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The behaviour of small moored vessels in surge and sway Taylor, Paul Steven 1983

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THE BEHAVIOUR OF SMALL MOORED VESSELS IN SURGE AND SWAY By PAUL STEVEN TAYLOR B.Eng. Technical U n i v e r s i t y of Nova S c o t i a Dip. Eng. U n i v e r s i t y of Pr i n c e Edward Island A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Ap p l i e d Science i n THE FACULTY OF GRADUATE STUDIES Department of C i v i l Engineering We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August 1983 © Paul S. Taylor 1983 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 her 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 ClVfC EflJ G f**lAJQ 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 SEPT&M&ER o3» , I9B3 DE-6 ( 3 / 8 1 ) i i ABSTRACT The behaviour of small moored vessels i n motions of surge and sway when subjected to a regular incident wave t r a i n has been examined. Model tests were performed i n a p a r t i a l l y standing wave system and experimental r e s u l t s compared to those generated by a computer model. Variables i n the analysis include wave height, wave length, wave period, p o s i t i o n i n a p a r t i a l standing wave, moorage material s t i f f n e s s , method of moorage, existence of protection fender, and amount of l i n e slackness. Results i n d i c a t e a frequency dependence with strong opportunity f o r resonance for soft moorage systems. Vessel response i s affected s i g n i f i c a n t l y by wave height, wave frequency and degree of r e f l e c t i o n . It i s recommended that c r i t e r i a for marina design should include consideration of a l l three wave c h a r a c t e r i s t i c s and suggested design values are presented. Also, to minimize the opportunity for resonant a m p l i f i c a t i o n , s t i f f moorage systems should be adopted by using s t i f f moorage material, short l i n e s and minimal slackness. TABLE OF CONTENTS ABSTRACT TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES ACKNOWLEDGEMENT CHAPTER 1. INTRODUCTION 1.1. General 1.2. Lit e r a t u r e Survey 1.3. Research J u s t i f i c a t i o n 1.4. Research Approach CHAPTER 2. MARINA VISITS CHAPTER 3. THEORY 3.1. General Body Motion 3.2. Hydrodynamic Factors A f f e c t i n g Motion . . . 3.2.1. Hydrodynamic Added Mass and Damping 3.2.2. Exciting Function 3.3. Hydrostatic S t i f f n e s s 3.4. S i m p l i f i c a t i o n for Surge and Sway of the Equation of Motion CHAPTER 4. ANALYSIS 4.1. Computer Program Logic 4.2. Dynamic Coe f f i c i e n t s 4.3. E x c i t i n g Function 4.4. V e r i f i c a t i o n of Time Step Solution . . . . TABLE OF CONTENTS (cont'd) CHAPTER 5. MOORAGE LINES AND FENDERS 5.1. Moorage Lines 5.2. Fenders CHAPTER 6. EXPERIMENTATION SET-UP 6.1. Wave Basin D e s c r i p t i o n . . . . 6.2. Moorage Dock Frame 6.3. Experimental S c a l i n g . . . . . 6.4. Vessel D e s c r i p t i o n 6.5. Instrumentation 6.6. Model Moorage L i n e s 6.7. Model Fenders CHAPTER 7. PRELIMINARY TESTS 7.1. Basin C h a r a c t e r i s t i c s 7.2. Response of Chart Recorder 7.3. C a l i b r a t i o n of Instruments 7.4. Tests to V e r i f y the Centre of F l o a t a t i o n . . . . 7.5. Tests to Determine Damping Induced by the Measurement System 7.6. Tests to Determine Added Mass and Damping of the Vessel 7.7. Tests to Determine the E f f e c t of Slackness on Natural Period CHAPTER 8. EXPERIMENTATION 8.1. Wave Period 8.2. Wave Height TABLE OF CONTENTS (cont'd) Page 8.3. Wave Reflec t i o n 8.4. Line S t i f f n e s s 8.5. Fender Condition 8.6. Slackness 8.7. Moorage Conditions CHAPTER 9. EXPERIMENTAL AND ANALYTICAL RESULTS 9.1. Observations 9.1.1. Surge 9.1.2. Sway 9.2. Measured Results 9.2.1. Comparison of Mathematical and Experimental Models 9.2.2. Unrestrained O s c i l l a t i o n s 9.2.3. Results of Surge Tests 9.2.4. Results of Sway Tests CHAPTER 10. RECOMMENDATIONS 10.1. Reducing Vessel Damage 10.2. Future Work CHAPTER 11. CONCLUSION BIBLIOGRAPHY APPENDIX I. COMPUTER LISTING OF PROGRAM "YACHT" APPENDIX I I . EXPERIMENTATION DATA v i LIST OF TABLES Table Page 1 Marinas and Yacht Clubs Examined 2 Physical Comparison Between Model and Prototype . . . 3 Results of Tests to Determine Measurement System Damping A Results of Tests to Determine Added Mass and Damping C o e f f i c i e n t s i n Surge 5 Model Vessel's Dynamic Behaviour to Beam Seas . . . . 6 P r o v i s i o n a l Recommended C r i t e r i a f o r Good Wave Climate i n Small Craft Harbours for Resisting Vessel Surge and Sway v i i LIST OF FIGURES Page FIGURE 1 Modes of Vessel Motion 2 Common Moorage P o s i t i o n s 3 Common Moorage Conditions 4 General Moorage S t i f f n e s s Curve 5 S i m p l i f i e d Vessel Shape 6 V e r i f i c a t i o n o f Time Step S o l u t i o n f o r Surge 7 V e r i f i c a t i o n of Time Step S o l u t i o n f or Sway 8 Nylon Rope Force-elongation Behaviour . . . 9 Prototype Balloon Fender 10 Fender F o r c e - D e f l e c t i o n Curve 11 Wave Basin Layout 12 Photo of Wave Basin 13 Model Moorage Frame 14 Photo of Moorage Frame 15 Photo of Model Vessel 16 Wave Probe System 17 Vessel Motion Measurement System 18 Photo of Measurement System V e s s e l Attachment 19 Photo of Measurement System LVDT 20 Model Moorage Lines 21 Moorage Lines Force Elongation Behavior . . 22 B a l l o o n Fender S t i f f n e s s v i i i LIST OF FIGURES (cont'd) Page FIGURE 23 P a r t i a l Standing Wave 24 Wave Basin Reflection C h a r a c t e r i s t i c s . . . . 25 Typ i c a l LVDT C a l i b r a t i o n Curve 26 Measurement System Damping Tests 2 7 Added Mass and Damping Tests 28 Ef f e c t of Slackness on Natural Period i n Surge 29 Photo of Vessel Moored f o r Surge Tests . . . 30 Generalized Surge Response 31 Photo of Vessel Moored f o r Sway Tests . . . . 32 Generalized Sway Response 33 Ty p i c a l Surge Time History Curve 34 Typical Sway Time History Curve 35 Unrestrained Vessel Motion 36 E f f e c t of Basin Ref l e c t i o n on Surge 37 E f f e c t of Wave Height on Surge 38 Ef f e c t of Slackness on Surge 39 Frequency Dependent Response i n Surge . . . . 40 E f f e c t of Line S t i f f n e s s on Surge 41 D i f f e r e n t Moorage Conditions i n Surge . . . . 42 Influence of Fender on Surge 43 E f f e c t of Wave Height on Sway 44 Ef f e c t of Slackness on Sway 45 Frequency Dependent Response i n Sway . . . . LIST OF FIGURES (cont'd) ix FIGURE Page 46 E f f e c t of Line S t i f f n e s s on Sway 47 Different Moorage Conditions i n Sway 48 Influence of Fender on Sway X ACKNOWLEDGEMENT In the preparation of th i s report, the author would l i k e to express h i s appreciation to Dr. M. Isaacson and Dr. M. Olson of the Department of C i v i l Engineering, U.B.C. for t h e i r indispensible and timely assistance. Also, Ms. N. Harry for her assistance i n the preparation of the subsequent text. F i n a l l y the work was made possible through a scholarship from the Natural Sciences and Engineering Research Council of the National Research Council of Canada. To t h i s group, I am very g r a t e f u l . 1 CHAPTER 1 INTRODUCTION Recreational vessels such as yachts, sailboats and powerboats are stored, when not i n use, almost e x c l u s i v e l y i n s p e c i a l l y protected marinas or harbours. The cost of these vessels and t h e i r potential for damage r e s u l t s i n the j u s t i f i a b l y intense e f f o r t that has gone into the design and construction of marinas to maximize the protection to these v e s s e l s . Yet, very l i t t l e has been written about how a boat owner or a marina designer may take steps to reduce the damage to a given vessel i n a given moorage berth. The necessity f o r such knowledge becomes doubly important when one considers that recent demand for moorage space has increased the boat density i n e x i s t i n g marinas and has thus increased the incidence of boat against boat damage. 1.1. General The movement of a vessel r e s t i n g on a free surface can be described as being composed of six modes of motion: surge, pi t c h , heave, sway, r o l l , and yaw (see F i g . 1). Of these, p i t c h , heave, yaw, and r o l l tend to be of a s e r v i c e a b i l i t y rather than damage concern. That i s , the comfort and a b i l i t y of occupants to work above or below deck i s of primary consideration. The remaining two, surge and sway, describe t r a n s l a t i o n a l motion and r e s u l t i n the bulk of the damage sustained by moored vessels. This does not imply that other modes are unimportant as they are of consideration i n t h e i r e f f e c t on the t r a n s l a t i o n modes and the moorage l i n e hawser forces. Vessel Moorage Damage can be characterized either by impact damage to the boat, dock, or another boat; or by hawser force damage caused by excessive loads on the moorage l i n e s r e s u l t i n g i n frayed or broken l i n e s or damaged moorage c l e a t s . Although i t i s clear that both types of damage may occur simultaneously, i t remains to be established the factors which independently or c o l l e c t i v e l y contribute to each manner of damage. 1.2. L i t e r a t u r e Survey A survey of l i t e r a t u r e on related topics has revealed l i t t l e work addressing t h i s s p e c i f i c subject. A great deal of research has gone into d e f i n i n g the behaviour of moored ships but f o r several reasons, i t cannot be d i r e c t l y applied to smaller vessels. The shape of a ship tends to be more regular, and hence, e a s i l y defined, than that of a yacht. This allows mathematical analyses to be conducted which e x p l o i t uniform cross-sections throughout the length of the vessel to s i m p l i f y the computational e f f o r t . However, t h i s advantage i s not available for the a n a l y s i s of small vessel motion. Extremes of motion which can be tolerated on a small vessel would be disastrous on a ship. Therefore, small motion assumptions used for ship analysis may become inappropriate for the vessels being presently examined. Also, the r a t i o of i n e r t i a terms to the other terms i n the dynamic equation of motion i s very d i f f e r e n t between ships and yachts, so that, once again, the assumptions necessary f o r s i m p l i f i c a t i o n may not be appropriate. Probably the most relevant and comprehensive attempt to understand the behavior of small vessels moored i n a marina environment was presented In a report prepared by Northwest Hydraulic Consultants Ltd. of North Vancouver, B.C. and submitted to the Small Craft Harbours 3 Branch of the Canadian Department of F i s h e r i e s and Oceans. In t h e i r work, Mercer & Isaacson (1981) re-evaluated the commonly used one foot wave height c r i t e r i a as an acceptable condition f o r marina design. Their intent was to determine what factors other than wave height contributed to an unfavourable v e s s e l response to incident waves. E s s e n t i a l l y , four approaches to the problem were taken: model tests were conducted; marinas were v i s i t e d ; a s i m p l i f i e d mathematical analysis was attempted; and measurements were taken on a f u l l scale v e s s e l under exposed conditions. Five of the degrees of freedom (excluding yaw) were examined and conclusions presented on the expected behaviour i n each mode of motion. Although p i t c h , r o l l , and heave were reasonably well understood as the r e s u l t of t h e i r e f f o r t s , the authors concluded that surge and sway, because of i n t r i c a c i e s introduced by moorage, could be predicted only to a high degree of uncertainty. It was f e l t that further e f f o r t , including a study of moorage c h a r a c t e r i s t i c s , was required to better deduce the factors a f f e c t i n g these t r a n s l a t i o n a l motions. The only other a v a i l a b l e Information on attempts to understand small vessel behaviour are discussed i n two papers by Raichlen (1966a, 1968a). The f i r s t , dated 1966, describes t e s t s performed on a si m p l i f i e d rectangular v e s s e l , restrained by a l i n e a r spring i n a standing wave, and permitted surge motion. The work was commissioned by the U.S. Army Corps of Engineers and was c l e a r l y intended to serve as a portion of a more comprehensive study. The model, a rectangular block of wood, was placed i n a standing wave created by a v e r t i c a l r e f l e c t i n g w all. Spring s t i f f n e s s , wave length and p o s i t i o n i n the standing wave were In turn examined. An a n a l y t i c a l solution was also attempted. No 4 attempt was made to r e l a t e the r e s u l t s to actual marina conditions. The second Raichlen study (1968a) examined the natural frequency of f u l l scale vessels i n surge. A number of moorage conditions and rope materials were studied by performing free o s c i l l a t i o n tests with a vessel on a calm water surface. Tests were performed with a 26 foot power boat moored with two bow l i n e s and two stern l i n e s . The motion was recorded by a movie camera and the r e s u l t s compared with an a n a l y t i c a l solution. Agreement was reasonable. 