r s g r e a t e r t h a n one t h e p r o c e d u r e f o r d e t e r m i n i n g t h e unknown m a t r i x r e q u i r e s t h e u s e o f s i n g u l a r v a l u e d e c o m p o s i t i o n methods w h i c h d e t e r m i n e t h e b e s t s o l u t i o n f r o m a n o v e r d e t e r m i n a n t s y s t e m . To a s s i s t w i t h t h i s a n a l y s i s t he s u b r o u t i n e SVD was t r a n s f e r r e d f r o m t h e MTS™ c o m p u t e r s y s t e m a t UBC t o t h e Vax™ 1 1 / 7 5 0 i n 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 . The r e s u l t s o f t h e a n a l y s i s a r e c o n t a i n e d i n T a b l e 5 . 2 . 4 . 1 . 1 - 1 . - 96 -T a b l e 5 . 2 . 4 . 1 . 1 - 1 R e s u l t s o f Dynamometer S t a t i c C a l i b r a t i o n U n i t Deg ree o f Max E r r o r # C o u p l i n g S u r g e Heave P i t c h (N) (%FS) (N) (%FS) (N-m) (%FS) 1 1 3 2 . 7 1.4 - - -2 1 4 4 . 0 2 . 0 1 2 1 8 . 8 0 . 8 - - 4 0 . 7 2 . 0 2 2 6 1 . 6 2 . 7 - - 5 9 . 8 2 . 9 1 3 2 8 . 0 1.2 5 2 4 . 8 4 . 2 4 3 . 9 2 . 1 2 3 7 2 . 0 3 . 2 4 6 1 . 2 3 .7 7 7 . 3 5 .1 T h e s e r e s u l t s i n d i c a t e t h a t t h e dynamometer d e s i g n a t e d as u n i t # 1 e x h i b i t e d t h e b e s t r e s u l t s on t h e s u r g e c h a n n e l when c o u p l e d w i t h p i t c h . The g r e a t e s t e r r o r was f o u n d t o be 0.8% o f f u l l s c a l e . The RMS e r r o r f o r t h e s u r g e f o r c e was f o u n d t o be 6 . 9 N (0 .3% FS) w i t h a s t a n d a r d d e v i a t i o n o f 8 . 5 N (0.4% F S ) . F u r t h e r m o r e , i t was f o u n d t h a t t h e h e a v e c h a n n e l p r o v i d e d o u t p u t t h a t was e s s e n t i a l l y m e a n i n g l e s s . The a d d i t i o n o f s t r a i n gages t o measu re h e a v e was a n a f t e r t h o u g h t t o t h e o r i g i n a l d e s i g n . I t was a n t i c i p a t e d t h a t t h e s e gages m i g h t p r o v i d e a weak and n o i s y o u t p u t s i g n a l b u t f o r t h e s l i g h t i n c r e m e n t a l c o s t i t was d e c i d e d t o i n s t a l l t h e s e gages and e v a l u a t e t h e i r m e r i t t h r o u g h a c a l i b r a t i o n . The r e s u l t s f r o m t h i s c a l i b r a t i o n i n d i c a t e d t h a t t h e r a t i o o f n o i s e , c r o s s t a l k and c r e e p t o t h e a c t u a l l o a d i n d u c e d s i g n a l was so h i g h i t was n o t p o s s i b l e t o i s o l a t e t h e l o a d i n g f o r c e f r o m t h e o t h e r e f f e c t s . On t h e b a s i s o f t h e s e r e s u l t s i t was d e c i d e d t h a t dynamometer #1 w o u l d be u s e d f o r a l l e x p e r i m e n t s . F u r t h e r m o r e , i t was assumed t h a t t h e s u r g e and p i t c h o u t p u t s i g n a l s a l o n e were c o u p l e d and c o u l d be d e c o u p l e d i n t o a s u r g e f o r c e and p i t c h moment b y t h e f o l l o w i n g m a t r i x e q u a t i o n . - 97 -" F " 2 3 3 . 2 1 2 0 . 1 0 " " V s s M - 3 . 0 0 2 2 0 . 5 8 V P P Where , F = s u r g e f o r c e , N ; M = p i t c h moment, N-m; p V = o u t p u t v o l t a g e f r o m s u r g e c h a n n e l ; s V = o u t p u t V o l t a g e f r om p i t c h c h a n n e l . The f o r c e and moments c o r r e s p o n d i n g t o z e r o v o l t s a r e assumed t o be z e r o . Any i n i t i a l v o l t a g e w h i c h was p r e s e n t was l a t e r removed b y a s u b r o u t i n e i n 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 . T h i s t o p i c i s f u r t h e r d i s c u s s e d i n S e c t i o n 6 . 2 . 1 . 4 . As a f u r t h e r c h e c k t h e dynamometer was a l s o c a l i b r a t e d d y n a m i c a l l y . T h i s was a l e s s e x h a u s t i v e c a l i b r a t i o n and s e r v e d e s s e n t i a l l y as a c h e c k o f t h e d a t a a n a l y s i s p r o c e d u r e and t h e s t a t i c c a l i b r a t i o n . The dynamic c a l i b r a t i o n c o n s i s t e d o f s e t t i n g up t h e c o m p l e t e m o t i o n g e n e r a t o r a s s e m b l y w i t h t h e s i n g l e c y l i n d e r mode l a t t a c h e d b u t t h e c y l i n d e r was t o t a l l y o u t o f t h e w a t e r . A t e s t was c o n d u c t e d i n t he n o r m a l manner a t two f r e q u e n c i e s , 1 .77 Hz and 2 . 5 5 H z . F o r e a c h c a s e t h e e r r o r i n measurement o f t h e s u r g e f o r c e was 4 . 4 N and 1.6 N w h i c h r e p r e s e n t s f u l l s c a l e e r r o r s o f .2% and .07% r e s p e c t i v e l y . The p h a s e a n g l e was w i t h i n two d e g r e e s o f t h e p o i n t o f maximum a c c e l e r a t i o n f o r b o t h c a s e s . T h e s e r e s u l t s i n d i c a t e t h a t t h e c a l i b r a t i o n was n o t f r e q u e n c y d e p e n d e n t f o r f r e q u e n c i e s up t o 2 . 5 5 H z . S i n c e t h e s t a t i c c a l i b r a t i o n was c o n d u c t e d u n d e r more c o n t r o l l e d c o n d i t i o n s t h e r e s u l t s o f i t s a n a l y s i s were u s e d f o r a l l h y d r o d y n a m i c t e s t s . - 98 -5 . 2 . 4 . 2 YO-YO POSITION TRANSDUCER The 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 mode l s was m e a s u r e d b y a y o - y o t ype p o s i t i o n t r a n s d u c e r . S i n c e t h e m o t i o n i s s i n u s o i d a l one c o u l d u s e the 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 = coX) and 2 a c c e l e r a t i o n (A = -w X) . The y o - y o p o t e n t i o m e t e r was f i x e d on t h e m o t i o n g e n e r a t o r f r ame and i t s r e c i p r o c a t i n g c a b l e was c o n n e c t e d t o t h e s l i d i n g p l a t e w h i c h was d i r e c t l y c o n n e c t e d t o t h e c y l i n d e r m o d e l . The c a l i b r a t i o n was c o n d u c t e d w i t h t h e c o m p l e t e s e t up 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 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 # 1 1 1 9 . I t was o r i g i n a l l y s o l d w i t h t h e p o t e n t i o m e t e r f o r m i n g a p a r t o f a W h e a t s t o n b r i d g e c i r c u i t . F o r t h i s a p p l i c a t i o n t he W h e a t s t o n b r i d g e c i r c u i t was b y - p a s s e d and t h e e x c i t a t i o n and s i g n a l were d i r e c t l y w i r e d t o t he p o t e n t i o m e t e r . I n a d d i t i o n t h e c a b l e was s h o r t e n e d f r o m i t s o r i g i n a l 1 .52 m l e n g t h t o a p p r o x i m a t e l y . 3 m due t o i t s d e t e r i o r a t i n g c o n d i t i o n . The 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 : - 99 -TABLE 5 . 2 . 4 . 2 - 1 YO-YO POTENTIOMETER SPECIF ICATIONS Maximum E x c i t a t i o n 15 VDC o r AC L i n e a r i t y ± .065% FS Maximum A c c e l e r a t i o n o f A c t u a t i n g C a b l e 4 g ' s C o l o r & P i n Code P i n 1 2 3 4 C o l o u r r e d b rown y e l l o w o r a n g e F u n c t i o n P+ S+ P - S -5 . 2 . 4 . 3 PRESSURE TRANSDUCERS A n a r r a y o f t h r e e d i a p h r a g m t y p e p r e s s u r e t r a n s d u c e r s was u s e d t o measu re p r e s s u r e a l o n g t h e t o p s i d e o f t h e h o r i z o n t a l s u r f a c e b e t w e e n t h e t o p mos t c y l i n d e r s e c t i o n and t h e l a r g e r c y l i n d e r s e c t i o n j u s t b e n e a t h i t . P r e s s u r e t r a n s d u c e r s were n o t u s e d f o r t h e s i n g l e c y l i n d e r t e s t s . The l o c a t i o n o f e a c h p r e s s u r e t r a n s d u c e r i s i n d i c a t e d i n F i g u r e 8 . 5 . The p r e s s u r e TM t r a n s d u c e r s u s e d were M o d e l AB u n i t s w i t h a r a n g e f r o m 0 t o 4 4 . 1 KPa (6 p s i g ) and a r e m a n u f a c t u r e d b y D a t a I n s t r u m e n t s I n c . O r i g i n a l l y t he p r e s s u r e t r a n s d u c e r s were l o c a t e d a l o n g t h e b a s e s u r f a c e o f e a c h t h e d o u b l e and t r i p l e c y l i n d e r mode ls b u t p r i o r t o c o n d u c t i n g any o f t h e s e t e s t s t he p r e s s u r e t r a n s d u c e r s were r e l o c a t e d t o a h i g h e r d r a f t so as t o g i v e g r e a t e r r e s o l u t i o n o f t h e h y d r o d y n a m i c p r e s s u r e . The p r i n c i p a l s p e c i f i c a t i o n s o f t h e p r e s s u r e t r a n s d u c e r s as s u p p l i e d by t h e m a n u f a c t u r e r a r e : 100 -TABLE 5 . 2 . 4 . 3 - 1 PRESSURE TRANSDUCER SPECIF ICATIONS S i g n a l o u t p u t E x c i t a t i o n V o l t a g e S e n s i t i v i t y 100 mV a t r a t e d p r e s s u r e , ± 1% 5 VDC 20 mV/V A c c u r a c y 1 % o f b e s t f i t l i n e R e s o l u t i o n i n f i n i t e C o l o r Code Red (P+ ) , G r e e n ( S + ) , W h i t e ( S - ) , B l a c k ( P - ) 5 . 3 . 4 . 4 TWO WIRE WAVE PROBE When two p a r a l l e l w i r e s a r e l o w e r e d v e r t i c a l l y i n t o w a t e r , t he c o n d u c t a n c e b e t w e e n them i s p r o p o r t i o n a l t o t h e d e p t h o f i m m e r s i o n and t he c o n d u c t i v i t y o f t h e w a t e r . The c o n d u c t a n c e a n d , h e n c e , i m m e r s i o n o r wave h e i g h t c a n be m e a s u r e d b y a p p l y i n g a p o t e n t i a l d i f f e r e n c e b e t w e e n t h e w i r e s and m e a s u r i n g t h e c u r r e n t w h i c h f l o w s . I n a l l t e s t s t h e same wave p r o b e was u s e d . H o w e v e r , f o r t h o s e t e s t s c o n d u c t e d i n O c t o b e r 1985 ( s e e T a b l e 2 . 2 - 1 ) , t h e wave p r o b e f o r m e d t h e f o r t h arm o f a W h e a t s t o n b r i d g e c i r c u i t . S l a d k e y (1985) r e p o r t e d w i t h 95% c o n f i d e n c e t h a t t h i s wave p r o b e d e s i g n was a c c u r a t e t o w i t h i n ± 2 . 