1.3. Research J u s t i f i c a t i o n From review of existing l i t e r a t u r e , i t has become clear that no work has been done which adequately evaluates small vessel behaviour i n surge and sway. The opportunity exists to compliment the work of Mercer and Isaacson (1981) by bringing the knowledge of surge and sway to the same l e v e l as that achieved for p i t c h , r o l l and heave. From this pooled e f f o r t , i t may be possible to more completely develop comprehensive guidelines for marina design, as well as defining more d i r e c t l y what i s meant by proper moorage p r a c t i c e s . By acknowledging the need for a more i n depth understanding of surge and sway behaviour, the objective of the research paper was established. Succinctly stated, i t i s to define the parameters which influence vessel t r a n s l a t i o n a l motion and evaluate the r e s u l t s of a l t e r i n g , i n turn, each of these parameters. Ultimately, recommenda-tions w i l l be forwarded which w i l l imply methods of reducing vessel motion and moorage l i n e hawser forces. The discussion and i n v e s t i g a t i o n w i l l apply to vessels of the order of r e c r e a t i o n a l yachts and power boats when subjected to incident waves as they may experience i n a 5 marina environment. The work examines only surge and sway, and does not attempt to Include an evaluation of a l l modes of motion to develop comprehensive guidelines. However, a l l attempts are made to make the r e s u l t s compatible with those developed by Mercer and Isaacson to f a c i l i t a t e such considerations i n the future. 1.4. Research Approach Two basic approaches were taken to s a t i s f y the objective. F i r s t l y , a series of model tests were performed which presented an opportunity to examine a large number of va r i a b l e s a f f e c t i n g vessel motion. Results are presented i n dimensionless form to o f f e r the opportunity to e a s i l y r e l a t e them to f u l l scale prototypes. Secondly, a s i m p l i f i e d mathematical solution was attempted to simulate surge and sway behaviour of small v e s s e l s . ultimately experimental data i s compared with the mathematical simulation. By way of background work leading up to subsequent experimentation, a number of peripheral tasks were performed. A program of marina v i s i t s was undertaken i n an e f f o r t to i d e n t i f y the v a r i a b l e s which might a f f e c t vessel motion. A des c r i p t i o n of these v i s i t s i s presented i n chapter 2. A survey of l i t e r a t u r e aimed at determining s t r u c t u r a l c h a r a c t e r i s t i c s of commonly used moorage rope was also conducted. As well, tests were performed on a commercially a v a i l a b l e balloon fender, once again to determine i t s str u c t u r a l c h a r a c t e r i s t i c s . These two matters are dealt with i n chapter 5. The subsequent text o u t l i n e s the manner of the experimentation as well as the mathematical analysis used to simulate vessel behaviour i n 6 surge and sway. Results are presented i n graphical form and these are discussed. F i n a l l y conclusions are forwarded and opportunities for future work presented. 7 CHAPTER 2 MARINA VISITS One of the e a r l i e s t tasks undertaken i n the preparation for t h i s research was the i d e n t i f i c a t i o n of common moorage practices and conditions c u r r e n t l y used for vessels of the type under consideration. To t h i s end, a number of v i s i t s were conducted to marinas i n the Vancouver area and i n Kingston, Ontario, and observations made as to the method of moorage, moorage materials, moorage p o s i t i o n , techniques, etc. The intent was not to attempt a s t a t i s t i c a l d e f i n i t i o n of moorage practices, but rather to es t a b l i s h what appear to be 'standard' mooring d e t a i l s among vessels of t h i s s i z e . Table 1 l i s t s the marinas and yacht clubs which were v i s i t e d , chosen for reasons of ease of access, and as a representative sample of va r i a b l e s such as vessel size and cost, exposure conditions, and moorage dock f a c i l i t i e s . As well, interviews were conducted with a number of vess e l owners and t h e i r comments noted. The fundamental and expected difference between the Vancouver and Kingston marinas was the t i d a l v a r i a t i o n s which e x i s t i n the coastal environment of Vancouver while no such d a i l y f luctuations occur on Lake Ontario. As a r e s u l t , the Kingston marinas incorporate a fixed dock condition while those i n Vancouver use a f l e x i b l e or f l o a t i n g arrangement to accommodate the, sometimes large, water l e v e l d i f f e r e n c e s . It was su r p r i s i n g to note that the marina exposure conditions did not dramatically a f f e c t either the method of moorage or the e f f o r t that went into securing the ve s s e l . For example, at the Sea Cove Marina i n 8 Deep Cove, B r i t i s h Columbia, vessels experience d i r e c t exposure over a fetch of approximately one and one half (1-1/2) kilometers, yet there appears to be l i t t l e , i f any, increased e f f o r t i n securing vessels than at the Thunderbird Yacht Club i n West Vancouver, B.C. which i s e s s e n t i a l l y unexposed. Also of note at the Sea Cove Marina, the docking arrangement i s such that some vessels are abeam to the p r i n c i p a l wave d i r e c t i o n while others are abow. Yet, there i s no notable difference between the respective methods of moorage. A more d e f i n i t i v e c r i t e r i o n which influenced the q u a l i t y and consistency of moorage i s undoubtedly the vessel size and cost. Not only did vessel owners i n marinas harbouring more expensive yachts pay greater attention to t h e i r i n d i v i d u a l moorage practices, but also there appeared to be a more consistent trend from vessel to v e s s e l . At the same time, however, i t was not obvious that a great deal of account was taken of the sea d i r e c t i o n i n securing even the more expensive vessels. Moorage position can be defined as the physical r e l a t i o n s h i p between the v e s s e l and the dock. The most common moorage p o s i t i o n was dockside, p a r a l l e l to the d i r e c t i o n of the dock (see f i g . 2a). Occasionally, a header dock (see f i g . 2b) would o f f e r a moorage point on the opposite side of the boat to the dock but the majority of vessel were moored on one side only. "U" shaped finger systems accommodating two vessels (see f i g . 2c) were occasionally encountered which would o f f e r a moorage arrangement very s i m i l a r to the lead boat i n a header system. One system encountered at the Kingston Yacht Club was a bow-in docking (see f i g . 2d). This system i s undoubtedly space e f f i c i e n t but i t presents a vessel access problem. In view of these observations, the bulk of the analysis w i l l be directed at pursuing the best s o l u t i o n 9 to these four commonly encountered moorage p o s i t i o n s . Note was also made of the moorage materials evident during these v i s i t s , the most important of these being the rope used. The most common rope encountered was a Double Braided nylon type with nominal si z e s between 12 and 25 mi l l i m e t e r s . Less frequently used products are 8 p l a i t nylon (hollow braided), twisted nylon, hemp, and cotton but i t seemed to be agreed among those interviewed that the double braided nylon was the most acceptable for vessels of this s i z e . Some vessel owners incorporated rubber s t r i p s i n t h e i r moorage l i n e s i n an attempt to d i s s i p a t e dynamic energy but the use of such products did not appear widespread. The dock moorage point was most commonly a metal dock c l e a t although wooden c l e a t s , wooden poles and wooden r a i l s were also encountered. At some of the v i s i t e d marinas the c l e a t s were of poor or neglected q u a l i t y and the problems of securing the l i n e s to the dock should be considered more c l o s e l y . As well as dock moorage points, anchors were sometimes used to keep the vessel away from the dock. These were usually permanent structures with a connection bouy to which the vessel was moored. Probably the most important purpose i n undertaking the marina v i s i t s was to e s t a b l i s h common methods of moorage. In a l l , a t o t a l of twenty moorage conditions were i d e n t i f i e d and of these, the twelve most common are i l l u s t r a t e d i n figure 3. Although subtle differences may e x i s t (such as between case II and case VII), i t i s the e f f e c t s of these differences on vessel response which are to be determined. From t h i s short l i s t of ten, the experimental and analysis program was set up to best evaluate each moorage condition for d i f f e r i n g environmental conditions and subsequently enter some recommendations for t h e i r use. 10 CHAPTER 3 THEORY A body f l o a t i n g on a free surface i s at l i b e r t y to undergo motion i n s i x degrees of freedom. C l a s s i c a l l y these have been defined as i l l u s t r a t e d on figure 1. Three of the motions are r o t a t i o n a l (yaw, r o l l & pitch) about some axis unique to the geometry of the body and the remaining three (heave, surge and sway) are t r a n s l a t i o n a l . Any or a l l degrees of motion may be i n i t i a t e d , sustained or restrained depending on the nature of the externally applied forces. In the case of the present discussion, the body i s a small c r a f t and the primary e x t e r n a l l y applied forces are those due to the incident waves and the r e s t r a i n i n g moorage l i n e s and fenders. 3.1. General Body Motion As a r e s u l t of the i n t e r a c t i o n of a wave t r a i n , a body can be taken to o s c i l l a t e harmonically i n i t s six degrees of freedom and may be expressed as follows: -iwt j — 1,2,...,6 (1) a j where: j i s the mode of motion 1 surge 2 sway 3 heave 4 r o l l 5 p i t c h 6 yaw a i s the displacement for mode j 11 £ i s the complex amplitude of motion f o r j u) i s the angular frequency of motion t i s time. For the subsequent discussion, a coordinate system as i l l u s t r a t e d i n f i g u r e 1 i s set up with the o r i g i n being at the so-called "centre of f l o a t a t i o n " . This i s the point on the vessel's waterplane surface about which i t pitches and r o l l s i n s t i l l water. The general dynamic equation f o r motion of a s i x degree of freedom system can be expressed i n matrix form as: o 1 V,.a. + C..a + k,.a. = F, f o r i = 1,2,...,6 j= x i j J i j i J J 1 i and j are subscript counters as defined e a r l i e r . For the case of a restrained body on a f l u i d surface: (2) V. . = m. . + a. . = v i r t u a l mass i j i j l j where m, . = vessel mass i j a_^ = hydrodynamic added mass c.. = + + c^.^ = t o t a l system damping i j i j i j i j where c ^ ^ = r a d i a t i o n damping (2) c = viscous damping (3) c = damping due to the moorage l i n e s + k (2) i j = t o t a l system s t i f f n e s s where hydrostatic s t i f f n e s s s t i f f n e s s due to moorage l i n e s and fenders and + F i (2) = hydrodynamic exc i t i n g function where I n e r t i a l component Drag component The vessel mass for each of the t r a n s l a t i o n a l modes of motion i s the actual body mass. For r o t a t i o n a l modes, i t i s a mass moment of i n e r t i a . Further discussion of the mass matrix i s available in Newman (1977). The hydrodynamic added mass and r a d i a t i o n damping are r e s p e c t i v e l y the acceleration and v e l o c i t y dependent influences of the f l u i d f i e l d on the body i f i t were forced to o s c i l l a t e i n s t i l l water. These may be determined by potential theory as w i l l be discussed l a t e r i n t h i s chapter. Viscous damping, i f s i g n i f i c a n t , has to be determined by empirical means. Moorage l i n e damping i s s i m i l a r to a s t r u c t u r a l damping and may be evaluated by the considering the energy los s per cycle represented by the area inside the h y s t e r s i s loop for a load-unload test on the moorage material. It w i l l be dealt with i n chapter 5. Hydrostatic s t i f f n e s s w i l l be discussed l a t e r in t h i s chapter. Moorage l i n e and fender s t i f f n e s s , discussed i n chapter 5, can be determined by considering the f o r c e - d e f l e c t i o n curve for dynamic load t e s t s on the s t i f f n e s s elements. The e x c i t i n g function i s analogous to the force exerted by a wave 13 t r a i n on a f i x e d body which includes a component of force ( i n e r t i a l ) i n phase with the f l u i d p a r t i c l e acceleration and a component (drag) i n phase with the v e l o c i t y . The i n e r t i a l component can be predicted by potential theory as w i l l be discussed below but the drag force i s u s u a l l y evaluated by empirical means. In the matrix equation, when i * j , the intent i s to recognize the influence on the mode of motion " i " of the motion of " j " . In some cases, such an e f f e c t i s zero and the two modes of motion are considered uncoupled, but generally such a cross-coupling e f f e c t e x i s t s . When i = j , i t i s the terms of the standard single degree of freedom which are being addressed. Attention w i l l now be directed towards the understanding of the r a d i a t i o n damping and hydrodynamic added mass c o e f f i c i e n t s and the i n e r t i a l e x c i t i n g function, a l l of which can be determined a n a l y t i c a l l y by p o t e n t i a l flow theory. 