7 mm. F o r a l l o t h e r t e s t s , a more s o p h i s t i c a t e d c i r c u i t d e s i g n was e m p l o y e d . T h i s d e s i g n d i s c u s s e d b y F r y e r and Thomas (1974) much i m p r o v e d t h e l i n e a r i t y o f t h e c a l i b r a t i o n . I t i s e s t i m a t e d t h a t t h i s wave p r o b e h a d a n a c c u r a c y o f ± 1 mm f o r t h e r a n g e o f wave h e i g h t s m e a s u r e d . The wave p r o b e i t s e l f was p o w e r e d b y a ±15 VDC e x c i t a t i o n s i g n a l . The o u t p u t s i g n a l w i t h o u t a m p l i f i c a t i o n c o u l d p r o v i d e an a c c e p t a b l e d e g r e e o f r e s o l u t i o n t o t he - 101 -MINC 11 m i n i - c o m p u t e r . N e v e r t h e l e s s , i t was n e c e s s a r y t o p a s s i t t h r o u g h the s i g n a l c o n d i t i o n e r , w i t h a g a i n s e t t i n g o f 1, t o c o n s e r v e p h a s e a n g l e r e l a t i o n s h i p s . - 102 -APPENDIX B 6. SOFTWARE USED IN EXPERIMENTS The s o f t w a r e u s e d i n t h e s e e x p e r i m e n t s may be d i v i d e d i n t o two d i s t i n c t g r o u p s : 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 and t h e 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 a t t h e OEC and was u s e d i n c o n j u n c t i o n w i t h t h e 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 a t BC R e s e a r c h . W h i l e t h e s y s t e m o p e r a t i o n a l l y w o r k e d v e r y w e l l t h e r e a r e some imp rovemen ts w h i c h t h i s a u t h o r w i l l recommend. These imp rovemen ts w i l l h e l p t o s a v e t i m e and i m p r o v e o v e r a l l e f f i c i e n c y i n c o n d u c t i n g e x p e r i m e n t s o f t h i s n a t u r e i n t h e f u t u r e . The d a t a a n a l y s i s s o f t w a r e was d e v e l o p e d o v e r s i x months a f t e r much e x p e r i m e n t a t i o n w i t h v a r i o u s t e c h n i q u e s . M o s t o f t h e p rog rams u s e d were w r i t t e n b y t h i s a u t h o r , w h i l e some u s e was made o f r o u t i n e s d e v e l o p e d b y o t h e r i n d i v i d u a l s . 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 f o r p r o c e s s i n g 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 he f i n a l r e s u l t s . I t was w r i t t e n t o r u n on t h e Vax™ 1 1 / 7 5 0 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 a t UBC. Two 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 . One 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 dynamic t e s t s 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 w h i l e t h e o t h e r p a c k a g e was d e s i g n e d t o p r o c e s s d a t a f r o m t h e i n c i d e n t wave t e s t s t o d e t e r m i n e t h e e x c i t i n g f o r c e on t h e s t r u c t u r e s . These two p a c k a g e s a r e q u i t e s i m i l a r and c a l l many o f t he same s u b r o u t i n e s . - 103 -6 . 1 DATA ACQUISIT ION SOFTWARE 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 r u n on a MINC 11 c o m p u t e r w h i c h has b e e n p r e v i o u s l y d e s c r i b e d i n S e c t i o n 5 . 2 . 2 . 2 . The d a t a was c o l l e c t e d on 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 f l o p p y d i s k e t t e s and was l a t e r t r a n s f e r r e d TM t o t h e V a x 1 1 / 7 5 0 i n t he 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 , where i t was a n a l y z e d . To g a t h e r t h e raw d a t a i t was n e c e s s a r y t o r u n t h e p rog ram ADMAIN f o r e v e r y t e s t . O t h e r p rog rams w h i c h were u s e f u l i n t h e d a t a c o l l e c t i o n p h a s e a r e a l s o d e s c r i b e d w i t h i n t h i s s e c t i o n . The d i s c u s s i o n w h i c h f o l l o w s i s n o t i n t e n d e d t o be 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 ; i t w i l l , h o w e v e r , g i v e an o v e r v i e w o f t h e p r o g r a m s u s e d . F u r t h e r i n f o r m a t i o n 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 BC R e s e a r c h . 6 . 1 . 1 ADMAIN PROGRAM 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 and i s c a p a b l e o f s a m p l i n g 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 p e c i f i e d f r e q u e n c y b e t w e e n 1 and 1000 H 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 . I n c l u d e d a r e t h e f o l l o w i n g : c a l i b r a t i o n f i l e name, 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 , samp le p e r i o d , d a t a f i l e s i z e ( w h i c h c o r r e s p o n d s t o t h e d u r a t i o n o f t h e t e s t ) , and a few o t h e r i n p u t s . I n c o n d u c t i n g t h e many h u n d r e d s o f t e s t s r e q u i r e d f o r t h i s p r o j e c t a l l b u t two i n p u t s r a r e l y c h a n g e d . T h i s 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 d a f t e r a l l t h e d a t a was g a t h e r e d f o r a s i n g l e t e s t . I t was a r a t h e r l a b o r i o u s p r o c e s s t o r e r u n t he p r o g r a m a n d r e e n t e r a l l t h e r e q u i r e d i n p u t s when , i n mos t c a s e s , o n l y t he d a t a f i l e name and t h e samp le p e r i o d c h a n g e d f r o m t e s t t o t e s t . I t w o u l d be d e s i r a b l e f o r a f u t u r e r e s e a r c h e r i n t h i s a r e a t o m o d i f y t h e s o f t w a r e t o - 104 -c o r r e c t t h i s i n a d e q u a c y . The 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 f o r m i n i m i z a t i o n o f s p a c e . To d e m u l t i p l e x t h e d a t a f r o m e a c h c h a n n e l i n t o r e a d a b l e ASCI code one must r u n a n o t h e r p r o g r a m , d e s c r i b e d s u b s e q u e n t l y , c a l l e d ADMUX. The ADMAIN p r o g r a m a l s o 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 s a m p l i n g . T h i s i s u s e f u l as a c h e c k t o e n s u r e each c h a n n e l i s r e c o r d i n g . W i t h t h i s u t i l i t y , h o w e v e r , t h e r e i s a l s o some d e b u g g i n g r e q u i r e d . The a x i s s c a l i n g i s i n c o r r e c t and i n a c a s e where the s i g n a l o u t p u t i s v e r y l o w t h e r e s o l u t i o n o f t h i s g r a p h i c u t i l i t y i s i n s u f f i c i e n t t o i n d i c a t e w h e t h e r o r n o t a s i g n a l i s p r e s e n t . I f i n d o u b t , h o w e v e r , i t i s p o s s i b l e t o d e m u l t i p l e x a c h a n n e l a f t e r t h e d a t a i s a c q u i r e d and c h e c k t h e o u t p u t q u i c k l y . 6 . 1 . 2 ADCAL PROGRAM ADCAL i s t h e 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 b e f o r e any a c t u a l d a t a a c q u i s i t i o n t a k e s p l a c e . 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 i s s i g n a l 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 5 . 2 . 2 . 2 . T h i s 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 e y a r e m e a s u r i n g . I n r u n n i n g t h i s p r o g r a m t h e i n s t r u m e n t a t i o n i s s e t up e x a c t l y as i t w o u l d be d u r i n g t e s t i n g . 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 t he a p p r o p r i a t e c h a n n e l number when p r o m p t e d f o r i t b y ADCAL. A t l e a s t 5 p o i n t s a r e r e q u i r e d f o r e a c h c a l i b r a t i o n . The o u t p u t o f ADCAL i s a 16x5 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 c h a n n e l s 0 t h r o u g h 15 c o n s e c u t i v e l y . The co lumns - 105 -f r o m l e f t t o r i g h t c o n t a i n t h e f o l l o w i n g p a r a m e t e r s : 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 he number o f s a m p l e s . ADCAL a l l o w s one 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 . F o r r u n n i n g ADMAIN o n l y , t h e i n f o r m a t i o n i n co lumns 1 and 2 i s u s e d . I f one h a s c o n d u c t e d a c a l i b r a t i o n w i t h o u t u s i n g ADCAL, a c a l i b r a t i o n f i l e c a n be m a n u a l l y e n t e r e d . F o r t h e t e s t s d e s c r i b e d i n t h i s document t h e s l o p e was s e t e q u a l to 1 a n d t h e i n t e r c e p t was s e t e q u a l t o z e r o . The c a l i b r a t i o n f a c t o r was i n t r o d u c e d l a t e r d u r i n g t h e d a t a a n a l y s i s p h a s e . T h i s a l l o w e d f o r maximum s i g n i f i c a n t f i g u r e s i n t h e d a t a f i l e s . 6 . 1 . 3 ADMUX PROGRAM ADMUX was 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 ADMAIN p rog ram and t o s t o r e t h e d a t a on u s e r 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 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 t o g i v e t h e o u t p u t i n u s e r u n i t s . The 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 t o d e m u l t i p l e x and t he 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 . E a c h c h a n n e l i s t o be d e m u l t i p l e x e d TM s e p a r a t e l y and t h i s p r o c e d u r e t a k e s c o n s i d e r a b l e t i m e on t h e MINC 1 1 . As o f May 1986 a new p r o g r a m was i n u s e a t t h e OEC w h i c h e x p e d i t e d t h i s p r o c e s s by a l l o w i n g one t o d e m u l t i p l e x a l l c h a n n e l s a t o n c e . T h i s p r o g r a m was r u n p e r i o d i c a l l y on d a t a f i l e s t o q u i c k l y c h e c k d a t a b e t w e e n t e s t s t o e n s u r e a l l c h a n n e l s were w o r k i n g . F o r t h e v a s t m a j o r i t y o f t e s t s h o w e v e r , t h i s p r o g r a m was n o t r u n . A t t h e d a t a a n a l y s i s p h a s e v e r y s i m i l a r s o f t w a r e was u s e d t o d e m u l t i p l e x e a c h c h a n n e l on t h e Vax 1 1 / 7 5 0 . - 106 -6 . 1 . 4 GRAPH PROGRAM The GRAPH 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 f r o m ADMUX on the v i d e o t e r m i n a l . T h i s a l l o w s one t o q u i c k l y see t h e q u a l i t y o f d a t a f r om any p a r t i c u l a r c h a n n e l . GRAPH c a n d i s p l a y X - Y d a t a o r e v e n l y s p a c e d Y d a t a w i t h a AX s p e c i f i e d . One c a n shade p o r t i o n s o f t h e g r a p h f i e l d t o t h e d a t a p o i n t s and t h e p r o g r a m c a n f i t a c u b i c s p l i n e t h r o u g h t h e d a t a p o i n t s . The u s e r c a n s p e c i f y a x i s l a b e l s , comments and a t i t l e i f d e s i r e d . The p r o g r a m a l s o has a u t o s c a l i n g w i t h t h e o p t i o n f o r t h e u s e r t o s p e c i f y t h e s c a l e . GRAPH t a k e s some t i m e t o r e a d t h e p o i n t s b u t w o r k s w e l l f o r a g e n e r a l l o o k a t raw d a t a . The a u t o s c a l i n g f e a t u r e does n o t u s u a l l y g i v e t h e maximum r e s o l u t i o n o f t h e r e s u l t s and manua l s c a l i n g i s f r e q u e n t l y n e c e s s a r y a f t e r t h e f i r s t v i e w o f t h e g r a p h i s d i s p l a y e d . 6 . 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 r e p r e s e n t s t h e r e s u l t o f e f f o r t s b y t h i s a u t h o r b e t w e e n November 1985 and A p r i l 1 9 8 6 . A p a r t i a l l y w o r k i n g v e r s i o n o f t h i s s o f t w a r e was i n p l a c e i n l a t e November b u t much e x p e r i m e n t a t i o n w i t h o t h e r t e c h n i q u e s was c o n d u c t e d b e f o r e a s a t i s f a c t o r y a n a l y s i s p r o c e d u r e was d e t e r m i n e d . T h i s f i n a l p r o g r a m was u s e d t o a n a l y z e a l l d a t a f i l e s . As was p r e v i o u s l y s t a t e d t h e d a t a a n a l y s i s f o r t h e dynam ic t e s t s and t h e i n c i d e n t wave t e s t s was c a r r i e d o u t b y 2 s e p a r a t e b u t s i m i l a r p r o g r a m s , c a l l e d DS ( f o r d a t a s o r t ) and DSEXF ( f o r d a t a s o r t , e x c i t i n g f o r c e ) , r e s p e c t i v e l y . E a c h p r o g r a m and s u b r o u t i n e i s documented i n d i v i d u a l l y . F o l l o w i n g i s a d e s c r i p t i o n o f how t h e p a c k a g e w o r k s as a u n i t , as w e l l as t h e f e a t u r e s o f t h e v a r i o u s - 107 -s u b r o u t i n e s u s e d . The r e a d e r i s r e f e r r e d t o F i g u r e s 8 . 9 and 8 . 1 0 f o r a f l o w c h a r t o f e a c h p r o g r a m p a c k a g e . 6 . 2 . 1 DS PROGRAM DS i s t h e m a i n p r o g r a m w h i c h p romp ts t h e u s e r f o r i n p u t and s e q u e n t i a l l y 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 romp ted f o r t h e f o l l o w i n g i n f o r m a t i o n : 1) Raw d a t a f i l e name 2) - C a l i b r a t i o n f i l e name 3) Name o f f i l e t o c o n t a i n r e s u l t s ( i . 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 ) 4 ) C y l i n d e r mode l t y p e ( i . e . s i n g l e , d o u b l e , 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 a s s o c i a t e d h a r d w a r e . 7) Mean d i s t a n c e f r o m t h e wave p r o b e t o t h e c y l i n d e r c e n t e r l i n e 8) F i l t e r f a c t o r , o p t i o n t o change f r o m d e f a u l t v a l u e o f f o u r (The m e a n i n g o f t h i s f a c t o r i s e x p l a i n e d i n S e c t i o n 6 . 2 . 1 . 6 ) 9) 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 i n p u t c o m b i n e d 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 the compu te 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 o t h e r p a r a m e t e r s . As h a s b e e n p r e v i o u s l y s t a t e d d u r i n g e a c h t e s t t h e f o l l o w i n g p a r a m e t e r s a r e b e i n g m e a s u r e d : c y l i n d e r d i s p l a c e m e n t , s u r g e f o r c e , p i t c h moment, wave h e i g h t , and t h e p r e s s u r e a t t h r e e p o i n t s on t h e c y l i n d e r s u r f a c e . To measure a l l o f t h e s e p a r a m e t e r s a t o t a l o f 7 c h a n n e l s a r e u t i l i z e d on t he MINC™ 11 m i n i - c o m p u t e r . - 108 -The u s e r s h o u l d n o t e t h a t o r i g i n a l l y t h i s p r o g r a m was s e t up t o sample 8 c h a n n e l s . H o w e v e r , i t was f o u n d t h a t t h e dynamometer was u n a b l e t o s a t i s f a c t o r i l y measure heave f o r c e . T h e r e f o r e , i n r u n n i n g t he d a t a c o l l e c t i o n p r o g r a m , i f u s i n g t h e recommended c h a n n e l a s s i g n m e n t s , t h e u s e r must assume t h a t t h e h e a v e c h a n n e l ( compu te r c h a n n e l #2) i s s a m p l i n g e v e n t h o u g h t h e d a t a g a t h e r e d on t h i s c h a n n e l i s n e v e r a c t u a l l y u s e d o r p r o c e s s e d i n any way. When ADMAIN p r o m p t s f o r t h e number o f c h a n n e l s u s e d , t h e u s e r must a l l o w f o r one e x t r a c h a n n e l t o e n s u r e t h a t a l l t h e c h a n n e l s a r e s a m p l e d , s i n c e t h e compute r assumes t h e c h a n n e l s u s e d s t a r t c o n s e c u t i v e l y f r o m 0 . I f t h e u s e r i s n o t u s i n g t h e recommended c h a n n e l a s s i g n m e n t s t h e n i t i s o n l y n e c e s s a r y t o e n s u r e t h a t t h e c h a n n e l s u s e d f o l l o w c o n s e c u t i v e l y f r o m 0 and t h a t t h e a s s i g n m e n t s u s e d may be e n t e r e d when DS i s r u n . The p r o g r a m assumes t h e u s e r has c o n n e c t e d e a c h t r a n s d u c e r t o a p a r t i c u l a r c h a n n e l on t h e c o m p u t e r as s p e c i f i e d i n T a b l e 6 . 2 . 1 - 1 . I f t he u s e r h a s n o t done so t h e n he o r she i s g i v e n t h e o p p o r t u n i t y t o e n t e r t h e new a s s i g n m e n t s . T a b l e 6 . 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 s U n i t s Assumed By " D S " Compute r 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 0 Y o - Y o P o s i t i o n T r a n s d u c e r C y l i n d e r P o s i t i o n mm 1 Dynamometer S u r g e O u t p u t S u r g e F o r c e N 3 Dynamometer P i t c h O u t p u t P i t c h Moment N-m 4 Wave P r o b e Wave H e i g h t mm 5 P r e s s u r e T r a n s d u c e r #1 H y d r o d y n a m i c P r e s s u r e P a 6 P r e s s u r e T r a n s d u c e r #2 H y d r o d y n a m i c P r e s s u r e P a 7 P r e s s u r e T r a n s d u c e r #3 H y d r o d y n a m i c P r e s s u r e P a A s T a b l e 6 . 2 . 1 - 1 i n d i c a t e s , t h i s p r o g r a m i s r u n w i t h t h e S I s y s t e m o f - 109 -u n i t s . A l l c a l i b r a t i o n s , t h e r e f o r e , s h o u l d c o n v e r t t h e o u t p u t s i g n a l i n t o t he u s e r u n i t s i d e n t i f i e d i n t h i s t a b l e . I f t h e c a l i b r a t i o n f i l e c r e a t e d f o r use w i t h 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 c o n v e r t s t he d a t a t o o t h e r u n i t s t h e n the s u b r o u t i n e CAL IB w i l l a l l o w one t o c o n v e r t t h e d a t a i n t o t h e recommended u n i t s . A s t h i s p r o g r a m i s r u n , i t 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 i n t e r m e d i a t e s t a g e s . These 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 l y e x t e n s i o n s t o t h e o r i g i n a l d a t a f i l e name. T a b l e 6 . 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 t h a t w o u l d be a p p l i e d t o a raw d a t a f i l e named DATA.DAT as c r e a t e d b y t h e d a t a a c q u i s i t i o n p r o g r a m , ADMAIN. T a b l e 6 . 2 . 1 - 2 F i l e Name E x t e n s i o n s U s e d b y " D S " 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 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 P o t e n t i o m e t e r DATA_SURG.DAT 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 DATA_PITC.DAT 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 DATA_WVPR.DAT D e m u l t i p l e x e d O u t p u t f r o m Wave P r o b e DATA_PRS1.DAT D e m u l t i p l e x e d O u t p u t f r o m P r e s s u r e T r a n s d u c e r #1 DATA_PRS2.DAT D e m u l t i p l e x e d O u t p u t f r o m P r e s s u r e T r a n s d u c e r #2 DATA_PRS3.DAT D e m u l t i p l e x e d O u t p u t f r o m P r e s s u r e T r a n s d u c e r #3 DATA_DISP_RAW.DAT Raw U n f i l t e r e d D a t a O r i g i n a l l y C o n t a i n e d i n DATA_DISP.DAT DATA_SURG_RAW.DAT Raw U n f i l t e r e d D a t a O r i g i n a l l y C o n t a i n e d i n DATA_SURG.DAT o o 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 . D A T A _ D I S P _ F F T . D A T 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 ime Domain D a t a C o n t a i n e d i n DATA_DISP.DAT DATA_SURG_FFT .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 Time Domain D a t a C o n t a i n e d i n DATA SURG.DAT o — o 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_TIME.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 he D a t a , where t h e M a g n i t u d e o f t he S u r g e F o r c e and P i t c h Moment i n Phase w i t h t h e V e l o c i t y and A c c e l e r a t i o n i s G i v e n . - 110 -DATA_FFT.DAT Summary o f t he 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 n a l y s i s C o n t a i n i n g t he Maximum R e c o r d e d A m p l i t u d e i n 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 Phase A n g l e A s s o c i a t e d w i t h t h i s Component . I n A d d i t i o n , t h i s F i l e C o n t a i n s t h e R e s u l t s o f R e a l T ime A n a l y s i s where t he M a g n i t u d e o f t h e S u r g e F o r c e and P i t c h Moment i n P h a s e w i t h t h e V e l o c i t y and A c c e l e r a t i o n i s G i v e n . 6 . 2 . 1 . 1 DEMUX SUBROUTINE DEMUX was d e v e l o p e d f r o m the ADMUX p r o g r a m 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 s o f t w a r e p a c k a g e . 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 . T h i s s u b r o u t i n e d e m u l t i p l e x e s , f r o m the o r i g i n a l raw d a t a f i l e , e a c h c h a n n e l s a m p l e d . 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 code t o 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 as s p e c i f i e d i n T a b l e 6 . 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 by i n c o r p o r a t i n g t h e c a l i b r a t i o n f i l e w h i c h was s p e c i f i e d when ADMAIN was o r i g i n a l l y r u n . 