3.2. Hydrodynamic Factors A f f e c t i n g Motion Newman (1977) reviews a procedure for evaluating the hydrodynamic factors which appear i n equation (2) by po t e n t i a l flow theory. The approach defines the v e l o c i t y p o t e n t i a l of the f l u i d flow as i t encounters the f l o a t i n g body. V e l o c i t y poential (denoted <j>) i s a representation of the character of motion for each f l u i d p a r t i c l e surrounding the body and has been defined i n three dimensions as: 3 cb v e l o c i t y i n the x d i r e c t i o n (u) = - -J*-3 x deb v e l o c i t y i n the y d i r e c t i o n (v) = -3di v e l o c i t y i n the z d i r e c t i o n (w) = - -jr-Once the v e l o c i t y potential has been evaluated, the e f f e c t of the 14 f l u i d on the body, and by inversion, the body on the f l u i d can then be determined. In order to take a v e l o c i t y potential approach to the problem, i t i s necessary to assume that the flow i s i r r o t a t i o n a l and the f l u i d i n v i c i d . Also, to simplify the analysis, a l i n e a r wave theory w i l l be adopted by considering the wave heights to be of small amplitude. Due to the assumption of l i n e a r i t y , the v e l o c i t y p o t e n t i a l created by a restrained but movable body on a wavy surface can be considered as the sum of three simpler components: that due to the incident waves alone; that due to the scattered waves generated by a fixed body; and that due to a body forced to o s c i l l a t e i n otherwise s t i l l water. Thus, the t o t a l v e l o c i t y potential i s expressed as: * = w^ + ^s + * f ( 3 ) where tj> i s incident wave v e l o c i t y p o t e n t i a l <|>s i s scattered wave v e l o c i t y p o t e n t i a l <j>£ i s forced body v e l o c i t y p o t e n t i a l . Each and a l l of these component po t e n t i a l s must s a t i s f y the following conditions as dictated by p o t e n t i a l theory ( l i n e a r i z e d where appropriate). Laplace Equation: V2<}> = 0 (4) bottom boundary condition: = 0 at the seabed (5) 15 free surface boundary c o n d i t i o n s : (6) 32<|> at z = 0 (7) where: z i s measured v e r t i c a l l y upward from the s t i l l water l e v e l n i s the e l e v a t i o n of the free surface g i s a c c e l e r a t i o n due to g r a v i t y As w e l l , cj>s and ty^ must s a t i s f y a r a d i a t i o n c o n d i t i o n to account f o r a d i s p e r s i o n of energy away from the body: where r i s the distance from the body, and i = / - T . The f i n a l c o n d i t i o n to be s a t i s f i e d , the boundary c o n d i t i o n on the body s u r f a c e , ensures that the v e l o c i t y of the f l u i d normal to the surface i s equal to the v e l o c i t y of the surface i t s e l f : l i m r J-+-00 i k * j = 0 (8) 3n 3n 9n = V on the surface (9) n where: ri i s the d i r e c t i o n normal to the s u r f a c e , V i s the surface v e l o c i t y i n the normal d i r e c t i o n . 16 By assuming the vessel motion to be small, equation (9) can be s i m p l i f i e d i n t o : 3d> 3<b w s + - 5 — = 0 on the body surface ( 1 0 ) 3n 3n and 3<|> ^ = V on the body surface (11) 3n n Equation (10) w i l l be used to evaluate the i n e r t i a l e x c i t i n g f u nction ( F ^ ) . Equation (11) can be used to determine the hydrodynamic added mass and damping of the system. 3.2.1. Hydrodynamic added mass and damping By way of s o l u t i o n to equation (11), we consider the normal v e l o c i t y at any point (x,y,z) on the surface to be composed of the v e l o c i t i e s associated with each of the s i x degrees of freedom. These can be superposed: 6 3a 6 V = ), n, = I -iwa.n.e 1 W t (12) Here n = n , n. = n , n„ = n 1 x 2 y 3 z n. = zn = yn 4 y J z n_ = xn - zn 5 z x n, = yn - xn 6 x y where n^, n^, n^ are the d i r e c t i o n cosines of ri with respect to the x, y and z axes r e s p e c t i v e l y . S i m i l a r l y , i t i s convenient to decompose the forced body v e l o c i t y p o t e n t i a l into those as associated with each of the s i x degrees of freedom, i . e . 17 • f - l a . • < « e " i W t (13) j=l by r e l a t i n g equations (11), (12) and (13) a v e l o c i t y p o t e n t i a l r e l a t i o n s h i p f o r each degree of freedom can be written independent of a .: 9 * ( . f ) - g ^ — = - itonj for j = 1,2 6 (14) on the body surface With the right hand side of equation (14) known, <j>^  can be solved, bearing i n mind that i t must also s a t i s f y the previously i d e n t i f i e d boundary conditions and Laplace's equation. In general, the so l u t i o n may be obtained by a Green's function approach or i n c e r t a i n cases by an a n a l y t i c a l s o l u t i o n involving i n f i n i t e s e r i e s techniques. Having determined the forced v e l o c i t y p o t e n t i a l (<t>^ ) for each mode of motion, the six corresponding components of force on the body, r e s u l t i n g from that p o t e n t i a l can be evaluated: = - lup I [j n dS) a. for i = 1,2,...,6 (15) j=l S J J  J o where: S i s the immersed body surface, o 18 That portion of this force which i s i n phase with the ac c e l e r a t i o n i s the added mass as defined i n equation ( 2 ) . That i n phase with the v e l o c i t y i s the hydrodynamic damping. Therefore, m 6 3 2a . . 3a. F j t ; - - I U i + c\\} j-3-) for i = l , 2 , . . . , 6 (16) j=l 3 3 t 2 3 It can be shown by combining equations (15) and (16) that the added mass and damping values f o r a body can be given by: a. . - - / I [<t>(.f) ] n. dS (17) i j ^ S m j i v ' o 4j} = - P / Re[<j»^f)j n j L dS (18) o where both the added mass and damping c o e f f i c i e n t s are frequency dependent. Im[ ] denotes the imaginary part of the expression, Re[ ] denotes the r e a l part. 3.2.2. E x c i t i n g function The f i n a l term associated with the f l u i d f i e l d i n equation (2), which remains to be solved by pot e n t i a l theory, i s the i n e r t i a l e x c i t i n g function. Equation (10), which related the incident and scatter v e l o c i t y p o t e n t i a l , i s now considered. By introducing a so-called 19 Hasklnd r e l a t i o n s h i p (Newman, 1962), the e x c i t i n g force can be expressed by: 3i|/ f) 3(j) F i e ) = p ' ^ w ' a l - " *±f) "airJ d S f o r 1 = 1»2»"-> 6 <19> s o It i s noted that the scattered p o t e n t i a l i s not e x p l i c i t l y required. <j> can be evaluated by l i n e a r wave theory as though the body w did not influence the wave t r a i n , and < t ) ^ ^ has been discussed above. Sarpkaya and Isaacson (1981) outline a further development which enables the e x c i t i n g force to also be determined by r e l a t i n g i t to the hydrodynamic damping c o e f f i c i e n t s but this w i l l not be discussed here. Equation (19) can be further s i m p l i f i e d by considering a "slender body approximation". Newman (1977) introduces the SBA by defining a slenderness parameter e such that: • - ! where B = vessel beam L = vessel waterplane length It i s demonstrated that for wave lengths of the order of "L" each of the force terms i n the dynamic equation of motion w i l l be of a magnitude related to a power function of e. For the case of surge, the primary f l u i d forces on a slender body, corresponding to the lowest power of "e" are those due to the undisturbed incident wave potential and body mass i t s e l f . A l l others are of a higher power of "e" and hence a lower order of magnitude and may be neglected. What th i s suggests i s that the 20 scattered v e l o c i t y p o t e n t i a l i n equation (10) can be ignored and the ex c i t i n g force on the body calculated as though the body did not disturb the wave motion. Such a force i s termed a Froude-Krylov force. For r o l l , the s i t u a t i o n i s s i m i l a r with only Froude-Krylov and hydrostatic forces being s i g n i f i c a n t . In the case of sway, the slender body approximation i s less accurate. Body forces for sway are of the same order as both mass forces and Froude-Krylov forces. By using the SBA for determining e x c i t i n g forces i n sway, a greater error i s introduced into the analysis than f o r surge. Nevertheless, Mercer, and Isaacson (1981) demonstrated that even for sway, the SBA i s reasonably accurate f o r c a l c u l a t i n g the wave e x c i t i n g force. By considering the SBA, equation (19) can reduce to: F $ e ) = P / • dS for i = 1,2,...,6 (20) S w o 3.3. Hydrostatic S t i f f n e s s The force exerted on a body by the f l u i d upon which i t i s at rest i s termed the hydrostatic force. In each of the s i x modes of motion, the force can be written. F5H ) = - P g // n z dS for i = 1,2,...,6 (21) S o Newman (1977) has demonstrated that motions from s t a t i c equilibrium w i l l introduce net r esultant forces on the body as follows: F ( H ) = " I r 5. for i = 3,4,5 (22) j=i 1 J J where: r33 = p 8 S r44 = P 8 S 2 2 + P g V z B - m g z G  r55 = P 8 S n + P g V z B - m g z G a l l other r ^ = 0 where S i s the waterplane area S = // x 2 dS s o s „ = // y 2 ds s o Zg = the elevation of the centre of bouyancy z^ = the elevation of the centre of g r a v i t y V = displaced volume of the v e s s e l . It i s these forces which w i l l form the hydrostatic s t i f f n e s s forces i n equation (2) above. 3.4. M o d i f i c a t i o n for Surge and Sway of the Equation of Motion The evaluation of motion i n surge and sway alone does not necessitate the solution to the six degree of freedom general system and can thus be s i m p l i f i e d . Since coupling terms for surge are zero or small a v e s s e l exhibits e s s e n t i a l l y s i n g l e degree of freedom behaviour i n surge and can be described as: ( ^ + a u ) x + ( c ( J > + c ^ + c < f ) W k < ^ - F ^ - f (23) I t should be noted that °as been omitted as a body has no inherent hydrostatic s t i f f n e s s against surge. To evaluate sway, the influence of coupled r o l l must be included. Therefore, a two degree of freedom system must be solved and may be expressed as: = F ^ 1 ) + F ^ 2 ) for i = 3,4 (24) Here: iS.1) = 0 for 1 * j * 4 l j It i s the solution of equations (23) and (24) which w i l l form the b a s i s of the computer analysis discussed i n chapter 4. 23 CHAPTER 4 ANALYSIS In conjunction with the experimental program to be discussed i n l a t e r chapters, a computer program has been developed to predict the surge and sway behaviour of small vessels. It i s based on a s i m p l i f i c a t i o n of the theory discussed e a r l i e r and u t i l i z e s a time step solution to the dynamic equations of motion for surge and sway/roll i n turn. A time step procedure i s used to accommodate n o i r l i n e a r s t i f f n e s s c o e f f i c i e n t s as well as permitting the evaluation, by considering Froude-Krylov forces, of the e x c i t i n g function. With a mathematical model, a much wider range of moorage conditions and h u l l shapes can be examined than could be possible with a purely experimentation approach. V e r i f i c a t i o n of the computer model i s accomplished by running the program with the dynamic c o e f f i c i e n t s and e x c i t i n g function determined for the model vessel and comparing the r e s u l t s of the computer analysis to those obtained through experimentation. 4.1. Computer Program Logic Appendix I contains the computer l i s t i n g of the program "YACHT" developed to predict surge and sway behaviour of small vessels. The program evaluates surge and sway/roll independently and an operator statement allows the user to choose one or both as desired. Input i s submitted as indicated i n appendix I and includes c h a r a c t e r i s t i c s of the wave t r a i n , moorage l i n e s , hydrodynamic and hydrostatic c h a r a c t e r i s t i c s of the v e s s e l , physical dimensions of the vessel, moorage l i n e p o s i t i o n i n g and slackness. A l l c o e f f i c i e n t s of the equations of motion are inputted but the exciting function i s cal c u l a t e d . Dealing with only displacement as a v a r i a b l e i n the time step s o l u t i o n , i t i s sought to solve for the dynamic equilibrium of forces at each time i n t e r v a l . In essence, the program, beginning with i n i t i a l conditions at time " t ", tests a series of displacement increments at time " t , " . Each o 1 displacement r e s u l t s i n a force value for each term i n the dynamic equation of motion and the displacement which best s a t i s f i e s dynamic equilibrium i s taken as the unique and true value for displacement at time " t ^ " . Acceleration, v e l o c i t y , and displacement at time " t ^ " are assigned as the i n i t i a l conditions f o r the next time increment for which the procedure i s repeated. The r e s u l t over a large number of time increments i s the time dependent response. For the most part, i t i s the steady state peak to peak motion which i s of i n t e r e s t and t h i s value i s obtained f o r each moorage and wave condition a f t e r a number of wave cy c l e s . For the degree of damping t y p i c a l f o r the vessels to be examined, ten cycles usually i s s u f f i c i e n t . Mathematical model increments have been chosen as follows: AT = 30 Ax = Ay = 3 x 10~ 6 A6 = 2 x i o - 6 rad where: AT = time increment Ax = surge displacement increment Ay = sway displacement increment 25 A9 = r o t a t i o n a l increments L = waterplane length of the vessel T = wave period The computer program, i n evaluating time dependent motion assumes l i n e a r behavior between succeeding time increments. Thus, f o r each time i n t e r v a l being analyzed, the following r e l a t i o n s h i p s hold: a - cx a = A T ° (25a) AT *1 = ' ^ T r ^ ^ o ( 2 5 b ) 2(a. - a ) 2a a - — ~ 2_ - _ £ (25c) AT 2 4(a, - a ) 4a °i - - ^ r ^ - T ? - « . <25J> AT where: AT = duration of the i n t e r v a l a Q = displacement at beginning of time i n t e r v a l a^ = deplacement at end a^ = v e l o c i t y at beginning a = average v e l o c i t y over the time increment a^ = v e l o c i t y at end a = acceleration at beginning o 26 a = average a c c e l e r a t i o n a, = acceleration at end From these equations, the average displacement, v e l o c i t y , and a c c e l e r a t i o n can be determined from an assumed value of ct^ since a l l other terms on the right hand side of the equation are known from the previous time i n t e r v a l . By s u b s t i t u t i n g equations (2 5) into equation (23), the dynamic equation of motion for surge can be written: ( m 1 1 + a i l ) 2(x rx o) Cffi + (ff> + <ff>) AT 1 AT 2 V + AT ( X _ X 0 ) + S x o * S x l = pJD + F ^ 2 ) (26) Here: S x q = moorage l i n e s t i f f n e s s force at the beginning of the time i n t e r v a l = moorage l i n e s t i f f n e s s force at the end of the time i n t e r v a l Sway motions, because they involve r o l l cross-coupling are s l i g h t l y more complicated. By applying equations (25) to eqution (24) for A 1 = 3, the dynamic equation of motion for sway can be written: 27 <°33 + °33> ( 2 < ^ o > 2- j , i f f - 4V * < f f l , AT L AT y o J AT ^ y yo} s + s . e_+e ... ... ... e +e, + .1° Zl+ ( m + a ) f 1 °1 + f c ( 1 ) + C ( 2 ) + C ( 3 ) l f ° 11 2 ^34 a34 ; L 2 J l L34 L34 °34 J 1 2 J = F^ 1 ) + F < 2 ) (27) Here: y = sway 6 = r o l l It should be noted that the terms S and S , include the yo y l moorage l i n e s t i f f n e s s against sway for both sway and r o l l motions. It i s therefore not necessary to Include an ad d i t i o n a l term f o r sway s t i f f n e s s corresponding to angular displacements i n r o l l . S i m i l a r l y , the equation of motion f o r r o l l can be written by l e t t i n g i = 4: (m.. + a..) 2 ( 6 - 6 ) ( c ^ + G\ ( 2 ) + G\ ( 3 )) 4 4A T 4 4 [ * ° - 26 ) + i - 4 4 4 4 (6 - 6 ) AT v AT oJ AT v o' s e o + s e i (yi+y0] f ( i ) (2) . co)]{yo+\ + 2 + (m43 + a43 } { ~ 2 ~ } + ^43 + C43 + C43 J l _ 2 ~ ~ J - F J X ) + F j 2 ) (28) Roll s t i f f n e s s moments (S„ , S Q.) are the sway forces discussed DO o l above times a moment arm which i s the height above the waterplane l e v e l at which the moorage l i n e s are attached to the v e s s e l . Although equations (2 7) and (2 8) could be solved simultaneously or through a number of i t e r a t i o n s , i t i s judged that the eff e c t of r o l l on sway i s not s i g n i f i c a n t enough to warrent t h i s computation. What i s done, however, i s to evaluate r o l l for the current time i n t e r v a l by solving equation (28) using values of vessel sway v e l o c i t i e s and accelerations determined from the previous time i n t e r v a l . The computed r o l l i s then used to evaluate sway i n equation (27) for the current i n t e r v a l and the process repeats i t s e l f f o r the next time i n t e r v a l . Attention w i l l now be directed towards determining the matrix c o e f f i c i e n t s i n the equation of motion. 4.2. Dynamic C o e f f i c i e n t s Each c o e f f i c i e n t i n the dynamic equations for surge, sway and r o l l depends on the unique geometry of the vessel under consideration. As discussed in chapter 3 some have to be determined empirically while others may be evaluated either through t h e o r e t i c a l development or by empirical means. For the most part, c o e f f i c i e n t s for mathematically modeling the vessel used i n the experimental program were determined emp i r i c a l l y although some reference to t h e o r e t i c a l l y predicted values were also made. The necessary mass components are given as: m n = m m33 = m m44 V1!! m43 • "34 = - m z G body mass coordinate of the centre of g r a v i t y mass moment of i n e r t i a about the x axis where: m ZG Added mass can be determined t h e o r e t i c a l l y by equation (17) or by empirical means. Chapter 7 describes tests performed to determine the added mass c o e f f i c i e n t s e m p i r i c a l l y by subjecting the vessel to a serie s of calm water t e s t s . As discussed i n chapter 3, damping c o e f f i c i e n t s can be expressed as the sum of three components. Radiation damping can be determined by equation (18) but viscous damping must be evaluated e m p i r i c a l l y . For the purposes of determining the above two hydrodynamic damping c o e f f i c i e n t s f o r the model ve s s e l , a s e r i e s of te s t s were performed and are described i n chapter 7. Based on these tests, i t i s not possible, nor necessary to separate r a d i a t i o n from viscous damping as they are always considered together. Moorage l i n e damping on prototype vessels i s discussed i n chapter 5. The moorage material used for the experimental model exhibits l i t t l e damping but the damping introduced by the motion measurement system can be considered as analogous to moorage (3) damping on a prototype and has thus been included as C ^ in the mathematical model. Evaluation of t h i s c o e f f i c i e n t i s discussed i n chapter 7. Hydrostatic s t i f f n e s s i s relevant f o r r o l l only. It can be evaluated t h e o r e t i c a l l y as discussed i n chapter 3 or by subjecting the ve s s e l to s t a t i c r o l l t e s t s . Mercer and Isaacson (1981) performed the l a t t e r tests on the model vessel and t h e i r value for r o l l s t i f f n e s s was used i n the computer model. The moorage element s t i f f n e s s forces are evaluated by determining the force corresponding to a prescribed displacement from a s t i f f n e s s curve as i l l u s t r a t e d i n fig u r e 4. The s t i f f n e s s curve Is made up of two parts: the slackness (A-B) and the 30 force elongation curve for the moorage material (B-C). The s t i f f n e s s term i n the equation of motion i s therefore generally non-linear. It i s not necessary to c a l c u l a t e a moorage l i n e s t i f f n e s s c o e f f i c i e n t as the f o r c e - d e f l e c t i o n curves are input d i r e c t l y Into the mathematical model. 4.3. E x c i t i n g Function The r i g h t hand side of equations (2 6), (2 7) and (28) are evaluated i n the computer model by applying the slender body approximation and c a l c u l a t i n g forces based on the undisturbed incident wave p o t e n t i a l . The wave p o t e n t i a l , together with r e s u l t i n g expressions for f l u i d pressure, v e l o c i t y , a c c e l e r a t i o n , etc. are based on l i n e a r theory and are available from any text on elementary wave theory (eg. Sorenson (1978) or the Shore Protection Manual (1977)). A s i m p l i f i c a t i o n to the model vessel i s made as i l l u s t r a t e d i n figure 5. The parameters defining the shape have been chosen to accommodate any r e c r e a t i o n a l v e s s e l . It i s important when simplifying the vessel shape for analysis that the displaced volume of the actual vessel and s i m p l i f i e d shape be the same. For surge, the i n e r t i a l component of the e x c i t i n g force on the body i s evaluated by considering the pressure of the incident waves acting over the s i m p l i f i e d body shape. Only the end projections of the vessel need be included as other forces cancel. The ends are divided into a s e r i e s of h o r i z o n t a l s t r i p s and the t o t a l force determined by summing up the forces acting on a l l the s t r i p s . Therefore: 41' - ip.» (29) 31 A = area of each s t r i p 1 _ cosh(ks) i • , v . « ,, , i P = ~2 P g cosh(kd) Lcos(kx " «t) + R cos(kx + tot)J where: p = wave pressure over the segment p = mass density of water g = acceleration due to g r a v i t y H = incident wave height k = 2n/wave length s = d + z R = r e f l e c t i o n c o e f f i c i e n t a) = angular frequency of the wave d = depth of water z = depth to the centroid of the segment The surge drag component i s evaluated from the expression: Fi 2 ) = 1 9 S A H u N ( 3 0 ) where: A = wetted area of the ve s s e l = empirical drag c o e f f i c i e n t dependent on Reynolds number (0.005-0.03) Newman (1977) u = the horizontal v e l o c i t y of the water p a r t i c l e s at the vessel's centre of bouyancy TTH cosh(k(z B+d) ) (cos(kx-wt) - Rcos(kx+wt)J where: T sinh(kd) 2TT , , = — = wave period z = z coordinate of centre of bouyancy (B) B For the vessel used i n the experiments, a drag c o e f f i c i e n t of 0.02 was used. However, the i n e r t i a l component of the e x c i t i n g function i s much greater than the drag component f o r surge. Therefore, the e x c i t i n g function i s not very sensitive to the choice of drag c o e f f i c i e n t . For sway, the i n e r t i a l component can be determined by assuming the displaced mass i s concentated at the vessels centre of bouyancy and determining the f l u i d a c c e leration at that point. Therefore: F o 1 ) = m „ • a. (31) 2 33 horz v ' where: a, = horizontal a c c e l e r a t i o n of the f l u i d horz 2gH cosh(k(z +d)) = , ... » Isln(kx-wt) + Rsln(kx+wt)l T2 sinh(kd) This force acts through the centre of bouyancy. Sway drag i s determined i n the same way as surge with only the drag c o e f f i c i e n t s and area d e f i n i t i o n changing. In the case of sway, the body shape i s s i m i l a r to that of a f l a t plate normal to the flow so values as obtained by Vugts (1968) for such a condition can be used. For the v e s s e l under consideration, a drag c o e f f i c i e n t f o r sway of 20 i s used. The area involved i s the vessel's side p r o f i l e area. Because of flow separation which occurs around the keel and rudder, the drag component of the ex c i t i n g function i s s i g n i f i c a n t for sway. The l i n e of a c t i o n f or t h i s force i s through the centre of area of the vessel i n p r o f i l e . R o l l can be considered as the couple created by the sway force about the vessel's center of f l o a t a t i o n . The i n e r t i a l component i s 33 given by: z B and drag by z cen where z = z coordinate of centroid of the vessel' s p r o f i l e area cen (D). 4.4. V e r i f i c a t i o n of the Time Step Solution To v e r i f y the performance of the computer program i n solving the dynamic equations of motion as defined e a r l i e r , i t was used to generate a s o l u t i o n f o r a system of a known a n a l y t i c a l s o l u t i o n . For surge, a problem involving a harmonic exc i t i n g function and u t i l i z i n g nonlinear s t i f f n e s s elements was set up and solved. The s o l u t i o n was then compared to a Ritz approximation so l u t i o n of the f i r s t order (Timoshenko et a l . , 1974) to the same problem and the two solutions i l l u s t r a t e d i n figure 6. It should be noted that only steady state behaviour i s a v a i l a b l e by the R i t z s o l u t i o n and i t i s t h i s feature of the response which i s i l l u s t r a t e d . The Damping term i n the equation of motion i s l i n e a r while the s t i f f n e s s term i s given by the following: k (x)»x = p(x ± ux 3) where p and p are constants. As can be seen, the differences are well within the accuracy demanded f o r t h i s a n a l y s i s . It should be emphasized that the v e r i f i c a t i o n means only that the dynamic equation of motion i s solved c o r r e c t l y and does not address d i r e c t l y the f l u i d structure i n t e r a c t i o n problem. The l a t e r w i l l be v e r i f i e d by comparison to the experimental r e s u l t s . For sway, a two degree of freedom system problem with l i n e a r s t i f f n e s s elements was used to v e r i f y the performance of "YACHT". Results of both the a n a l y t i c a l steady state s o l u t i o n and the time step solution are i l l u s t r a t e d i n figure 7. The a n a l y t i c a l solution was obtained by solving the two dynamic equations of motion (sway and r o l l ) simultaneously i n a manner as outlined by Clough & Penzien (1975). As was the case f o r surge, the sway r e s u l t s are within the accuracy demanded for th i s analysis. 35 CHAPTER 5 MOORAGE LINES AND FENDERS An unrestrained vessel resting on a free surface has no inherent dynamic s t i f f n e s s against surge and sway. For a moored vessel, the s t i f f n e s s elements which tend to return the boat to a centering p o s i t i o n are the moorage l i n e s and protection fenders. It i s these elements which therefore define the dynamic c h a r a c t e r i s t i c s of the moored system and i t i s to the understanding of these that t h i s chapter Is devoted. 5.1. Moorage Lines In an attempt to include the moorage l i n e behaviour i n the present i n v e s t i g a t i o n , knowledge of two s p e c i f i c c h a r a c t e r i s t i c s are required. That i s , the force-elongation r e l a t i o n s h i p and the si z e and shape of the hysteresis loop for damping under c y c l i c loading. Although there e x i s t s a wide range of rope types as well as s i z e s , i t i s conceded that by f a r the most common for moorage purposes i s a double braided nylon type produced by a number of manufacturers i n North America. It was therefore decided that t h i s material would be emphasized for the purposes of t h i s discussion and no major s p e c i f i c attempt was made to compare i t s performance to that of other rope types. A number of sources have examined and reported on the behaviour of double braided nylon including Raichlen (1968a) and Wilson (1967). For the most part, these research i n v e s t i g a t i o n s dealt with larger diameter specimens such as those for use with ships but the information remains v a l i d as the nature of the manufacturing process s t r i v e s to make f i b r e 36 strength non-dependent on area. In add i t i o n to the published sources a number of North American manufacturers were contacted for t h e i r t e c h n ical assistance. From these discussions, i t was agreed that the manufacturing process for double braided nylon rope d i f f e r e d l i t t l e between each company and that reasonable uniformity i n the fi n i s h e d product exists across the industry. A l l agreed that the most comprehensive information on nylon rope could be obtained from the testing f a c i l i t i e s of the I n s t i t u t e of Ocean Sciences i n Bedford, Mass. The i n s t i t u t e i s a support group to Sampson Ocean Industries of Boston, Mass. From Sampson a technical report e n t i t l e d , "Strength, Elongation, and Energy Absorption of Synthetic Fibre Ropes" was obtained. The behaviour of nylon rope i s characterized by a permanent set r e s u l t i n g both from a squeezing of f i b r e s and a realignment of the long chain polymer molecular strands to orient themselves i n the d i r e c t i o n of the force. The permanent set i s followed by an e l a s t i c behaviour. The axes of the hysteresis loops are p a r a l l e l and achieve a s t a b i l i t y a f t e r a number of r e p e t i t i o n s (usually of the order of 100) and do not creep to greater s t r a i n . It therefore follows that the h i s t o r y of a rope specimen i s of major i n t e r e s t . As much as a 10% ultimate strength reduction can be experienced by a wet rope but i t i s not clear the e f f e c t on the force elongation that can be expected. Another condition which influences the behavior of nylon rope i s moisture content. For the purposes of t h i s work, only dry s t a b i l i z e d material subjected to c y c l i c loading w i l l be considered. The force elongation curve for t h i s m a t erial i s i l l u s t r a t e d i n fi g u r e 8. The damping introduced by subjecting a nylon l i n e to c y c l i c loading 37 i s s i m i l a r to a s t r u c t u r a l damping. Nevertheless, i t can be expressed as an equivalent viscous damping by considering the energy dissipated per load/unload cycle and converting i t to an equivalent damping c o e f f i c i e n t to recognize the same energy. This can be done by evaluating the area of the h y s t e r s i s loop and may be expressed by: KPL C E Q TTA 2U> where: c g ^ = equivalent viscous damping c o e f f i c i e n t K = average width of the h y s t e r s i s loop i n percent of s t r e t c h = 0.02 for DB nylon P = load range L = length of rope A = cross s e c t i o n a l area to = angular frequency of cycles For double braided nylon l i n e s the damping i s t y p i c a l l y between one and fi v e per cent of c r i t i c a l damping. 