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 he number o f samp le p o i n t s , t h e samp le p e r i o d (msec) 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 e x p r e s s e d as an 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 . 6 . 2 . 1 . 2 CAL IB SUBROUTINE CAL IB a l l o w s one t o make f u r t h e r m o d i f i c a t i o n s t o t h e d a t a i n t he d e m u l t i p l e x e d d a t a f i l e s . I t m u l t i p l i e s t h e d a t a i n e a c h f i l e b y f a c t o r s w h i c h a r e s p e c i f i e d i n t he c a l i b r a t i o n f i l e . The m a i n p r o g r a m , DS, p romp ts t h e u s e r f o r t h e name o f t h i s c a l i b r a t i o n f i l e . - I l l -The d a t a s a m p l e d i s p e r i o d i c f o r a l l t h e e x p e r i m e n t s d i s c u s s e d i n t h i s t h e s i s . T h e r e f o r e , f o r l o n g samp le p e r i o d s t he d a t a w i l l have a mean v a l u e v e r y n e a r t o z e r o . T h i s means t h a t , i n c o n d u c t i n g a c a l i b r a t i o n , one need o n l y be c o n c e r n e d w i t h t h e s l o p e o f t h e l i n e a r r e l a t i o n s h i p r e l a t i n g the o u t p u t s i g n a l t o t h e m e a s u r e d p a r a m e t e r . I t i s n o t n e c e s s a r y t o know the y - i n t e r c e p t s i n c e t h i s r e p r e s e n t s a c o n s t a n t s i g n a l o f f s e t w h i c h i s removed b y t h e s u b r o u t i n e , TREND. 6 . 2 . 1 . 3 DYNO SUBROUTINE DYNO i s c a l l e d t o c o n v e r t t h e v o l t a g e o u t p u t f r o m t h e s u r g e and p i t c h c h a n n e l s o f t h e dynamometer t o S I f o r c e u n i t s ( i . e . N e w t o n s ) . The dynamometer was c a l i b r a t e d w i t h t h e a s s u m p t i o n t h a t t h e s u r g e and p i t c h o u t p u t s a r e l i n e a r l y c o u p l e d . T h e r e f o r e , r a t h e r t h a n a s t r a i g h t l i n e a r c a l i b r a t i o n c o n t a i n i n g a s i n g l e f a c t o r , a c a l i b r a t i o n m a t r i x i s u s e d t o d e c o u p l e s u r g e and p i t c h and c o n v e r t them t o t h e d e s i r e d u n i t s . D e t a i l s o f t h e dynamometer c a l i b r a t i o n a r e p r o v i d e d i n S e c t i o n 5 . 2 . 4 . 1 . 1 . T h i s s u b r o u t i n e r e a d s d a t a f r o m t h e f i l e s c o n t a i n i n g t h e s u r g e and p i t c h d a t a i n v o l t a g e u n i t s and w r i t e s t h e t r a n s f o r m e d d a t a b a c k i n t o f i l e s o f t h e same name i n u n i t s o f N e w t o n s . 6 . 2 . 1 . 4 TREND SUBROUTINE A s was s t a t e d p r e v i o u s l y i n t h e d e s c r i p t i o n o f t h e s u b r o u t i n e C A L I B , a l l o f t h e d a t a s a m p l e d s h o u l d have 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 d a t a was - 112 -s a m p l e d f o r a s u f f i c i e n t l y l o n g i n t e r v a l . I n t h e s e e x p e r i m e n t s a l l f i l e s were s a m p l e d a t a n o m i n a l r a t e o f 25 s a m p l e s p e r c y c l e o v e r 20 c y c l e s . TREND removes any t r e n d s f r o m the d a t a . T h i s i s c a r r i e d o u t by 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 l i n e f r o m t h e s e same d a t a p o i n t s . The 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 h o r i z o n t a l l i n e . H e n c e , one i s e s s e n t i a l l y r e m o v i n g a c o n s t a n t "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 h o r i z o n t a l 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 by n o t s t a r t i n g and s t o p p i n g a t t h e same p o i n t i n a c y c l e . The m a g n i t u d e o f t h i s e r r o r i s 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 and i s o f 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 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 "DC o f f s e t " due t o 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 . F o r c o m p a r i s o n p u r p o s e s , some d a t a was p r o c e s s e d b y s u b t r a c t i n g t h e mean o f a l l t h e d a t a f r o m e a c h d a t a p o i n t . The r e s u l t s o f t h i s method o f a n a l y s i s were compared w i t h t h e t r e n d r e m o v a l method and t h e r e s u l t s d i f f e r e d b y an i n s i g n i f i c a n t amount o f l e s s t h a n . 1 % . T h i s i n d i c a t e d t h a t t h e r e was no s t r o n g t r e n d s i n t h e d a t a p r o c e s s e d b y b o t h m e t h o d s . On t h e b a s i s o f t h i s i n v e s t i g a t i o n , i t was d e c i d e d t h a t t r e n d r e m o v a l , w h i l e n o t l i k e l y t o y i e l d s i g n i f i c a n t l y d i f f e r e n t r e s u l t s f r o m t h o s e o b t a i n e d b y s i m p l e mean v a l u e c o r r e c t i o n , was a more s o p h i s t i c a t e d and w i d e l y a c c e p t e d d a t a p r o c e s s i n g p r o c e d u r e . 6 . 2 . 1 . 5 COPY SUBROUTINE The COPY s u b r o u t i n e c o p i e s d a t a f r o m an i n p u t f i l e t o an o u t p u t f i l e w i t h a s p e c i f i e d name. The i n p u t f i l e must c o n t a i n d a t a w h i c h i s f o r m a t t e d i n - 113 -t h e same way as t h e o u t p u t f r o m DEMUX. I t i s u s e d t o copy d a t a b e f o r e f i l t e r i n g , so t h a t t h e raw d a t a o u t p u t i s k e p t . 6 . 2 . 1 . 6 F I L T E R SUBROUTINE 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 . D e t a i l s o f t h e f i l t e r i n g method u s e d a r e p r o v i d e d b y A u b a n e l and Oldham ( 1 9 8 5 ) . I n c l u d e d i n t h i s r e f e r e n c e i s a l i s t i n g o f a B a s i c p r o g r a m , w h i c h i n c o r p o r a t e s t h i s p r o c e d u r e . G e r r y R o h l i n g l a t e r c o n v e r t e d t h i s B a s i c p r o g r a m i n t o a F o r t r a n p r o g r a m and t h i s a u t h o r 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 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 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 i n t h e same f o r m a t as t h e o u t p u t f r o m DEMUX. I n a d d i t i o n , t h e u s e r must p r o v i d e a f i l t e r f a c t o r r a n g i n g b e t w e e n 2 and N , where 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 . By e x p e r i m e n t a t i o n w i t h t i m e doma in s i g n a l t r a c e s , i t was f o u n d t h a t a f i l t e r f a c t o r o f b e t w e e n 3 and 4 was most a p p r o p r i a t e . To 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 , b y d e f a u l t t o u s e a f i l t e r f a c t o r o f 4 b u t t he u s e r i s g i v e n t h e o p t i o n t o r e s p e c i f y i t s v a l u e i n i t i a l l y when DS i s r u n . I t was f o u n d t h a t t h i s f i l t e r f a c t o r was 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 Hz 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 rom 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 i t s v e r s i o n number i n c r e m e n t e d b y o n e . 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 . - 114 -6 . 2 . 1 . 7 FOURT SUBROUTINE 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 s u b r o u t i n e F F T , i n t r o d u c e d i n t he n e x t s e c t i o n , c a l l s FOURT. I t 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 f r o m the UBC TM TM MTS m a i n f r ame c o m p u t e r t o t he Vax 1 1 / 7 5 0 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 s u b r o u t i n e i s w e l l documen ted and more d e t a i l e d d o c u m e n t a t i o n 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 c o m p u t i n g c e n t e r a t UBC. B e f o r e t h i s s u b r o u t i n e o r any o t h e r FFT p r o g r a m c a n be a p p l i e d t o p e r i o d i c d a t a , t h e d a t a must be p r e p r o c e s s e d c o r r e c t l y f o r t h e o u t p u t t o be m e a n i n g f u l . I t i s i m p e r a t i v e t h a t t h e u s e r c l e a r l y u n d e r s t a n d t h e e f f e c t s o f t h e v a r i o u s d a t a p r e p a r a t i o n p r o c e d u r e s . M o s t d o c u m e n t a t i o n p r o v i d e d w i t h FFT p a c k a g e s f a i l s t o u n d e r l i n e t h e i m p o r t a n c e o f t h i s f a c t , so a b r i e f e x p l a n a t i o n i s p r o v i d e d h e r e . The F a s t F o u r i e r T r a n s f o r m i s 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 t a k e s a f i n i t e t i m e s e r i e s and assumes i t t o be r e p r e s e n t a t i v e o f a n i n f i n i t e l y l o n g s a m p l e . I n so d o i n g , i t assumes t h a t e a c h samp le i s r e p e a t e d f o r e v e r . F o r e x a m p l e , i f a samp le were o f 10 s e c o n d d u r a t i o n , t he 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 i n t e r v a l f r o m s e c o n d 0 t o s e c o n d 10 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 10 s e c o n d i n t e r v a l i s assumed t o be i d e n t i c a l t o t h e f i r s t . I f t h e a m p l i t u d e s o f t he end p o i n t s a r e n o t e q u a l , t h e n t h e jump d i s c o n t i n u i t i e s w h i c h e x i s t a t t h e s e p o i n t s i n t r o d u c e e r r o r s i n t o t he - 115 -t r a n s f o r m a t i o n . I f , i n f a c t , 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 m a t h e m a t i c a l l y p r o v e n t h a t t h e FFT g e n e r a t e s an i d e n t i c a l s p e c t r u m t o t he 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 n e x c e l l e n t d i s c u s s i o n o f t h e FFT i s c o n t a i n e d i n the 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 ) . To s o l v e t h i s p r o b l e m o f end e f f e c 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 s e d . I n t h i s c a s e , 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 whe reby t h e d a t a i s t a p e r e d 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 and t o p u t i n c r e a s i n g l y 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 t o w a r d 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 w i l l c l o s e l y a p p r o x i m a t e 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 . R a m i r e z (1985) t a b u l a t e s many o f t h e window f u n c t i o n s w h i c h a r e commonly u s e d and shows t h e e f f e c t e a c h h a s on t h e r e s u l t i n g s p e c t r u m . I t i s n o t n e c e s s a r y t o w indow d a t a c o n t a i n i n g an i n t e g e r number o f c y c l e s s i n c e b e g i n n i n g and end p o i n t s w i l l have 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 d o m i n a n t f r e q u e n c y , i t i s a s i m p l e m a t t e r o f o b s e r v i n g z e r o c r o s s i n g , 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 does n o t f o r m a c o m p l e t e c y c l e . F o r more c o m p l i c a t e d d a t a , h a v i n g more t h a n one p r e d o m i n a n t f r e q u e n c y , t h i s p r o c e d u r e w i l l wo rk l e s s e f f e c t i v e l y and w i n d o w i n g s h o u l d be be u s e d . - 116 6 . 