5.2. Fenders Protective fenders f o r vessels encountered i n a marina may be a manufactured unit or makeshift design. The l a t t e r includes rubber t i r e s , rope meshes, and offshore bouys to name a few. The former i s usually an a i r f i l l e d polymer unit which may be either pressurized or under atmospheric pressure. No manufacturer's technical information could be located as to the compressive behaviour of these manufactured units nor has any work been done to characterize the home-fashioned versions. In the absence of any commercial information, i t was decided that to obtain the desired data for dynamic c h a r a c t e r i s t i c s , a t y p i c a l commercially a v a i l a b l e specimen would be tested. The unit examined was a Norweigen polyform fender model F3 manufactured by ALESUND CORPORATION. Its dimensions are shown on figu r e 9a and i t was test loaded as i l l u s t r a t e d by figure 9b. It was capable of accommodating pressures of twenty to f o r t y (20-40) MPa. The tes t s were run at t h i r t y -f i v e (35) MPa and a constant s t r a i n rate of twelve and one half (12.5) mm/min. The r e s u l t i n g force d e f l e c t i o n curve i s i l l u s t r a t e d by f i g u r e 10. There was l i t t l e evidence of a hy s t e r s i s loop although the s t r a i n rate may have been too small to get a meaningful unloading curve. It i s emphasized that no attempt was made to rela t e the speciment or test r e s u l t s s t a t i s t i c a l l y to a l l yacht fenders. The test specimen chosen remains merely a t y p i c a l example. 39 CHAPTER 6 EXPERIMENTATION SET-UP Experimentation was performed i n the Coastal engineering laboratory at the Un i v e r s i t y of B r i t i s h Columbia over a period of two months i n the spring of 1983. The following i s a d e s c r i p t i o n of some of the d e t a i l s involved i n preparing f o r the experiment t r i a l s . 6.1. Wave Basin Description The U.B.C. hydraulics wave basin, i n which a l l of the tests were conducted i s a 12 m long by 5 m wide concrete block structure as i l l u s t r a t e d by fig u r e 11. Figure 12 i s a photograph of the wave basin and experiment layout. The f l o o r of the basin i s smooth concrete with only enough slope to provide drainage and can be considered h o r i z o n t a l . Waves are generated at one end of the basin by a hinged paddle driven by a 2 hp motor through a v a r i - b e l t transmission providing frequencies between 0.4 Hz and 2.0 Hz. The paddle had a maximum excursion of 23 degrees about i t s hinge point and d i f f e r e n t sized waves could be generated by adjusting t h i s v a r i a b l e . High Frequency e f f e c t s were reduced by a double f i l t e r i n s t a l l e d 0.5 m from the wave generator. Side r e f l e c t i o n was controlled by i n s t a l l i n g horse hair mats as energy d i s s i p a t o r s along the side of the basin. A d i s s i p a t i o n beach of double layer horsehair on a slope of 1:9 was i n s t a l l e d at the far end of the basin. Preliminary t e s t s to evaluate the degree of wave r e f l e c t i o n w i l l be described i n chapter 7. The mere existence of the wave r e f l e c t i o n d i d not, however, present a problem. The c h a r a c t e r i s t i c s of the p a r t i a l standing wave set up i n the basin i s peculiar to the p a r t i c u l a r wave frequency. After being evaluated and positioned, the basin wave r e f l e c t i o n provided a rather r e a l i s t i c simulation of actual marinas which often experience wave r e f l e c t i o n , and, as such, offered an i n t e r e s t i n g extension to the o r i g i n a l scope of the research. No attempt was made to evaluate the existence of any transverse standing waves. A l l tests examined i n - l i n e behaviour only so that any transverse motion was not of concern. However, the experiments were conducted i n the l o n g i t u d i n a l c e n t e r l i n e of the basin. It was judged that measurements taken at t h i s point were s u f f i c i e n t l y d i s t a n t to avoid any side wall interference on the f l u i d v e l o c i t y p o t e n t i a l as i t was created by the vessel motion. 6.2. Moorage Dock Frame The model vessel used i n the tests was moored to an experimental dock as i l l u s t r a t e d i n figure 13 and photographed i n figure 14. Mercer and Isaacson (1981) determined that f o r s i m i l a r experiments the d i f f e r i n g influence between a fixed and f l o a t i n g dock on the vessel behaviour was i n s i g n i f i c a n t . It was therefore decided that a s t r u c t u r a l l y r i g i d framework would be fabricated for the experiments which could be attached to the underside of a moveable carriage as i l l u s t r a t e d i n the figure. By securing the framework to the carriage, t e s t s could be performed at any p o s i t i o n along the length of the basin. The f i n a l design provided a structure s t i f f enough that i t did not behave dynamically to influence the v e s s e l motion. Four wooden "docks" 4 1 were attached which offered the desired a b i l i t y to test the vessel i n a number of varied positions, accommodating both head and beam seas. The b o l t s which secured the docks to the framework were located such that they would be the attachment points for the moorage l i n e s . This ensured that the moorage l i n e s were attached r i g i d l y to the frame and no dynamic interference by the connection to a wooden dock could be expected. The e l e v a t i o n of the frame was established high enough so that the lowest parts of the structure did not encounter the incident waves. Furthermore, the frame had to be positioned low enough such that the top of the dock and thereby the point of moorage was at the same l e v e l as the deck of the moored model v e s s e l . This was necessary to keep the moorage l i n e s horizontal and thereby eliminate s i g n i f i c a n t force components i n the l i n e s from vessel heave. Although generally moored vessels are not moored i n a horizontal plane, t h i s approach was taken to avoid experimental influences of heave. Ultimately, there was a small but acceptable margin to work within these two constraints at the prescribed water depth. The docks themselves were of wooden construction with no attempt made to model the e l a s t i c c h a r a c t e r i s t i c s of a r e a l wooden dock. This was unnecessary as modeled fenders would be i n s t a l l e d between the dock and the vessel so the dock behaviour did not influence the vessel d i r e c t l y . The e n t i r e assembly, of aluminum construction, was secured as a ca n t i l e v e r to the underside of the movable carriage and s a t i s f i e d a l l of the s e r v i c e a b i l i t y o b j e c t i v e s . 42 6.3. Experimental Scaling The matter of modeling f o r f l u i d structure i n t e r a c t i o n , i n t h i s case the structure being the dynamic moored ves s e l , i s a subject of on-going discussion. The problem a r i s e s from the r e a l i t y that the Reynold's number, a standard among f l u i d mechanics modeling, cannot be maintained as constant from prototype to model. Such a problem i n e v i t a b l y raises questions as to whether the model tests are able to adequately predict prototype behavior when subjected to a viscous f l u i d . Nevertheless, alternate techniques have been developed to address f l u i d s t r ucture modeling and i t i s upon the basis of these that the experimentation has been established. A d e s c r i p t i o n of the r a t i o n a l e behind dynamic s i m i l a r i t y can be obtained from Sarpkaya and Isaacson (1981) where the various dimensionless numbers are presented and discussed. It i s convenient, for the purposes of r e l a t i n g model to prototype, to use scale factors i n d e f i n i n g r e l a t i v e q u a n t i t i e s . A scale f a c t o r , designated "K", can be defined as the r a t i o of a value i n the model to the same value i n the prototype. That i s : b K, = -rr- for quantity "b" D D m For the purposes of t h i s work, as with v i r t u a l l y a l l wave motion modeling s i t u a t i o n s , the following requirements w i l l be met: a) K = 1 P b) K = 1 g K 2 43 where: p = f l u i d d e n s i t y g = g r a v i t a t i o n a l a c c e l e r a t i o n u = v e l o c i t y L = c h a r a c t e r i s t i c length From these a s s o c i a t i o n s a l l of the f o l l o w i n g r e l a t i o n s h i p s can be d e r i v e d : K K„ K a K P K K M K„ = — /S 1 /S 1 = s = s3 = s" Here "S" i s the length scale f a c t o r , K . Li For a s c a l e f a c t o r of 10, as i s our case, the f o l l o w i n g m u l t i p l i c a t i o n f a c t o r s w i l l r e l a t e the model to the prototype: v e l o c i t y (u) 3,162 p e r i o d (T) 3,162 frequency ( f ) 0,316 a c c e l e r a t i o n (a) 1 pressure (p) 10 f o r c e (F) 1000 moment (M) 10000 length (L) 10 6.4. V e s s e l D e s c r i p t i o n The v e s s e l used i n the t e s t s was the same as the one used by Mercer and Isaacson i n t h e i r 1981 experiments. S p e c i f i c a l l y , i t i s a 44 m o d i f i c a t i o n to a 900 mm long f i n - k e e l racing sloop commercially a v a i l a b l e . Alternations were made to s a t i s f y dynamic simlitude to a Swiftsure 24, t h i s being a 24 foot long s a i l b o a t . These included the removal of 150 mm from the midsection and 90 mm from keel as well as a b a l l a s t reduction of 1.5 kg. Mast and rigg i n g were s i m i l a r l y arranged to model such a f u l l scale v e s s e l . Figure 15 i l l u s t r a t e s the f i n a l p h y s i c a l c h a r a c t e r i s t i c s of the ves s e l and table 2 compares some key quantities with those of the Swiftsure 24. The scaling factor in t h i s case i s 10 as discussed i n the previous s e c t i o n . 6.5. Instrumentation Probably the most important component of conducting successful and meaningful tests i s the design and f a b r i c a t i o n of a s u i t a b l e instrumentation system. For the te s t s , two quantities were measured: wave p r o f i l e and vessel surge (or sway). Wave height was measured by a Robert Shaw capacitance type water l e v e l transducer (see f i g . 16). It included a v e r t i c a l wave probe which provided a l i n e a r l y varying voltage output as the water l e v e l f l u ctuated. Input voltage was a constant 26.6 v o l t s from a power supply. The signal was patched through a 'Linear' horizontal s t r i p chart recorder and attenuated to the desired scale. In addition to the wave probe output, a v i s u a l check was kept on the wave heights as the e l e c t r o n i c system was subject to some d r i f t . Measuring vessel motion presented what proved to be a se n s i t i v e problem as the vessel was susceptible to even the smallest of induced forces by any measurement system. The f i n a l design i s i l l u s t r a t e d i n f i g u r e 17. 45 It c o n s ists of a balsa wood "Z" shaped frame hinged at points "C" and "B" (see f i g . 18). It i s attached to the vessel by a greased b a l l j o i n t (point "A") at a s p e c i f i c point, chosen to eliminate the interference of other modes of motion on the desired t r a n s l a t i o n a l measurements. This point d i f f e r e d for surge and sway. In surge, i t wa<s positioned on the v e r t i c a l c e nterline of the vessel at the centre of f l o a t a t i o n . I t s l o n g i t u d i n a l p o s i t i o n was determined by permitting the vessel to p i t c h in s t i l l water and observing i t s behaviour. In sway, defined on the s t i l l water l e v e l as discussed i n chapter 3, the point oif attachment was once again on the v e r t i c a l c e nterline and the plane of f l o a t a t i o n of the v e s s e l . However, i t s l o n g i t u d i n a l p o s i t i o n was established by placing i t at the center of yaw, being that v e r t i c a l axis about which the vessel yaws when subjected to waves. T h i s point too was i d e n t i f i e d by preliminary t e s t s . The end r e s u l t i s that although the v e s s e l was permitted to p i t c h (or r o l l ) f r e e l y no component of p i t c h (or r o l l ) was read as surge (or sway) on the instrumentation. Heave was permitted by i n s t a l l i n g a hinge at point "B", but by f i x i n g the angle at "F" the p o s i t i o n of B could remain defined by the vessel's forward motion. The geometry i s such that a heave of ± 25 mm introduces only a 5% error on the surge (or sway) measurements. There was enough f l e x i b i l i t y b u i l t i n t o the j o i n t s at "C" and "D" to permit sway i n head seas (or surge i n beam seas). Yaw was not a problem i n head seas as the point "A", being on the vessel c e n t e r l i n e no component was transfered to the surge measurements. In beam seas however, yaw could influence the measurements so point "A" was positioned on the v e r t i c a l axis of yaw. The f i n a l system permitted motion i n a l l degrees of freedom but only surge (or sway i n beam seas) would r e g i s t e r on the measurement. The end r e s u l t i s that point "B" would exhibit t r a n s l a t i o n a l movement i n surge (or sway) exactly l i k e point "A". It therefore remains to f i n d an e f f i c i e n t means to measure the motion of point "B". This was accomplished by attaching a lightweight but s t i f f polyester thread at point "E", passing i t under the moveable carriage, under a r o l l e r at "G", over a second r o l l e r at "H" and attached to the core of a v e r t i c a l l y mounted l i n e a r voltage displacement transducer (LVDT) which would record the motion. The LVDT was a Textronics type with a maximum excursion of ± 75 mm and i s photographed i n figure 19. It uses a 10 v o l t d.c. power source and the s i g n a l was patched into the same chart recorder, on which the wave height and motion were recorded simultaneously. The weight of the core (0.1 N) acting v e r t i c a l l y tended, under s t a t i c conditions, to tug the vessel astern. To counteract t h i s a second weight of 0.2 N was attached over a r o l l e r to point "D" on the v e r t i c a l member C-D such that the vessel rested i n s t a t i c equilibrium. The r o l l e r s at G, H ii J were l u b r i c a t e d d a i l y with a l i g h t WD10 o i l and t h e i r performance was very e f f i c i e n t . The t o t a l e f f e c t i v e h o r i z o n t a l mass of the system, including the balsa wood, LVDT core, and counterweight, when transfered to point "A" was 30 g, a mere one percent of the vessel weight, and equivalent to 30 kg on a f u l l scale v e s s e l . With the net s t a t i c force being zero and the mass-acceleration product being i n s i g n i f i c a n t , the only forces induced on the vessel undergoing dynamic motion was the viscous damping created by the r o l l e r s and the f r i c t i o n of the LVDT core. Tests outlined i n a l a t e r chapter have evaluated these terms as being approximately 1% of c r i t i c a l 47 damping. This f i g u r e represents only 4% of t o t a l v essel damping i n sway but i s as high as 25% in surge. This was, however, deemed acceptable because i t was included as representing part of the damping created by moorage l i n e s on a r e a l v e s s e l . Also, as the computer analysis w i l l v e r i f y , f o r l i g h t damping the actual magnitude of damping w i l l not s e r i o u s l y a f f e c t the peak to peak displacements except at resonance conditions. 6.6. Model Moorage Lines In the search f o r an appropriate material to represent the nylon l i n e s which we seek to model, not only need the force elongation behaviour be assimilated but also the degree of i n t e r n a l damping must be considered. In t h i s case, a material which has a very narrow h y s t e r s i s loop i s sought. In addition to possessing the required e l a s t i c c h a r a c t e r i s t i c s , the modeling material should be convenient to use during the actual experiments. Several material types and s i z e s were examined and the f i n a l system established to s a t i s f y these objectives. It resulted i n an arrangement c o n s i s t i n g of 1/2 mm diameter braided nylon twine which could be considered as p e r f e c t l y s t i f f (see f i g . 20). E l a s t i c i t y was provided by i n s e r t i n g double strands of 0.5 mm x 5 mm rubber s t r i p s at the dock end of the composite moorage l i n e s . The connection to the vessel was made by ty i n g f i s h hooks to the nylon l i n e s , thereby o f f e r i n g convenient attachments. The hooks also avoided knot damping through chafing which could not be evaluated and would have affected the r e s u l t s . The use of f i s h hooks also avoided slippage i n the l i n e s . The nylon braids were attached to the rubber s t r i p s by a squeeze j o i n t where the force could 48 be d i s t r i b u t e d uniformly i n t o the rubber strands. The connection to the dock was accomplished by s l i p p i n g a cotter pin over the rubber s t r i p s and squeezing i t by a wing nut between two washers. This again introduced a uniform force into the strands. The e l a s t i c i t y of the l i n e s was adjusted by a l t e r i n g the length of the rubber s t r i p s and the t o t a l length of the moorage l i n e set by f i x i n g the length of the braided nylon l i n e . The r e s u l t was a system which was f u l l y and e a s i l y adjustable and offered well defined point to point support with no opportunity f o r s l i p . The materials e l a s t i c behaviour was determined by s t a t i c load t e s t s . These were conducted by loading a known length of rubber s t r i p with graduated weights and the observing elongations. Unloading was also observed and a hy s t e r s i s loop determined. The rubber l i n e s offered a very narrow h y s t e r s i s loop as required and exhibited f a i r l y reasonable e l a s t i c behaviour for low stresses. The material exhibits some permanent s t r e t c h but a f t e r 100 cycles i t achieves s t a b i l i t y . One concern with the e l a s t i c behaviour, however, i s that the general shape of the curve for the rubber s t r i p i s concave downward while that f o r the prototype ropes are concave upward. While t h i s i s less than i d e a l , i f the stress l e v e l s are kept within l i m i t s d i c t a t e d by the model l i n e s (on model vessel less than 6 N) then the deviations should not be s i g n i f i c a n t and the model l i n e s w i l l be acceptable. Figure 21 i l l u s t r a t e s the dimensionless comparison between model and prototype moorage l i n e s f o r a number of l i n e s i z e s . It should be noted that e l a s t i c i t y (parameter E) of the model l i n e s i s defined as a c h a r a c t e r i s t i c s t i f f n e s s by the r a t i o of the s t i f f to e l a s t i c lengths of the components in the composite l i n e s . 6.7 . Model Fenders Modeling f o r the balloon fenders was achieved by pressurizing ordinary household balloons to r e a l i z e the required e l a s t i c behaviour. Two balloons were used, positioned such that no yaw motion was introduced by the impact of the vessel against the dock. C a l i b r a t i o n was accomplished by squeezing the model fenders between a f l a t plate and wooden block with the addition of c a l i b r a t e d weights. Forces and d e f l e c t i o n s were recorded. Figure 22 i l l u s t r a t e s the f o r c e - d e f l e c t i o n curves for the two balloons used i n the experiments together with that of the prototype. Some concern was rai s e d over the p o s s i b i l i t y of a i r leakage over the course of the t r i a l s . However, c a l i b r a t i o n s a f t e r the experiments were concluded indicates that t h i s was not the case. 50 CHAPTER 7 PRELIMINARY TESTS By way of preparation for the subsequent experiments, a series of preliminary t e s t s were performed. These were concerned e i t h e r with c a l i b r a t i n g the instruments or with es t a b l i s h i n g the c a p a b i l i t i e s or l i m i t a t i o n s of the experimental systems. A l l were of v i t a l concern as they provided a benchmark against which the ensuing t r i a l s could be meaningfully interpreted. 7.1. Basin C h a r a c t e r i s t i c s As has been discussed i n a previous chapter, any wave r e f l e c t i o n encountering the incident wave t r a i n can have a wide ranging e f f e c t on the v e s s e l behavior. However, i f the r e f l e c t i o n c h a r a c t e r i s t i c s of the basin are evaluated and included i n the present analysis, an a d d i t i o n a l i n t e r e s t i n g parameter may be considered. To t h i s end, a se r i e s of tes t s were conducted to define the amount of r e f l e c t i o n within the basin from the f a r beach. This r e f l e c t i o n i s known to be frequency dependent. The wave generator was activated to generate a wave t r a i n with a p a r t i c u l a r wave height and period. After s u f f i c i e n t time f o r r e f l e c t i o n to be b u i l t up i n the basin, the wave probe was positioned at one point i n the basin and a recording made of the wave heights encountered. If the wave heights remain constant with time, i t can be concluded that any e x i s t i n g standing wave f o r that frequency i s i n f a c t motionless. The probe i s then moved at a slow constant speed i n the d i r e c t i o n of wave propagation and a recording made. An envelope of waves w i l l r e s u l t on the chart recorder as indicated i n figure 2 3. The amount of wave r e f l e c t i o n which, by d e f i n i t i o n , i s the height of the r e f l e c t e d wave divided by the height of the incident wave can be determined by: 51 H - H min K max (x 100) (30) r H + H min max where: K r = r e f l e c t i o n c o e f f i c i e n t i n per cent H max = maximum wave height i n the envelope H min = minimum wave height i n the envelope The procedure i s repeated for several wave heights and wave periods and a t o t a l p i c t u r e of basin r e f l e c t i o n can be established. Such a group of tests was performed both before the experiments were begun and a f t e r they were concluded. This was done to v e r i f y the s t a b i l i t y of the wave basin c h a r a c t e r i s t i c s over the duration of the t r i a l s . Based upon the measurement techniques, I t was judged that basin r e f l e c t i o n could be evaluated to within ±5 percent of the true value. Results from both sets of t e s t s were consistant. In a d d i t i o n to the frequency dependent r e f l e c t i o n c o e f f i c i e n t , a knowledge i s required of the p o s i t i o n on the wave envelope at which the t r i a l s are performed. For t h i s reason, a serie s of envelope curves were drawn and from these curves, also frequency dependent, the experimental pos i t i o n s were established. The e f f e c t of wave height on basin r e f l e c t i o n was also examined. Although i t was expected that there would be an influence of wave height, measurements indi c a t e such an influence f a l l s within the tolerances as discussed above. This parameter could therefore be neglected f o r these purposes. 52 The wave period, on the other hand had a dramatic e f f e c t on the r e f l e c t i o n c o e f f i c i e n t . As i s i l l u s t r a t e d by figure 24, no r e f l e c t i o n e x i s t s f o r periods up to 1.0 second. A f t e r that there i s a reasonable constant r i s e to 46% at a period of 2.5 seconds. This represents the range within which the experimental t r i a l s were run. 7.2. Response of Chart Recorder Some concern was raised as to the consistent accuracy of the pen chart recorder for higher frequency s i g n a l s , p a r t i c u l a r l y for large excursions of the pen. A test was therefore run to evaluate the frequency response of the recording system. A s i g n a l generator was presented as input to the recorder and attenuation adjusted to cause a ± 5 cm response. The frequency, beginning at 0.2 Hz was gradually increased as the pen response was monitored. It was not u n t i l frequencies exceeded 5 Hz that response exaggeration was noted, and at that i t represented only 5% of the reading. It was therefore concluded that response amplitude was s u f f i c i e n t l y independent of frequency to f u l f i l l the present requirements. 7.3. C a l i b r a t i o n of Instruments The l i n e a r voltage displacement transducer (LVDT) i s an e l e c t r o n i c a l l y stable instrument which was not expected to d r i f t over the course of time. Nevertheless to account f o r the p o s s i b i l i t y that the d i a l s were inadvertantly adjusted, a c a l i b r a t i o n test was performed each day of the experiments. The c a l i b r a t i o n procedure involved moving the vessel through a s e r i e s of measured steps, c l o s i n g back on the i n i t i a l p o s i t i o n , while the chart recorder kept account of the movement. The r e s u l t s were then plotted up and the c a l i b r a t i o n constant was determined. Figure 25 i l l u s t r a t e s a t y p i c a l such pl o t , i n t h i s case for the date of May 31. As can be seen, the instrument behaves as l i n e a r l y as could be expected from the c a l i b r a t i o n procedure. As testimony to the s t a b i l i t y of the instrument, the c a l i b r a t i o n constant was c o n s i s t e n t l y within the range of 3.7 to 3.8 (mm of motion per unit readout scale) for the bulk of the t e s t s . 7.4. Tests to V e r i f y the Centre of F l o a t a t i o n A v i t a l consideration of the experimental procedure revolved around the c e r t a i n t y with which the measurement of t r a n s l a t i o n a l motions (surge and sway) could be i s o l a t e d from the r o t a t i o n a l behaviour ( p i t c h and r o l l ) . It was therefore important to locate the axis of f l o t a t i o n i n both p i t c h and r o l l from which the measurements could be made. To t h i s end, a series of simple tests were run whereby the unrestrained v e s s e l was positioned f or surge measurements ( l a t e r sway) and allowed to o s c i l l a t e i n p i t c h ( r o l l for sway). Any component of p i t c h registered on the instruments would be periodic while surge motion would be aperiodic. Examination of the r e s u l t s yielded no periodic behaviour and therefore i t could be concluded that indeed the center of p i t c h had been located. For the associated sway tests i t was expected that even i f r o l l was not being contributed to the instrumentation, the dynamic cross coupling between r o l l and sway would cause the vessel to sway p e r i o d i c a l l y . This, however, d i d not prove to be the case as the recorded r e s u l t yielded no periodic component whatsoever. It was 54 therefore obvious that not only had the vessel's centre of r o l l been located but the cross coupling of sway by r o l l was minimal. 