2 . 1 . 8 FFT SUBROUTINE FFT p e r f o r m s t h r e e f u n c t i o n s : i t f u r t h e r p r e p a r e s d a t a f o r u s e b y the FOURT s u b r o u t i n e , i t c a l l s t he FOURT s u b r o u t i n e and i t 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 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 e o u t p u t f r o m DEMUX. T h i s d a t a h a s p r e d o m i n a n t l y one f r e q u e n c y . F o r r e a s o n s 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 t i s , t h e r e f o r e , n o t n e c e s s a r y t o w indow t h e d a t a . I n s t e a d , FFT r e a d s t h e d a t a and t h e l o c a t i o n o f z e r o c r o s s i n g s i s n o t e d . N e x t , 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 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 a n a r r a y o f c o m p l e x n u m b e r s . The 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 r u n c a t i o n o f t he o r i g i n a l d a t a . T h i s o u t p u t i s t h e n 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 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 ) . 6 . 2 . 1 . 9 REALTIME SUBROUTINE REALTIME u s e s a r i g o r o u s a p p r o a c h 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 s u r g e and p i t c h 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 . S i n c e t h e m o t i o n was s i n u s o i d a l w i t h 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 p o s s i b l e t o c l e a r l y i d e n t i f y p o i n t s o f maximum a c c e l e r a t i o n and v e l o c i t y f rom t h e d i s p l a c e m e n t t r a c e . As was p r e v i o u s l y s t a t e d , a t s u c h 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 he e f f e c t s o f added mass and d a m p i n g , s i n c e when a c c e l e r a t i o n i s maximum the v e l o c i t y i s z e r o and v i c e - v e r s a . T h i s s u b r o u t i n e - 117 -l o c a t e s a l l z e r o c r o s s i n g s ( i . e . p o i n t s o f maximum v e l o c i t y ) f r om the d i s p l a c e m e n t d a t a f i l e and r e a d s t h e c o r r e s p o n d i n g r e a d i n g s f r o m the s u r g e and p i t c h f i l e s . S i m i l a r l y , i t l o c a t e s p o i n t s o f maximum and minimum d i s p l a c e m e n t ( i . e . p o i n t s o f minimum and maximum a c c e l e r a t i o n ) and d e t e r m i n e s t h e c o r r e s p o n d i n g r e a d i n g s f r o m t h e s u r g e and p i t c h f i l e s . REALTIME w i l l i n t e r p o l a t e b e t w e e n p o i n t s t o more p r e c i s e l y l o c a t e p o i n t s o f i n t e r e s t . I n t h e c a s e o f d e t e r m i n i n g maxima o r m i n i m a , 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 was r e q u i r e d . I t was f o u n d a f t e r much e x p e r i m e n t a t i o n t h a t t h i s method y i e l d e d t h e mos t s a t i s f a c t o r y r e s u l t s . T h i s was p a r t i c u l a r l y e v i d e n t when the component i n p h a s e w i t h t h e maximum v e l o c i t y was n e a r z e r o . 6 . 2 . 1 . 1 0 BIG SUBROUTINE The s u b r o u t i n e , B I G , s c a n s e a c h o f t he o u t p u t f i l e s f r o m FFT and 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 and phase 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 f i l e and t h e r e s u l t i n g o u t p u t w h i c h summar i zes t h e FFT r e s u l t s i s c o n t a i n e d i n a u s e r s p e c i f i e d f i l e . The d a t a s a m p l e d b y t h e wave p r o b e r e q u i r e s s p e c i a l t r e a t m e n t i n t h a t i t i s l o c a t e d some d i s t a n c e away f r o m t h e c y l i n d e r c e n t e r l i n e . A f u r t h e r c o r r e c t i o n must be a p p l i e d t o t h i s p h a s e a n g l e o f t h e w a t e r waves m e a s u r e d . The s u b r o u t i n e s BIGWAVE and DELAY a r e u s e d f o r t h i s p u r p o s e . I n a d d i t i o n , t h e s u b r o u t i n e BIG w i l l 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 a n g l e b e t w e e n t h e d i s p l a c e m e n t c h a n n e l and e a c h o f t h e o t h e r c h a n n e l s . The s u r g e and p i t c h d a t a i s t r e a t e d i n a s i m i l a r manner b u t BIG a l s o w r i t e s t h e r e s u l t s f r o m REALTIME i n t o t h e o u t p u t f i l e . - 118 -6 . 2 . 1 . 1 1 BIGWAVE SUBROUTINE The s u b r o u t i n e , BIGWAVE', i s v e r y s i m i l a r t o B I G , e x c e p t i t i s s p e c i f i c a l l y d e s i g n e d t o r e t u r n t h e maximum wave a m p l i t u d e and t he c o r r e s p o n d i n g wave f r e q u e n c y and p h a s e a n g l e . The r e t u r n e d p a r a m e t e r s a r e n o t w r i t t e n t o a f i l e , b u t a r e t r a n s f e r r e d as v a r i a b l e s w h i c h a r e l a t e r p a s s e d t o DELAY t o c o r r e c t t h e wave p h a s e f o r t h e wave p r o b e p o s i t i o n . 6 . 2 . 1 . 1 2 DELAY SUBROUTINE A s p r e v i o u s l y s t a t e d , t h e s u b r o u t i n e , DELAY, d e t e r m i n e s t h e p h a s e l e a d o r l a g w h i c h a r i s e s f r o m t h e wave p r o b e b e i n g l o c a t e d a t some d i s t a n c e f r om t h e c y l i n d e r c e n t e r l i n e . L i n e a r wave t h e o r y i s a p p l i e d t o d e t e r m i n e t h e wave number . The p r o d u c t o f t h i s number and t h e mean 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 and t h e wave p r o b e i s e q u a l t o t h e p h a s e a n g l e c o r r e c t i o n . 6 . 2 . 1 . 1 3 COEF SUBROUTINE The s u b r o u t i n e , COEF, d e t e r m i n e s t h e f i n a l r e s u l t s f r o m t h e o u t p u t f i l e c r e a t e d b y BIG a n d f r o m i n f o r m a t i o n p r o v i d e d i n t h e i n i t i a l p r o m p t s b y DS. 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 d e t e r m i n e d b y a l g o r i t h m s d e s c r i b e d i n E q u a t i o n s 6 . 2 . 1 . 1 3 - 1 and 6 . 2 . 1 . 1 3 - 2 . - 119 -a 11 ( 6 . 2 . 1 . 1 3 - 1 ) Where , F b . . . ( 6 . 2 . 1 . 1 3 - 2 ) u> X a = added mass c o e f f i c i e n t ; n b = damping c o e f f i c i e n t ; F = su rge , 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 = s u r g e 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 as 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 = d i s p l a c e m e n t a m p l i t u d e as m e a s u r e d ; m = c y l i n d e r m a s s . 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 z e d i n t he a b u s u a l manner as — and ——— . p V pvw Where , p = d e n s i t y o f f l u i d medium; V = v o l u m e o f f l u i d d i s p l a c e d . The o p e r a t i n g f r e q u e n c y i s t a k e n f r o m 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 t h i s s i g n a l i s v i r t u a l l y n o i s e f r e e . The m e c h a n i c a l component w h i c h d e t e r m i n e s t h i s a m p l i t u d e i s a c c u r a t e t o t h o u s a n d t h s o f an i n c h . The f i l t e r i n g p r o c e s s , a l o n g w i t h o t h e r e x p e r i m e n t a l e r r o r s , does n o t a l l o w t h e t r a n s d u c e r t o measu re t h e d i s p l a c e m e n t a m p l i t u d e s t o t h i s d e g r e e o f a c c u r a c y . F o r t h i s r e a s o n , t h e d i s p l a c e m e n t a m p l i t u d e s , as d e t e r m i n e d b y t h e FFT p r o c e s s , i s - 120 -r o u n d e d t o t h e n e a r e s t 5 mm, s i n c e i t was p r e c i s e l y m a n u f a c t u r e d ( ± . 0 2 5 mm) t o h a v e s e t t i n g s w h i c h a r e i n t e g e r m u l t i p l e s o f 5 mm. COEF c a l l s t h e s u b r o u t i n e EXCITE t o d e t e r m i n e t h e damp ing c o e f f i c i e n t u s i n g W e h a u s e n ' s f o r m u l a t i o n . The o u t p u t i s w r i t t e n t o t h e f i l e w h i c h was s p e c i f i e d when DS was i n i t i a l l y r u n . I n a d d i t i o n t o 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 e o u t p u t f i l e a l s o c o n t a i n s i n f o r m a t i o n s u c h as t h e wave a m p l i t u d e and l e n g t h . 6 . 2 . 1 . 1 4 EXCITE SUBROUTINE As a l l u d e d t o i n t h e p r e v i o u s s e c t i o n , t h e s u b r o u t i n e , E X C I T E , d e t e r m i n e s t h e damp ing c o e f f i c i e n t f r o m W e h a u s e n ' s f o r m u l a t i o n . F o r an a x i s y m m e t r i c shape i n p u r e s u r g e , t h i s f o r m u l a t i o n may be s t a t e d as f o l l o w s : i i ( 6 . 2 . 1 . 1 4 - 1 ) 8 . pg V Where , k = wave number ; g = a c c e l e r a t i o n due t o g r a v i t y ; V — g roup v e l o c i t y o f t h e wave ; s A = wave a m p l i t u d e . 121 -6 . 2 . 