7.5. Tests to Determine Damping Induced by the Measurement System A ser i e s of tests was conducted to evaluate the e f f e c t that the c a l i b r a t i o n system would have on the motion of the v e s s e l . As discussed e a r l i e r , the only net e f f e c t , apart from a minimal a d d i t i o n a l mass, was the damping created by the f r i c t i o n on r o l l e r s and the LVDT core. By way of evaluating t h i s damping, a decrement test was set up where the v e s s e l , suspended on v e r t i c a l l i n e s i n a i r (see f i g . 26a), was equated to a standard single degree of freedom (SDOF) spring-mass-dashpot system by the i n c l u s i o n of a spring at e i t h e r end of the v e s s e l . The true spring constant, i t can be shown, i s the sum of both the system springs and the pendulum action of the v e r t i c a l support l i n e s . The vessel was then pulled to one side and permitted to f r e e l y o s c i l l a t e while natural period and damping c h a r a c t e r i s t i c s were observed. It should be noted that a mass of 30 g was added to the v e s s e l to account f o r the equivalent mass of the measurement system. From a log decrement analysis, the equivalent damping of the springs as a percentage of c r i t i c a l damping was c a l c u l a t e d . This was done for three d i f f e r e n t spring constants to y i e l d r e s u l t s at three d i f f e r e n t frequencies. The a d d i t i o n a l mass was then removed and the measurement system i n s t a l l e d as i t would operate for the subsequent experiments (see f i g . 26b). The v e s s e l was again allowed to o s c i l l a t e and the d i f f e r e n c e i n the damping c o e f f i c i e n t s for the two cases yielded the amount of damping introduced by the measurement system. The r e s u l t s obtained from these t e s t s are presented i n table 3. As can be seen from the r e s u l t s , the system damping shows quite a strong frequency dependence. As discussed e a r l i e r , the magnitude of the damping i s s u f f i c i e n t l y small to expect that r e s u l t s from subsequent experiments w i l l be unaffected by the influence of t h i s system. The system damping, however, w i l l be considered i n the f i n a l analysis as part of the moorage l i n e damping. 7.6. Tests to Determine Added Mass and Damping of the Vessel A s i m i l a r s e r i e s of t e s t s as described i n 7.5 were run to evaluate the frequency dependent added mass and hydrodynamic damping c o e f f i c i e n t s p e c u l i a r to t h i s v e s s e l . The damping includes both r a d i a t i o n and viscous damping. As i l l u s t r a t e d by f i g u r e 27, the wave basin i s flooded and the water allowed to support the vessel In l i e u of the v e r t i c a l l i n e s . Without the pendulum e f f e c t of these l i n e s , the e f f e c t i v e s t i f f n e s s of the system w i l l be d i f f e r e n t from the previous t e s t s . Once again the vessel i s pulled to one extreme and allowed to f r e e l y o s c i l l a t e . Data obtained from the chart recorder i s used to evaluate the desired parameters. In the case of surge, both a bow tug and a stern tug i s tested but there was no discernable d i f f e r e n c e between the two. Table 4 l i s t s the r e s u l t s from these surge t e s t s . On a related subject, Kim (1965) attempts to obtain added mass and r a d i a t i o n damping c o e f f i c i e n t s f o r shapes s i m i l a r to the model vessel discussed herein. To t h i s end, he outlines a Green's function solution to determine these hydrodynamic c h a r a c t e r i s t i c s f o r a number of body shapes. The closest approximation for reference to the vessel under consideration i s a surging e l l i p s o i d with a length over width r a t i o of 56 four. Curves f o r t h i s shape and for d r a f t over h a l f - l e n g t h r a t i o s of 4 and 8 are presented by Kim. Comparing experimental r e s u l t s with those determined a n a l y t i c a l l y by Kim for surge in d i c a t e s the behaviour of the vessel ( d r a f t over half-length r a t i o of 5.7) as f a l l i n g within the two a n a l y t i c a l values. It can be therefore concluded that a n a l y t i c a l values for added mass and damping could be used i n l i e u of experimental r e s u l t s . Also, since Kim's r e s u l t s are f o r r a d i a t i o n damping only, i t can be concluded that, for t h i s v e s s e l , viscous damping does not s i g n i f i c a n t l y a l t e r the t o t a l damping c o e f f i c i e n t . In sway, the matter was s l i g h t l y d i f f e r e n t . The existence of the ke e l , as expected, s i g n i f i c a n t l y affected the problem to such an extent that i t was not p r a c t i c a l to compare i t to Kim's theory for swaying bodies. The experimental r e s u l t s would therefore have to stand on t h e i r own. Also i n sway, the cross-coupling of r o l l had to be addressed. To t h i s end, the test procedure was a l t e r e d s l i g h t l y to include r o l l , r o l l and sway, and sway only. The former was accomplished by merely observing free r o l l o s c i l l a t i o n s and noting the damping and natural periods. Only one frequency was tested as i t was concluded that r o l l i s only of secondary concern f or the sway an a l y s i s and any e f f e c t of frequency dependence would not have a large impact on sway. The intermediate case was tested and the r e s u l t s analyzed i n the same manner as surge. Very few o s c i l l a t i o n s were possible before the vessel came to a r e s t owing to the high damping c o e f f i c i e n t s . The l a t t e r required the vessel to be prevented from r o l l i n g while being tested. This was accomplished by securing tefther l i n e s to the mast which kept the mast v e r t i c a l and performing tests analogous to the surge t e s t s . Results from these sway and r o l l t e s t s are l i s t e d i n table 5 where i t can be seen that the values are an order of magnitude above those of surge. The d i s c u s s i o n entered into above u t i l i z e s terms and techniques for dynamic analysis common to most texts on introductory dynamics such as Clough & Penzien (1975). 7.7. Tests to Determine the E f f e c t of Slackness on Natural Period A f i n a l preliminary test was performed to e s t a b l i s h the s e n s i t i v i t y of natural period to slackness. In t h i s series of t e s t s , the vessel was allowed to f r e e l y o s c i l l a t e with the spring s t i f f n e s s being provided by the actual moorage l i n e s and the natural periods recorded. The r e s u l t s are presented i n f i g u r e 28 and as expected indicate an increasing natural period with increasing amounts of slackness. This can be r a t i o n a l i z e d by considering that as the slackness i s increased, the percentage of the time per cycle during which the vessel i s being forced by the s t i f f n e s s elements i s reduced. This r e s u l t s i n a reduced natural frequency accounting for the increased natural period. 5 8 CHAPTER 8 EXPERIMENTATION A t o t a l of 231 t r i a l s were performed during the main body of the experimentation program. Appendix II l i s t s the p a r t i c u l a r s of each test run together with the raw data r e s u l t s and these r e s u l t s placed i n dimensionless form. Dimensionless r e l a t i o n s h i p s w i l l become more meaningful as we l a t e r Interpret the experimental data. In a preface to appendix II i s the coding system and d e s c r i p t i o n of units which w i l l explain the data. It remained before the experimentation was set up to define those parameters which i t was judged would influence the vessel's surge and sway behavior, together with the range through which each parameter should be varied i n order to obtain meaningful and comprehensive r e s u l t s . To serve as a basis f o r comparison, a Response Amplitude Operator (R.A.O.) i s defined to compare the e f f e c t s of a l t e r i n g s the parameters being examined. R.A.O. i s the r a t i o of some d e t a i l of the vessel response to the e x c i t a t i o n . For the purposes of t h i s discussion, the ves s e l response i s surge or sway and the e x c i t a t i o n i s incident wave height. That i s : R.A.O. (surge or sway) = ^ ~ n. 1 where: A = twice the motion amplitude H = incident wave height. 59 A dimensional analysis indicates that the R.A.O. depends on several dimensional parameters which may be selected as follows: 2 H R.A.O. = f[ 2L& , j l , 4, K r, y, E, FC, VL, MCJ where T = wave period g = g r a v i t a t i o n a l a c c e l e r a t i o n L = vessel waterplane length d = water depth K = r e f l e c t i o n c o e f f i c i e n t r x = p o s i t i o n i n the standing wave X = wave length E = l i n e s t i f f n e s s (see section 6.6) FC = Fender Condition VL = vessel type SL = slackness MC = Moorage Condition The f i r s t three parameters describe the c h a r a c t e r i s t i c s of a progressive wave t r a i n . The next two include the e f f e c t of wave r e f l e c t i o n . The f i n a l describe d e t a i l s of the vessel moorage and upon solu t i o n o f f e r a basis upon which comparisons for e f f e c t i v e moorage pra c t i c e s can be made. Of these ten i d e n t i f i e d parameters three; water depth, r e f l e c t i o n c o e f f i c i e n t , and vessel type were kept constant. The others were examined through the experimentation program. They are discussed below with r e s u l t s presented i n chapter 9. It should be noted that a l l attempts were made to con t r o l remaining parameters while the one under consideration was being examined. 8.1. Wave Period In an e f f o r t to asce r t a i n the wave climate acceptable f or moorage of small vessels, a paramount part of the research e f f o r t went into examining a number of wave conditions which may be experienced i n a marina. To t h i s end, the wave period was altered from a minimum of 0.5 sec. to a maximum of 2.5 seconds. This represents on a f u l l scale vessel a range of 1.6 seconds to 8 seconds. The lower value, as w i l l be shown by the experimental r e s u l t s i s below the threshold of energy to cause the vessel to move i n surge and sway. The 8 second upper l i m i t i s well above the maximum expected i n a small marina where t y p i c a l maximum values of 4-5 seconds are common. Only through very exposed conditions with a f e t c h exceeding 50 km could such an 8 second wave propagate. Although the r e f l e c t i o n i n the basin i s dependent on the wave period, the former has no e f f e c t on the l a t t e r . 8.2. Wave Height The second parameter i n defining a wave climate i s , as mentioned, the wave height. It i s important here to note the d i f f e r e n c e between the incident wave height and the l o c a l wave height. The former describes that wave generated before i t i s influenced by basin r e f l e c t i o n and i t remains unchanged over the length of the basin and the duration of the t e s t . The l a t t e r , as discussed, i s influenced by the degree of r e f l e c t i o n and varies over the length of the wave basin i f a standing wave i s present. It i s therefore the incident wave height that we seek to examine as i t alone describes the character of the wave t r a i n . The incident wave height was varied from 5 mm (scaled = 50 mm) to 61 50 mm (scaled = 500 mm) through what was again considered as a r e a l i s t i c range which could be encountered In a marina environment. It could not be assumed that the wave height and wave period behave independently to influence vessel motion so a matrix of tests were conducted which r e f l e c t e d various combinations of both parameters. 8.3. Wave R e f l e c t i o n To discover how a r e f l e c t e d wave influences vessel motion, a serie s o f t e s t s were performed at various p o s i t i o n s on the standing wave while maintaining the same wave period and incident wave height. For a purely progressive wave t r a i n , the motion would not d i f f e r as a function of po s i t i o n . Any v a r i a t i o n , therefore, i n the test r e s u l t s over the length of the basin could be a t t r i b u t e d e n t i r e l y to basin r e f l e c t i o n . For a l l other t r i a l s , the vessel was positioned at the node of the standing wave, i d e n t i f i e d f o r each frequency by the preliminary t e s t s described i n an e a r l i e r chapter. A note was taken of the p o s i t i o n i n the basin at which the t r i a l s were run but i n a l l cases t h i s p o s i t i o n corresponded to the node of the standing wave. 8.4. Line S t i f f n e s s To evaluate the e f f e c t of using moorage l i n e s of various s t i f f n e s s e s , corresponding to d i f f e r e n t sizes and types of rope ma t e r i a l , the s t i f f n e s s of the model moorage l i n e s were changed through a se r i e s of t r i a l s . Three d i f f e r e n t s t i f f n e s s e s were examined with fo r c e elongation behavior as i l l u s t r a t e d i n chapter 6. For the most part, however, 13 mm diameter double braided nylon l i n e was modeled and used throughout the bulk of the t r i a l s . 62 8 . 5 . Fender Condition The existence or absence of a balloon fender between the vessel dock was considered. For surge t e s t s , the fender was usually omitted; for sway te s t s , i t was usually included. 8 . 