2 DSEXF PROGRAM The p r o g r a m , DSEXF, i s q u i t e s i m i l a r t o t h e DS p r o g r a m d e s c r i b e d i n the l a s t s e c t i o n . The f u n d a m e n t a l d i f f e r e n c e b e t w e e n t he two p r o g r a m s i s t h a t DS d e t e r m i n e s b o t h added mass and damping c o e f f i c i e n t s , w h i l e DSEXF d e t e r m i n e s o n l y t h e damp ing c o e f f i c i e n t . The d a t a p r o c e s s i n g t e c h n i q u e s u s e d b y b o t h p r o g r a m s i s v e r y s i m i l a r w i t h one e s s e n t i a l d i f f e r e n c e : f o r t h e dynamic t e s t s , p h a s e r e l a t i o n s a r e v e r y i m p o r t a n t . H e n c e , t h e p r o g r a m DS u s e s t he s u b r o u t i n e REALTIME t o c o n s e r v e p h a s e r e l a t i o n s b e t w e e n t h e s u r g e f o r c e t r a c e and t h e d i s p l a c e m e n t r e c o r d . The f o r m u l a t i o n u s e d b y DSEXF t o d e t e r m i n e t he damp ing c o e f f i c i e n t f r o m t h e wave i n d u c e d s u r g e f o r c e i s n o t p h a s e d e p e n d e n t . I n a d d i t i o n , t h e r e i s no c y l i n d e r m o t i o n d u r i n g t h e i n c i d e n t wave t e s t s . H e n c e , t h e d i s p l a c e m e n t c h a n n e l d a t a i s n o t u s e d and i t i s n o t n e c e s s a r y t o c a l l t h e s u b r o u t i n e , REALTIME. The f l o w c h a r t s o f F i g u r e s 8 . 1 0 and 8 .11 i l l u s t r a t e t h e s i m i l a r i t i e s and d i s s i m i l a r i t i e s b e t w e e n t h e p r o g r a m s , DS and DSEXF. The h a r d w a r e s e t up f o r t h i s p r o g r a m i s t h e same as f o r DS . The 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 , a r e as g i v e n i n T a b l e 6 . 2 . 1 - 1 . I f u s i n g t h e d e f a u l t a s s i g n m e n t s t h e c o m p u t e r w i l l r e a d t h e d a t a on c h a n n e l 0 and 2 b u t t h i s d a t a i s n e v e r p r o c e s s e d i n any way . When r u n n i n g ADMAIN, t h e u s e r must a l l o w f o r two e x t r a dummy c h a n n e l s , s i n c e t h e compu te r assumes t h e y s t a r t c o n s e c u t i v e l y f r o m 0 . T h i s p r o g r a m , l i k e DS, p romp ts t h e u s e r f o r t h e r e q u i r e d i n p u t and s e q u e n t i a l l y 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 i n f o r m a t i o n : - 122 -1) Raw d a t a f i l e name 2) C a l i b r a t i o n f i l e name 3) Name o f f i l e t o c o n t a i n r e s u l t s ( i . e . damp ing c o e f f i c i e n t ) 4 ) C y l i n d e r mode l t y p e ( i . e . s i n g l e , d o u b l e , 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) Mean d i s t a n c e f r o m t h e wave p r o b e t o t h e c y l i n d e r c e n t e r l i n e 7) F i l t e r f a c t o r , o p t i o n t o change f r o m d e f a u l t v a l u e o f 4 (The mean ing o f t h i s f a c t o r i s e x p l a i n e d i n S e c t i o n 6 . 2 . 1 . 6 ) 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 i n p u t , c o m b i n e d 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 s , i s s u f f i c i e n t t o a l l o w t h e compu te 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 damping c o e f f i c i e n t and e x c i t i n g f o r c e a m p l i t u d e . D u r i n g e a c h i n c i d e n t wave t e s t , t h e f o l l o w i n g p a r a m e t e r s a r e b e i n g m e a s u r e d : s u r g e f o r c e , p i t c h moment, wave h e i g h t , and t h e p r e s s u r e a t t h r e e p o i n t s on t h e c y l i n d e r s u r f a c e . To measure a l l o f t h e s e p a r a m e t e r s i t was n e c e s s a r y t o u t i l i z e a t o t a l o f 6 c h a n n e l s on t h e MINC™ 11 m i n i - c o m p u t e r . A s T a b l e 6 . 2 . 2 - 1 i n d i c a t e s , t h i s p r o g r a m i s r u n w i t h t h e SI s y s t e m o f u n i t s . A l l c a l i b r a t i o n s , t h e r e f o r e , s h o u l d c o n v e r t t h e o u t p u t s i g n a l i n t o t he u s e r u n i t s i d e n t i f i e d i n t h i s t a b l e . I f t h e c a l i b r a t i o n f i l e c r e a t e d t o use w i t h 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 c o n v e r t s t he d a t a t o o t h e r u n i t s , t h e n the s u b r o u t i n e CALIB w i l l a l l o w one t o c o n v e r t t h e d a t a i n t o t h e recommended u n i t s . A s t h i s p r o g r a m i s r u n , i t 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 t he d a t a u s e d i n i n t e r m e d i a t e s t a g e s . These 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 l y e x t e n s i o n s t o the o r i g i n a l d a t a f i l e name. T a b l e 6 . 2 . 2 - 1 g i v e s an - 123 -examp le o f t h e name e x t e n s i o n s t h a t w o u l d be a p p l i e d t o a raw d a t a f i l e named DATA.DAT, as c r e a t e d b y t h e d a t a a c q u i s i t i o n p r o g r a m ADMAIN. T a b l e 6 . 2 . 2 - 1 F i l e Name E x t e n s i o n U s e d by "DSEXF 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_SURG.DAT 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 DATA_PITC.DAT 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 DATA_WVPR.DAT D e m u l t i p l e x e d O u t p u t f r o m Wave P r o b e DATA_PRS1.DAT D e m u l t i p l e x e d O u t p u t f r o m P r e s s u r e T r a n s d u c e r #1 DATA_PRS2.DAT D e m u l t i p l e x e d O u t p u t f r o m P r e s s u r e T r a n s d u c e r #2 DATA_PRS3.DAT D e m u l t i p l e x e d O u t p u t f r o m P r e s s u r e T r a n s d u c e r #3 DATA_SURG_RAW.DAT Raw U n f i l t e r e d D a t a O r i g i n a l l y C o n t a i n e d i n DATA_SURG.DAT DATA_PITC_RAW.DAT Raw U n f i l t e r e d D a t a O r i g i n a l l y C o n t a i n e d i nDATA_PITC.DAT o o o 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_SURG_FFT.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 ime Domain D a t a C o n t a i n e d i n DATA_SURG.DAT DATA_PITC_FFT.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 ime Domain D a t a C o n t a i n e d i n DATA_PITC.DAT o o o 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_FFT.DAT 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 n a l y s i s 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 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 Phase A n g l e A s s o c i a t e d w i t h t h i s Component . The s u b s e q u e n t s u b - s e c t i o n s w i l l i n c l u d e a d e s c r i p t i o n o f the s u b r o u t i n e s u s e d w i t h DSEXF t h a t a r e u n i q u e t o DSEXF. Those s u b r o u t i n e s w h i c h a r e common t o b o t h DSEXF and DS have b e e n p r e v i o u s l y d i s c u s s e d i n S e c t i o n s 6 . 2 . 1 t o 6 . 2 . 1 . 1 4 . 6 . 2 . 2 . 1 B IGEXF SUBROUTINE BIGEXF i s s i m i l a r t o t h e s u b r o u t i n e , B I G , p r e v i o u s l y d e s c r i b e d i n S e c t i o n 6 . 2 . 1 . 1 0 . T h i s s u b r o u t i n e s c a n s e a c h o u t p u t f i l e f r o m FFT and - 124 -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 g r e a t e s t a m p l i t u d e . The f r e q u e n c y , a m p l i t u d e and 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 i s p o i n t a r e w r i t t e n t o a u s e r s p e c i f i e d o u t p u t f i l e . The wave p r o b e p h a s e i s a d j u s t e d t o c o r r e c t f o r t he p h a s e l a g w h i c h a r i s e s f r o m t h e wave p r o b e b e i n g l o c a t e d u p s t r e a m f r o m t h e c y l i n d e r . T h i s s u b r o u t i n e a l s o d e t e r m i n e s 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 t h e p o i n t where t h e wave c r e s t p a s s e s t h e c y l i n d e r c e n t e r l i n e and t h e p o i n t o f maximum a m p l i t u d e f o r e a c h o f t h e o t h e r c h a n n e l s . 6 . 2 . 2 . 2 COEFEXF SUBROUTINE COEFEXF i s t h e s i s t e r s u b r o u t i n e t o COEF, t h i s s u b r o u t i n e d e t e r m i n e s t he f i n a l r e s u l t s f r o m t h e i n d u c e d wave t e s t s . T h i s p r o g r a m u s e s t h e s u b r o u t i n e EXCITE t o a c t u a l l y c a l c u l a t e t h e damping c o e f f i c i e n t f r o m t h e f o r m u l a t i o n d e s c r i b e d i n E q u a t i o n ( 6 . 2 . 1 . 1 4 - 1 ) . The o u t p u t f r o m COEFEXF i n c l u d e s t he n o r m a l i z e d s u r g e i n d u c e d f o r c e , t h e c a l c u l a t e d damp ing c o e f f i c i e n t and o t h e r p a r a m e t e r s s u c h as wave a m p l i t u d e and wave l e n g t h . 6 . 2 . 3 OTHER SOFTWARE USED 6 . 2 . 3 . 1 ANGLE SUBROUTINE The s u b r o u t i n e , ANGLE, c o n v e r t s any a n g l e , c4, i n d e g r e e u n i t s t o an e q u i v a l e n t d e g r e e measurement s u c h t h a t -180 < <f> < 1 8 0 . - 125 -6 . 2 . 3 . 2 EZ PROGRAM The p r o g r a m , E Z , t a k e s t he o u t p u t f r o m t h e FFT s u b r o u t i n e and w r i t e s the f r e q u e n c y and a m p l i t u d e d a t a i n t o s e p a r a t e u s e r s p e c i f i e d f i l e s . T h i s TM p r e p a r e s t h e d a t a so t h a t i t c a n be p l o t t e d b y t h e V a x l i b r a r y p r o g r a m , EZGRAF. 6 . 2 . 3 . 3 HCPLOT PROGRAM The p r o g r a m , HCPLOT, t a k e s t h e o u t p u t f r o m t h e COEF s u b r o u t i n e and w r i t e s t h e f r e q u e n c y , added mass and damp ing c o e f f i c i e n t s i n t o s e p a r a t e TM f i l e s . T h i s p r e p a r e s t h e d a t a so t h a t i t c a n be p l o t t e d b y t h e V a x l i b r a r y p r o g r a m , EZGRAF. 6 . 2 . 3 . 4 LABEL SUBROUTINE The s u b r o u t i n e , L A B E L , i s u s e d t o m o d i f y f i l e names b y a l l o w i n g t h e u s e r t o a t t a c h a s u f f i x a n d / o r a new e n d i n g t o a g i v e n f i l e name. I t was c a l l e d e x t e n s i v e l y t h r o u g h o u t DS . 6 . 2 . 3 . 5 L INREG SUBROUTINE The s u b r o u t i n e , L I N R E G , p e r f o r m s a l e a s t s q u a r e s l i n e a r r e g r e s s i o n a n a l y s i s on a n a r r a y o f d a t a and r e t u r n s t h e s l o p e and i n t e r c e p t o f t h e r e s u l t i n g l i n e . - 126 -6 . 2 . 3 . 6 XX PROGRAM The p r o g r a m , X X , t a k e s 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 and w r i t e s t h e t i m e and a m p l i t u d e d a t a i n t o s e p a r a t e u s e r - s p e c i f i e d f i l e s . T h i s p r e p a r e s t h e d a t a so t h a t i t c a n be p l o t t e d b y t h e Vax™ l i b r a r y p r o g r a m , EZGRAF. 127 -APPENDIX C 7. PHOTOGRAPHS - 128 -FIGURE 7.1 - GULF CANADA'S ' K U L L U K ' 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 - 129 -FIGURE 7 . 2 - EXTERIOR VIEW OF OCEAN ENGINEERING CENTRE A t BC R e s e a r c h i n V a n c o u v e r , B r i t i s h C o l u m b i a - 130 -FIGURE 7 .3 - INTERIOR VIEW OF OCEAN ENGINEERING CENTRE Show ing t h e Tow ing Tank and M a n o e u v e r i n g B a s i n - 131 -FIGURE 7.4 - ENERGY ABSORBING BEACH MATERIAL Placed Between Motion Generator and Wave Maker End of Tank during Hydrodynamic Tests - 132 -FIGURE 7.5 - WAVE MAKER U s e d D u r i n g E x c i t i n g F o r c e T e s t s O n l y 133 -FIGURE 7 .6 - OVERHEAD HOIST Designed and I n s t a l l e d by Author i n J u l y 1985. Used to L i f t Equipment i n and out of the Tank - 134 -FIGURE 7.7 - OVERVIEW OF EXPERIMENTAL SETUP Show ing M o t i o n G e n e r a t o r , A u x i l i a r y B r i d g e , V i d e o Camera and Tow ing C a r r i a g e - 135 -FIGURE 7 .8 - OVERVIEW OF TOWING CARRIAGE SETUP 136 FIGURE 7.9 - MOTION GENERATOR SINGLE CYLINDER TEST SETUP - 137 -FIGURE 7 . 1 0 - MOTION GENERATOR T R I P L E CYLINDER TEST SETUP - 138 -FIGURE 7 .11 - OVERVIEW OF WAVE PATTERN PRODUCED BY HYDRODYNAMIC TEST Show ing t h e R a d i a l Wave p a t t e r n and t h e E f f e c t o f t h e W a l l - 139 -FIGURE 7 .12 - SINGLE CYLINDER AND Show ing t h e A d a p t e r B l o c k , T h r e e T h r e a d e d Rod MOTION GENERATOR CONNECTIONS C h a n n e l Dynamometer and t h e S t i c k - u p 140 -FIGURE 7 . 1 3 - DOUBLE CYLINDER MODEL W i t h A d a p t e r B l o c k A t t a c h e d - 141 -FIGURE 7 . 1 4 - T R I P L E CYLINDER MODEL W i t h A d a p t e r B l o c k A t t a c h e d - 142 -FIGURE 7 . 1 5 - DYNAMOMETER, ADAPTER BLOCK DETAIL S c a l e a t Top o f P h o t o i s u s e d f o r C a l i b r a t i n g Y o - Y o P o t e n t i o m e t e r - 143 -FIGURE 7 . 1 6 - HYDRAULIC POWER UNIT - 144 -FIGURE 7 .17 - DATA ACQUISITION HARDWARE Show ing From l e f t M ine 11 M i n i Computer and VT 105 T e r m i n a l . S m a l l Box Above VT 105 i s t h e ST41B S i g n a l C o n d i t i o n e r . - 145 -- 146 -FIGURE 7 .19 - TWO WIRE WAVE PROBE Moun ted on C a l i b r a t i o n F i x t u r e 147 -FIGURE 7 . 2 0 - YO-YO POSITION TRANSDUCER - 148 -FIGURE 7 .21 - DYNAMOMETER STATIC CALIBRATION SETUP Show ing T u n i s - O l s e n U n i v e r s a l T e s t i n g M a c h i n e w i t h Dynamometer i n C a l i b r a t i o n J i g . - 149 -FIGURE 7 . 2 2 - DYNAMOMETER WITHOUT PROTECTIVE - 150 -APPENDIX D 8. FIGURES 151 -FIGURE 8 , - COORDINATE SYSTEM USED AND DEFINITION OF MOTIONS - 152 -- 154 -F I G U R E 8.4- - G E O M E T R Y O F T R I P L E C Y L I N D E R M O D E L - 155 -- 156 -T - a > 2 - a — > 3 STEP DRAFT )k-FIGURE 8.6 - SUBDIVISION OF TRIPLE CYLINDER FLUID DOMAIN FOR MATCHING TECHNIQUE THEORY - 157 -A X I S OF SYMMETRY FREE SURFACE Tl I I I I I I I I I I I T'TTT I I I I I I I I I I. I I I I I I CYLINDER MODEL TTTTT TT : s . BOTTOM OF TANK R -RADIATION SURFACE -> -n FIGURE 8.7 - DISCRETIZED CONTROL SURFACE USED IN BOUNDARY ELEMENT METHOD FORMULATION - 158 -(START)  / INPUT RAW DATA AND PARAMETERS ("DS")/ DEMULTIPLEX RAW DATA ("DEMUX"; CONVERT DATA TO DESIRED UNITS ("CALIB") DECOUPLE DYNAMOMETER SURGE S PITCH CHANNELS ("DYNO") REMOVE TREND IN DATA ("TREND": FILTER DATA ("FILTER") PREPARE DATA FOR FFT -REMOVE INCOMPLETE CYCLES ("FFT") PERFORM FFT ("FOURT") CORRECT WAVE PROBE DATA FOR PHASE DELAY ("BIGWAVE", "DELAY") PERFORM REALTIME ANALYSIS TO DETERMINE MAGNITUDE OF SURGE FORCE IN 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 COEFFICIENTS ("COEF1 RERUN ^ \ FOR ANOTHEi FREQ FIGURE B . 9 - FLOW CHART FOR "DS" PROGRAM - 160 -.START; */ INPUT RAW DATA AND PARAMETERS ("DSEXF")/ i ! DEMULTIPLEX RAW DATA ("DEMUX") CONVERT DATA TO DESIRED UNITS ("CALIB": DECOUPLE DYNAMOMETER SURGE £ PITCH CHANNELS ("DYNO") A. REMOVE TREND IN DATA ("TREND"! FILTER DATA ("FILTER") PREPARE DATA FOR FFT -REMOVE INCOMPLETE CYCLES ("FFT": PERFORM FFT ("FOURT": CORRECT WAVE PROBE DATA FOR PHASE DELAY ("BIGWAVE". "DELAY") J L . jFIND MAX AMPLITUDE AND CORRESPONDING FREQ. AND PHASE ANGLE FROM FFT RESULTS FOR EACH CHANNEL ("BIGEXF") I , i , DETERMINE WEHAUSEN DAMPING COEFFICIENT S NON DIM. EXCITING FORCE ("COEFEXF") FIGURE B.10 - FLOW CHART FOR "DSEXF" PROGRAM - 161 -A P P E N D I X E 9. G R A P H I C A L P R E S E N T A T I O N O F R E S U L T S - 162 -9 .1 NOMENCLATURE FOR GRAPHS a = S u r g e added mass c o e f f i c i e n t 11 ° b = S u r g e 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 medium V = Vo lume o f d i s p l a c e d f l u i d w = f r e q u e n c y o f o s c i l l a t i o n ( r a d i a n s / s e c ) R = Maximum r a d i u s o f c y l i n d e r mode l max 2 g = a c c e l e r a t i o n due t o g r a v i t y ( 9 . 8 0 7 m / s ) F = Wave i n d u c e d s u r g e f o r c e s A => Wave a m p l i t u d e - 163 CYLINDER DISPLACEMENT TRACE SAMPLE PLOT FROM TEST S D F 4 DISPLACEMENT AMPLITUDE vs TIME P 4 0 . - i T I M E ( s e c ) FIGURE 9.1 F I L T E R E D CYLINDER DISPLACEMENT TRACE -SAMPLE PLOT - 164 -E E bJ Q D h J Q_ < h Z uJ UJ 0 < _ l Q_ if) Q CYLINDER DISPLACEMENT SPECTRUM SAMPLE PLOT FROM TEST SDF4 DISPLACEMENT AMPLITUDE vs FREQUENCY 4 0 . - i 3 5 . -3 0 . -2 5 . 2 0 . -I 1 5 . 1 0 . H 5 . 0 . 0 . 0 1 . 0 2 . 0 3 . 0 4 . 0 FREQUENCY (Hz) FIGURE 9 .2 ' FILTERED CYLINDER DISPLACEMENT SPECTRUM SAMPLE PLOT - 165 -UNFILTERED CYLINDER SURGE TRACE SAMPLE PLOT FROM TEST SDF4 SURGE FORCE vs TIME 0 . 0 1 . 0 2 . 0 3 . 0 4 . 0 5 . 0 6 . 0 T I M E ( s e c ) FIGURE 9.3 UNFILTERED SURGE FORCE TRACE -SAMPLE PLOT - 166 -CYLINDER SURGE TRACE S A M P L E P L O T F R O M T E S T SDF4 S U R G E F O R C E v s T I M E 2 . 4 . 6 . 8 . 1 0 T I M E ( s e c ) FIGURE 9.4 FILTERED SURGE FORCE TRACE -SAMPLE PLOT - 167 -z hi 0 Q : o LL LI 0 LZ D 00 DYNAMOMETER SURGE CHANNEL SPECTRUM SAMPLE PLOT FROM TEST SDF4 SURGE FORCE AMPLITUDE vs FREQUENCY 60. -. 50. -40. 30. -20. -10. -0. 0.0 1.0 2.0 3.0 4.0 FREQUENCY (Hz) F I G U R E 9.5 F I L T E R E D S U R G E F O R C E S P E C T R U M -S A M P L E P L O T - 168 -D O U B L E C Y L I N D E R H Y D R O D Y N A M I C T E S T R E S U L T S NON-DIM. ADDED MASS vs NON-DIM. FREQUENCY 0.90-1 0.80-0.70-0.60-0.50-0.40-0.30 S T E P D R A F T s 2 4 7 m m LEGEND • EXPERIMENT - AMP= 10mm BEM THEORY — MT THEORY CJ2R g FIGURE 9.6 ORIGINAL 'UNSMOOTHED' ADDED MASS RESULTS FROM BEM AND MT THEORIES - 169 -D O U B L E C Y L I N D E R H Y D R O D Y N A M I C T E S T R E S U L T S NON-DIM. DAMPING COEF. vs NON-DIM. FREQUENCY STEP DRAFT=247mm 0.30-1 0.25-0.20 0.15-0.10-0.05 0.00 LEGEND • EXPERIMENT - AMPalOmm —BEM THEORY •— MT THEORY co2R /g FIGURE 9.7 ORIGINAL 'UNSMOOTHED' DAMPING COEF. RESULTS FROM BEM AND MT THEORIES - 170 -S I N G L E C Y L I N D E R H Y D R O D Y N A M I C T E S T R E S U L T S NON-DIM. ADDED MASS vs NON-DIM. FREQUENCY STEP DRAFT=211mm LEGEND • EXPERIMENT - A M P » 1 0 m m A EXPERIMENT - A M P = 2 5 m m O EXPERIMENT - A M P = 3 5 m m —BEM THEORY CJ2R / g FIGURE SINGLE CYLINDER 9 .8 - ADDED MASS - DRAFT = 211mm - 171 -S I N G L E C Y L I N D E R H Y D R O D Y N A M I C T E S T R E S U L T S NON-DIM. ADDED MASS vs NON-DIM. FREQUENCY STEP DRAFT=218mm 0.0 1.0 2.0 3.0 4.0 5.0 co2R / g FIGURE 9.9 SINGLE CYLINDER - ADDED MASS - DRAFT = 218mm - 172 -DOUBLE CYLINDER HYDRODYNAMIC TEST RESULTS NON-DIM. ADDED MASS vs NON-DIM. FREQUENCY S T E P D R A F T s 1 7 l m m 0.90-1 0.80-0.70-0.60-0.50-0.40 LEGEND • EXPERIMENT - AMP= 1 3 m m A EXPERIMENT - A M P = 2 5 m m —BEM THEORY MT THEORY 0.30 0.0 CJ2R / g FIGURE 9.10 DOUBLE CYLINDER - ADDED MASS - OVERALL DRAFT = 625mm - 173 -D O U B L E C Y L I N D E R H Y D R O D Y N A M I C T E S T R E S U L T S N O N - D I M . A D D E D M A S S vs NON-DIM. F R E Q U E N C Y STEP DRAFT=236-247mm 0.90-! 0.80-0.70-> 0.60-0.50-0.40-0.30 \ A 0.0 1^ 1.0 r 2.0 LEGEND • EXP.(247) - AMPaiOmm A EXP.(2S6) - AMP=15mm O EXP.(2S6) - AMP»25mm BEM THEORY (247) — MT THEORY (247) T 3.0 CJ2R / g I 4.0 n 5.0 FIGURE 9.11 DOUBLE CYLINDER - ADDED MASS - OVERALL DRAFT = 690-701mm - 174 -T R I P L E C Y L I N D E R H Y D R O D Y N A M I C T E S T R E S U L T S NON-DIM. ADDED MASS vs NON-DIM. FREQUENCY S T E P DRAPTsr 1 OOmm 1.20-1 1.10-1.00-0.90-0.80-0.70-0.60 0.50H 0.40 LEGEND • EXPERIMENT - A M P » 1 0 m m A EXPERIMENT - A M P = 2 9 m m BEM THEORY — MT THEORY 0.00 —I T " 1.00 2.00 I 3.00 co2R g FIGURE 9.12 TRIPLE CYLINDER - ADDED MASS - OVERALL DRAFT = 867mm - 175 -T R I P L E C Y L I N D E R H Y D R O D Y N A M I C T E S T R E S U L T S NON-DIM. ADDED MASS vs NON-DIM. FREQUENCY STEP D RAFT = 171mm 1.20-1.10-1.00-0.90-0.80-0.70-0.60-0.50 0.40 0.00 • A • L E G E N D N • EXPERIMENT - A M P = 1 0 m m A EXPERIMENT - A M P = 2 3 m m BEM THEORY ( 1 7 1 m m ) — MT THEORY ( 1 6 2 m m ) • A T T 1.00 2.00 co2R / g 1^ 3.00 FIGURE 9.13 TRIPLE CYLINDER - ADDED MASS - OVERALL DRAFT = 938mm - 176 -TRIPLE CYLINDER HYDRODYNAMIC TEST RESULTS NON-DIM. ADDED MASS vs NON-DIM. FREQUENCY 1.20-1 1.10-1.00 0.90-0.80-0.70-0.60-0.50-0.40 A A • • 0.00 STEP DRAFT=218mm A • " T " 1.00 LEGEND • EXPERIMENT - A M P = 1 0 m m A EXPERIMENT - A M P = 1 5 m m BEM THEORY — - MT THEORYA n T " 2.00 CJ2R / g " T " 3.00 FIGURE 9.14 TRIPLE CYLINDER - ADDED MASS - OVERALL DRAFT = 985mm - 177 -T R I P L E C Y L I N D E R H Y D R O D Y N A M I C T E S T R E S U L T S NON-DIM. ADDED MASS vs NON-DIM. FREQUENCY STEP DRAFT=322mm 1.20-1 1.10-1.00-0.90-0.80-0.70-0.60 0.50-0.40 A A 0.00 • A LEGEND • EXPERIMENT^ - A M P = 1 0 m m A EXPERIMENT - A M P = 1 5 m m BEM THEORY — MT THEORY ""I 1"" 1.00 2.00 co2R / g I 3.00 FIGURE 9.15 TRIPLE CYLINDER - ADDED MASS - OVERALL DRAFT = 1089mm - 1 7 8 -T R I P L E C Y L I N D E R H Y D R O D Y N A M I C T E S T R E S U L T S NON-DIM. ADDED MASS vs NON-DIM. FREQUENCY STEP DRAFT=41 O m m 1.20q 1.10-1.00 0.90-0.80-0.70-0.60 0.50-1 0.40 9 0.00 4 LEGEND • EXPERIMENT - A M P = 1 0 m m A EXPERIMENT - A M P = 1 5 m m O EXPERIMENT - A M P = 2 5 m m BEM THEORY — MT THEORY - i i — 1.00 2.00 w2R / g I 3.00 FIGURE 9.16 TRIPLE CYLINDER - ADDED MASS - OVERALL DRAFT 1177mm - 179 -T R I P L E C Y L I N D E R H Y D R O D Y N A M I C T E S T R E S U L T S NON-DIM. ADDED MASS vs NON-DIM. FREQUENCY STEP DRAFT=420mm 1.20-1 1.10-1.00-0.90-0.80-0.70-0.60-0.50-0.40 6 6 • A LEGEND • E X P E R I M E N T - A M P s l O m m A E X P E R I M E N T - , - A M P = 1 5 m m BEM THEORY — MT THEORY 0.00 m r" 1.00 2.00 co2R / g T " 3.00 FIGURE 9.17 TRIPLE CYLINDER - ADDED MASS - OVERALL DRAFT = 1187mm - 180 -SINGLE CYLINDER HYDRODYNAMIC TEST RESULTS NON-DIM. DAMPING COEF. vs NON-DIM. FREQUENCY S T E P DRAPT=211mm 0.0 1.0 2.0 3.0 4.0 5.0 co2R /g m a x FIGURE 9.18 SINGLE CYLINDER - DAMPING COEF. - DRAFT = 211mm - 181 -S I N G L E C Y L I N D E R H Y D R O D Y N A M I C T E S T R E S U L T S N O N - D I M . D A M P I N G C O E F . va N O N - D I M . F R E Q U E N C Y > CI 3 0.60 -I 0.50 -0.40 -0.30 -0.20 -0.10 -0.00 S T E P DRAFT=218mxn LEGEND • EXPERIMENT - A M P - 1 0 m m A EXPERIMENT - A M P = 2 S m m O EXPERIMENT - A M P » 3 S m m — B E M THEORY CJ2R FIGURE 9.19 SINGLE CYLINDER - DAMPING COEF. - DRAFT = 218mm - 182 -D O U B L E C Y L I N D E R H Y D R O D Y N A M I C T E S T R E S U L T S NON-DIM. DAMPING COEF. vs NON-DIM. FREQUENCY STEP DRAFT= 171mm 0.0 1.0 2.0 3.0 4.0 5.0 CJ2R / g -max FIGURE 9.20 DOUBLE CYLINDER - DAMPING COEF. - OVERALL DRAFT = 625mm - 183 -D O U B L E C Y L I N D E R H Y D R O D Y N A M I C T E S T R E S U L T S NON-DIM. DAMPING COEF. vs NON-DIM. FREQUENCY STEP DRAFT=236-247mm 0.30-n 0.25-0.20-0.15-0.10-0.05-0.00 0.0 LEGEND • EXP.(247) - A M P » 1 0 m m A EXP.(2S6) - A M P = 1 3 m m O E XP.(236) - A M P « 2 3 m m 0 BEM TjHEORY ( 2 4 7 ) — MT THEORY ( 2 4 7 ) CJ2R g FIGURE 9.21 DOUBLE CYLINDER - DAMPING COEF. - OVERALL DRAFT = 690-701mm - 184 -T R I P L E C Y L I N D E R H Y D R O D Y N A M I C T E S T R E S U L T S NON-DIM. DAMPING COEF. vs NON-DIM. FREQUENCY STEP DRAFTs 1 OOmm LEGEND • EXPERIMENT - AMP=»10mm A EXPERIMENT - A M P = 2 5 m m — BEM THEORY MT THEORY 0.00 0.00 T r 1.00 2.00 CJ2R /g 3.00 FIGURE 9.22 TRIPLE CYLINDER - DAMPING COEF. - OVERALL DRAFT = 867mm - 185 -T R I P L E C Y L I N D E R H Y D R O D Y N A M I C T E S T R E S U L T S NON-DIM. DAMPING COEF. vs NON-DIM. FREQUENCY STEP DRAFT= 171mm 0.30-1 0.25-0.20-0.15-0.10-0.05-LEGEND • EXPERIMENT - A M P = 1 0 m m A EXPERIMENT - A M P = 2 5 m m BEM THEORY ( 1 7 1 m m ) — MT THEORY ( 1 6 2 m m ) 0.00 0.00 2.00 co2R / g 3.00 FIGURE 9.23 TRIPLE CYLINDER - DAMPING COEF. - OVERALL DRAFT = 938mm - 186 -TRIPLE CYLINDER HYDRODYNAMIC TEST RESULTS NON-DIM. DAMPING COEF. vs NON-DIM. FREQUENCY STEP DRAFT=218mm 0.30 0.25-0.20 0.15-0.10-0.05-LEGEND • EXPERIMENT - A M P n I O m m A EXPERIMENT - A M P = 1 3 m m —BEM THEORY — MT THEORY 0.00 0.00 1.00 3.00 co2R / g FIGURE 9.24 TRIPLE CYLINDER - DAMPING COEF, - OVERALL DRAFT = 985mm - 187 -> 3 -Q TRIPLE CYLINDER HYDRODYNAMIC TEST RESULTS NON-DIM. DAMPING COEF. vs NON-DIM. FREQUENCY STEP DRAFT=322mm 0.30-1 0.25-0.20-0.15-0.10-0.05-0.00 LEGEND • EXPERIMENT - A M P » 1 0 m m A EXPERIMENT - A M P = 1 5 m m BEM THEORY — M T THEORY • 0.00 1.00 2.00 3.00 CJ2R g FIGURE 9.25 TRIPLE CYLINDER - DAMPING COEF. - OVERALL DRAFT = 1089mm - 188 -T R I P L E C Y L I N D E R H Y D R O D Y N A M I C T E S T R E S U L T S NON-DIM. DAMPING COEF. vs NON-DIM. FREQUENCY STEP DRAFT=410mm 0.30-1 0.25-^ 0.20-0.15-0.10-0.05-LEGEND • EXPERIMENT - A M P = 1 0 m m ^ E X P E R I M E N T - AM P= 1 3 m m O O EXPERIMENT - A M P = 2 S m m BEM THEORY — MT THEORY 0.00 0.00 1.00 CJ2R / g FIGURE 9.26 TRIPLE CYLINDER - DAMPING COEF. - OVERALL DRAFT = 1177mm - 189 -TRIPLE CYLINDER HYDRODYNAMIC TEST RESULTS NON-DIM. DAMPING COEF. vs NON-DIM. FREQUENCY STEP DRAFT=420mm 0.30-1 0.25-0.20-0.15-0.10-0.05-0.00 • A LEGEND • EXPERIMENT - A M P d O m m A EXPERIMENT - AMP=15mm BEM THEORY — MT THEORY • 0.00 1.00 u2R g FIGURE 9.27 TRIPLE CYLINDER - DAMPING COEF. - OVERALL DRAFT = 1187mm - 190 -SINGLE CYLINDER TEST RESULTS WAVE INDUCED EXCITING FORCE TESTS NON-DIM. EXCITING FORCE vs NON-DIM. FREQUENCY 6.00-1 5.00-4.00-3.00-2.00-1.00-0.00 0.0 STEP DRAFT=211-218mm • A * • • A A LEGEND • EXPERIMENT ( 211 ) A EXPERIMENT (218 ) O EXPERIMENT ( 218X ) BEM THEORY • T 0.5 I 1.0 CJ2R / g I 1.5 n 2.0 FIGURE 9.28 SINGLE CYLINDER - EXCITING FORCE - DRAFT = 211-218mm - 191 -DOUBLE CYLINDER TEST RESULTS WAVE INDUCED EXCITING FORCE TESTS NON-DIM. EXCITING FORCE vs NON-DIM. FREQUENCY STEP DRAFT= 171mm 3.00 «i 2 . 5 0 -2 . 0 0 -1.50-1.00-0.50' 0.00 • • LEGEND • EXPERIMENT — B E M THEORY • 1 1 1 1 0.0 0.5 1.0 1.5 2.0 co2R / g FIGURE 9.29 DOUBLE CYLINDER - EXCITING FORCE - OVERALL DRAFT = 625mm D O U B L E C Y L I N D E R T E S T R E S U L T S WAVE INDUCED EXCITING FORCE TESTS NON-DIM. EXCITING FORCE vs NON-DIM. FREQUENCY STEP DRAFT=236mm < 3.00-1 2.50-2.00-1.50 1.00-0.50-0.00 B • 0.0 S • • • • LEGEND • EXPERIMENT — B E M THEORY n 1 r 0.5 1.0 1.5 CJ2R / g I 2.0 FIGURE 9.30 DOUBLE CYLINDER - EXCITING FORCE - OVERALL DRAFT = 690mm - 193 -D O U B L E C Y L I N D E R T E S T R E S U L T S WAVE INDUCED EXCITING FORCE TESTS NON-DIM. EXCITING FORCE vs NON-DIM. FREQUENCY STEP DRAFT=247mm 0.0 0.5 1.0 1.5 2.0 w2R / g max FIGURE 9.31 DOUBLE CYLINDER - EXCITING FORCE - OVERALL DRAFT = 701mm - 194 -T R I P L E C Y L I N D E R T E S T R E S U L T S WAVE INDUCED EXCITING FORCE TESTS NON-DIM. EXCITING FORCE vs NON-DIM. FREQUENCY STEP DRAFT= 100mm 0.0 0.5 1.0 1.5 2.0 w 2R / g max FIGURE 9.32 TRIPLE CYLINDER - EXCITING FORCE - OVERALL DRAFT = 867mm - 195 -TRIPLE CYLINDER TEST RESULTS WAVE INDUCED EXCITING FORCE TESTS NON-DIM. EXCITING FORCE vs NON-DIM. FREQUENCY STEP DRAFT= 169mm 2.50-1 2.00-1.50-1.00-0.50-0.00 • • • • • 0.0 " T " 0.5 LEGEND • EXPERIMENT BEM THEORY 1.0 I 1.5 co2R n 2.0 FIGURE 9.33 TRIPLE CYLINDER - EXCITING FORCE - OVERALL DRAFT = 936mm - 196 -TRIPLE CYLINDER TEST RESULTS WAVE INDUCED EXCITING FORCE TESTS NON-DIM. EXCITING FORCE vs NON-DIM. FREQUENCY STEP DRAFT=410mm 2.50-1 2.00-1.50-1.00-0.50-0.00 • • s • 0.0 T" 0.5 • 1 ^ 1.0 LEGEND • EXPERIMENT — B E M THEORY 1 ^ 1.5 co2R / g FIGURE 9.34 TRIPLE CYLINDER - EXCITING FORCE I 2.0 OVERALL DRAFT = 1177mm - 197 -S I N G L E C Y L I N D E R T E S T R E S U L T S CALCULATED BY WEHAUSEN FORMULATION WAVE INDUCED EXCITING FORCE TESTS NON-DIM. DAMPING COEF. vs NON-DIM. FREQUENCY STEP DRAFT=211mm O.8O-1 0.70-0.00 LEGEND • EXPERIMENT BEM THEORY CJ2R / g FIGURE 9.35 SINGLE CYLINDER - WEHAUSEN CALCULATED DAMPING COEF. - DRAFT = 211mm - 198 -SINGLE CYLINDER TEST RESULTS CALCULATED BY WEHAUSEN FORMULATION WAVE INDUCED EXCITING FORCE TESTS NON-DIM. DAMPING COEF. vs NON-DIM. FREQUENCY O.8O-1 0.20 0.10 0.00 STEP DRAFT=218mm LEGEND • EXPERIMENT A EXPERIMENT (X) — B E M THEORY co2R g FIGURE 9.36 SINGLE CYLINDER - WEHAUSEN CALCULATED DAMPING COEF. - DRAFT = 218mm - 199 -D O U B L E C Y L I N D E R T E S T R E S U L T S CALCULATED BY WEHAUSEN FORMULATION WAVE INDUCED EXCITING FORCE TESTS NON-DIM. DAMPING COEF. vs NON-DIM. FREQUENCY STEP DRAFT=s 171mm 0.50-1 0.40-0.30-0.20-0.10 0.00 LEGEND • EXPERIMENT BEM THEORY — MT THEORY CJ2R / g FIGURE 9.37 DOUBLE CYLINDER - WEHAUSEN CALCULATED DAMPING COEF. - OVERALL DRAFT = 625mm - 200 -D O U B L E C Y L I N D E R T E S T R E S U L T S CALCULATED BY WEHAUSEN FORMULATION WAVE INDUCED EXCITING FORCE TESTS NON-DIM. DAMPING COEF. vs NON-DIM. FREQUENCY 0.50-1 0.40-0.30-0.20-0.10-0.00 STEP DRAFT=236mm • • • LEGEND • EXPERIMENT BEM THEORY — MT THEORY OJ2R / g FIGURE 9.38 DOUBLE CYLINDER - WEHAUSEN CALCULATED DAMPING COEF. - OVERALL DRAFT = 690mm - 201 -D O U B L E C Y L I N D E R T E S T R E S U L T S CALCULATED BY WEHAUSEN FORMULATION WAVE INDUCED EXCITING FORCE TESTS NON-DIM. DAMPING COEF. vs NON-DIM. FREQUENCY STEP DRAFTs247mm 0.50-1 0.40-0.30-0.20-0.10-0.00 0.0 • • • LEGEND • EXPERIMENT BEM THEORY — MT THEORY CJ2R / g FIGURE 9.39 DOUBLE CYLINDER - WEHAUSEN CALCULATED DAMPING COEF. - OVERALL DRAFT = 701mm - 202 -T R I P L E C Y L I N D E R T E S T R E S U L T S CALCULATED BY WEHAUSEN FORMULATION WAVE INDUCED EXCITING FORCE TESTS NON-DIM. DAMPING COEF. vs NON-DIM. FREQUENCY 0.40-1 0.00 STEP DRAFT=100mm LEGEND • EXPERIMENT —BEM THEORY MT THEORY co2R / g FIGURE 9.40 TRIPLE CYLINDER - WEHAUSEN CALCULATED DAMPING COEF. - OVERALL DRAFT = 867mm - 203 -TRIPLE CYLINDER TEST RESULTS CALCULATED BY WEHAUSEN FORMULATION WAVE INDUCED EXCITING FORCE TESTS NON-DDI. DAMPING COEF. vs NON-DIM. FREQUENCY 0.40-1 0.35-0.00 STEP DRAFTS 169mm LEGEND • EXPERIMENT — B E M THEORY — MT THEORY co2R / g FIGURE 9.41 TRIPLE CYLINDER - WEHAUSEN CALCULATED DAMPING COEF. - OVERALL DRAFT = 936mm - 204 -TRIPLE CYLINDER TEST RESULTS CALCULATED BY WEHAUSEN FORMULATION WAVE INDUCED EXCITING FORCE TESTS NON-DIM. DAMPING COEF. va NON-DIM. FREQUENCY STEP DRAFT=410mm 0.25-1 0.20-0.15-0.10 0.05-0.00 LEGEND • EXPERIMENT —BEM THEORY MT THEORY co2R / g FIGURE 9.42 TRIPLE CYLINDER - WEHAUSEN CALCULATED DAMPING COEF. - OVERALL DRAFT = 1177mm - 205 -APPENDIX F 10. BESSEL FUNCTIONS AND RELATED FORMULAE i) Bessel function of the f i r s t kind of order n OO J ( X ) - J n L, - ( - l ) k ( x / 2 ) 2 " - " k! T(k+l-n) k = o where the Gamma function i s defined as: F(n) - r t"" 1 e"fc dt i i ) Bessel function of the second kind of order n i . J ( x ) COS (D7r) - J ( X ) Y ( X) = l l m — 2 — n p-»n s i n (pw) i i i ) Modified Bessel function of the f i r s t kind of order n I (x) - - i J (x) = e"1"'2 J (x) n n n iv) Modified Bessel function of the second kind of order n n p->n 2. Sin Cp7T; -p p v) Hankel function of the f i r s t kind of order n H (x) = J (x) + iY (x) n n n v i ) Derivatives of Bessel functions £ (x) - 2 (x) - I 2 (x) n n-1 X n - 206 or of s p e c i f i c i n t e r e s t i n t h i s report f o r n - l , I (x) - S (x) - - 2 (x) 1 0 X 1 where 2 denotes J , Y, I, K or H. v i i ) Complete e l l i p t i c i n t e g r a l of the f i r s t kind F(k dd 0 f~. . 2 . 2 , / 1 - k s i n ( i{ i + i 2 k2 • + 1 3 2 4 k* + 1 3 5 2 4 6 i i x ) Complete e l l i p t i c i n t e g r a l of the second kind E(k,ir/2) f ' Y J o H1 i 2 • 2a k s i n 6 de 1 3 2 4 2 k 4 1 3 5 2 4 6 - 207 

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