6 . Slackness The amount of slackness i n the l i n e s i s defined as the free movement e i t h e r seaward or leeward from a centering p o s i t i o n . The amount of slackness has a remarkable e f f e c t on the dynamic c h a r a c t e r i s t i c s of the motion and for that reason, a great deal of time was devoted to understanding i t s influence. Slackness i n a moorage condition comes in t o play when one considers that some slackness must exist when mooring a vessel to accommodate heave i n p a r t i c u l a r , but also r o l l , and to a l e s s e r extent p i t c h . It remains to discover what, i f any, would be the slackness most e f f e c t i v e i n l i m i t i n g v e s s e l motion. Values of slackness were alt e r e d through a number of steps from taut to about 5 per cent of the vessel length. (Note t h i s i s a t o t a l f r e e movement of 10%.) The lower l i m i t i s obvious. The upper l i m i t was chosen as i t represented the maximum range of the instruments, but i s , nevertheless, considered more than adequate to formulate a complete a n a l y s i s . 8 . 7 . Moorage Conditions A t o t a l of s i x moorage conditions were tested, chosen from the short l i s t of twelve i l l u s t r a t e d i n chapter 2. These six were judged s u f f i c i e n t to determine the effectiveness of a l l moorage conditions i d e n t i f i e d from the marina v i s i t s , as t h e i r c h a r a c t e r i s t i c s can be considered as representative. The majority of surge t r i a l s were c a r r i e d out under condition I (see appendix II for an i l l u s t r a t i o n ) while those of sway were moored as i n condition I I . It i s upon the r e l a t i v e and respective influences of these seven parameters that the experimental t r i a l s were based. 64 CHAPTER 9 EXPERIMENTAL AND ANALYTICAL RESULTS This chapter w i l l discuss the d e t a i l s of moored vessel behaviour learned from the experimental program and compared to the computer model. The f i r s t part of the chapter w i l l examine the q u a l i t a t i v e trends and tendencies which were observed. Later, q u a n t i t a t i v e r e s u l t s w i l l be presented and discussed. It should be noted that recordings f o r each t r i a l were made a f t e r the dynamic system was permitted to achieve steady state response. This implies both that the waves were afforded s u f f i c i e n t time to b u i l d up a p a r t i a l standing wave i n the basin, and that the vessel was permitted to o s c i l l a t e i n t h i s environment long enough f o r the transient c h a r a c t e r i s t i c s of the response to have d i s s i p a t e d . The decision to approach the experimentation i n t h i s manner was made, as the necessary i n c l u s i o n of time dependence for transient responses i n the measurements, and ultimately the analysis would make the r e s u l t s d i f f i c u l t to i n t e r p r e t . As well, i n recognition of the objective of t h i s research, to provide a comparative understanding of the parameters under study, i t was judged that steady state behaviour could best s a t i s f y t h i s goal. 9.1. Observations As each t r i a l was being conducted, observations of the boat motion provided i n t e r e s t i n g i n s i g h t into the behaviour of such a moored ve s s e l . In some ways, the i n t e r p r e t a t i o n of the i n d i v i d u a l t r i a l s offered the 65 opportunity to generalize about the influence on vessel motion of one or more of the parameters being examined. The f i r s t half of the t r i a l s were conducted for surge while the second h a l f d e a l t with sway. 9.1.1. Surge For the bulk of the surge tests, the vessel was moored as per moorage condition " I " of figure 3 as photographed i n f i g u r e 29. This was a symmetrical moorage arrangement which afforded the opportunity to examine surge behaviour without the complicating influences of sway or yaw. For the most part, the vessel response i n surge was periodic, repeatable and predictable. The shape of the response curve showed a dramatic dependence on the frequency of the waves fo r c i n g the v e s s e l . Tests run at low frequencies resulted i n a response curve best characterized by a square wave (see f i g . 30a). The nature of the water p a r t i c l e motion was such that i t could be considered as a p e r i o d i c a l l y reversing uniform flow. The wave would move the v e s s e l to tautness of moorage l i n e s i n one d i r e c t i o n and hold i t there at s t a t i c equilibrium as the wave passed. On the opposite phase of the wave, the vessel was forced to tautness i n the opposite d i r e c t i o n and once again held i n s t a t i c equilibrium u n t i l returned to i t s o r i g i n a l p o s i t i o n . The period of motion being high, there was l i t t l e vessel impact on the moorage l i n e s and hence the square wave form was established. Figure 33a i l l u s t r a t e s a measured example. As the frequency was increased, the slamming which was absent i n the low frequency tests began to appear. The vessel would be thrown i n t o the moorage l i n e s which would elongate and bring the body to a r e s t . The moorage l i n e s would then bring the vessel back to a s t a t i c a l l y stable p o s i t i o n while the wave passed. On the return motion of the ve s s e l , the slamming seldom occurred and the response curve was squared o f f as before (see f i g . 30b). At t h i s frequency, i t could be characterized by the previously described square wave with a peak appearing on one h a l f of the wave c y c l e . An e f f o r t was made to determine the cause of asymmetry evident i n the response curve at t h i s period range. By a l t e r i n g the p o s i t i o n of the v e s s e l i n the node of the standing wave, the response could be made to exhibit this peaked behavior i n either d i r e c t i o n . However, i t would not create peaks i n both d i r e c t i o n s on the same t r i a l . In addition, the vessel's motion i n p i t c h seemed to be i n some way responsible as the peaks occurred only i n the d i r e c t i o n i n which the bow or stern was forced by the wave out of the water. It was concluded that the slamming motion of the vessel causing the peaks i n the response curve, was related to the vessels behavior i n p i t c h . Furthermore, the nature of the vessel response i n pitc h at t h i s frequency did not permit the vessel to have i t s extremity out of the water for both d i r e c t i o n s of motion. By increasing the frequency yet again, a further c h a r a c t e r i s t i c of motion became apparent. As the vessel slammed into the moorage l i n e s and the l i n e s rebounded the vessel to s t a t i c s t a b i l i t y , i t would pass the equilibrium point and be brought back to p o s i t i o n by the wave, e f f e c t i v e l y o s c i l l a t i n g about equilibrium. This resulted i n a secondary p e r i o d i c i t y appearing i n the response curve as two peaks were evident i n one d i r e c t i o n during one cycle (see f i g s . 30c and 33b). Once again the behaviour did not repeat i t s e l f during the return motion. By t h i s frequency the response curve has become much smoother than for lower frequencies where an abrupt square wave pattern was evident. The 67 smoothing o f f can be explained by considering the f a c t that the vessel i s permitted to rest on the moorage l i n e s for a shorter period of time at higher frequencies. This implies that i t spends more time during the cycle a c t u a l l y moving In the wave flow. By continuing to increase the wave frequency, a reduction i n the d e f i n i t i o n of the second peak on the response curve w i l l occur (see f i g . 30d). This i s explained by noting that as the moorage l i n e s rebound the vessel i n t h e i r natural frequency, the wave w i l l have passed and w i l l force the v e s s e l on the return phase before i t has an opportunity to rest on the taut moorage l i n e s . At t h i s frequency, the vessel response begins to e x h i b i t some resonant a m p l i f i c a t i o n as the moorage l i n e s and wave forces begin to aid each other i n t h e i r e f f o r t s to move the v e s s e l . Figure 33c i l l u s t r a t e s a measured example. A further increase i n frequency w i l l force the vessel into resonant behaviour (see f i g s . 30e and 33d). The f i n a l mode of vessel response occurs when the frequency i s increased to such a point as to produce waves of length less than or equal to the waterplane length of the v e s s e l . At t h i s point, the forces at the bow of the boat e s s e n t i a l l y cancel the forces at the stern, and the f o r c i n g function i s reduced markedly. There may s t i l l be a periodic response but i t s magnitude i s seldom large enough to exceed the free movement of the moorage l i n e s (see f i g . 30f). The s i x modes of response motion described above i s a general d e s c r i p t i o n of those observed during experimentation. Their magnitude, d e f i n i t i o n , and frequency dependence i s determined by the complicated r e l a t i o n s h i p between wave length, wave height and slackness that w i l l be discussed l a t e r i n t h i s chapter. Nevertheless they w i l l a l l be evident 68 to varying degrees as the vessel i s subjected to waves of d i f f e r e n t frequencies. For non-symmetrical moorage conditions, the e f f e c t i s the same. However, the issue i s confused somewhat by the introduction of yaw and sway i n t o the vessel's motion. Although the brest l i n e s are slack and the vessel i s subjected to head seas alone, the brest l i n e s and fenders come into play. The eccentric mooring of the spring l i n e s for non-symmetric moorage conditions causes the vessel to undergo a yaw motion which i s r e s i s t e d by the fenders and the brest l i n e s . This yaw motion has the c a p a b i l i t y of a f f e c t i n g the vessels motion by hydrodynamic damping, f r i c t i o n from the fenders, and e l a s t i c behaviour of the fenders. One f i n a l frequency dependent c h a r a c t e r i s t i c observed i s the degree of uniformity evident i n the response curve. During low frequencies when the vessel response i s most dependent on the wave frequency, the behaviour i s very periodic. However, for higher frequencies, the d i f f e r e n c e i n the peak to peak magnitudes between cycles becomes greater. This i s undoubtedly due to the degree of pitching and heaving which occurs as a r e s u l t of the higher frequency waves. 9.1.2. Sway For examining behaviour i n sway, the bulk of the tests were conducted with the vessel moored as i n moorage condi t i o n " I I " . This included the i n s t a l l a t i o n of two balloon fenders to r e s i s t dockward motion as well as short brest l i n e s to keep the vessel from moving away from the dock. Spring l i n e s were also included but these did not influence the vessel's motion i n sway. In general, there was more movement evident i n sway than surge. 69 This was due to the f a c t that the keel and vessel's side p r o f i l e creates a higher drag c o e f f i c i e n t than the streamlined surge body. The vessel would therefore be expected to more accurately describe the water p a r t i c l e s ' o r b i t a l motion i n sway. As with the case of surge, the general v e s s e l behavior i s frequency dependent. However, the number of d i s t i n c t stages into which the response motion can be categorized i s fewer. At low frequencies, the vessel rebounds o f f the fenders (see f i g . 31a) and i s c a r r i e d by the wave to the l i m i t of the moorage l i n e s . The flow i s r e l a t i v e l y uniform as described above. At t h i s time, the i n e r t i a of the vessel c a r r i e s i t s center of g r a v i t y further from the dock while the moorage l i n e s r e s t r a i n the vessel at deck l e v e l . The force couple r e s u l t s In r o l l being introduced into the motion (see f i g . 31b). For p a r t i c u l a r l y high periods, flow separation occurs about the keel and rudder. There i s l i t t l e r o l l or heave r e s u l t i n g from the waves themselves at th i s frequency, as the waves tend to be r e l a t i v e l y f l a t . The r e s u l t i n g response of the vessel i n sway can be characterized by a square or rounded periodic function as described for surge (see f i g . 32a). For d i f f e r e n t frequencies i n t h i s range, the response changes l i t t l e i n form, although there are great magnitudal d i f f e r e n c e s . Figure 34a i l l u s t r a t e s a measured example. The next highest frequency range recognizes an increased r o l e to be played by the fenders and l i n e s . When the vessel i s slammed against a r e s t r a i n t , i t rebounds to a centering p o s i t i o n where i t waits u n t i l the wave returns to slam i t i n the opposite d i r e c t i o n . As a r e s u l t , the ve s s e l e x h i b i t s sway behaviour as i l l u s t r a t e d i n figure 32b. As the frequency i s increased s t i l l further, the same mechanism appears to e x i s t but because the response i s near resonance, motion i s exaggerated dramatically. The issue becomes complicated by a great deal of dock interference together with a dramatic r o l l response. Nevertheless, the motion for frequencies near resonance can be characterized by the curve I l l u s t r a t e d i n f i g u r e s 32c and 34c. The si z e and number of loops depends on the r e l a t i o n s h i p between the s t i f f n e s s element's natural frequency and the frequency of the waves, as does the actual magnitude of the motion. At resonant conditions, there appears to be an envelope of response with a period of 2-4 cycles as the system adjusts i t s e l f i n and out of phase with the waves. The e f f e c t of changing the method of moorage i s l e s s dramatic for sway than for surge. This i s because the symmetry of the system i s not a l t e r e d but only the s t i f f n e s s of the moorage system to sway. The dynamic system i t s e l f i s therefore not changed so there i s no reason to expect a d i f f e r e n t general response. The influence of wave height, wave period and slackness, however, are every b i t as important for the case of sway as f o r surge. 9.2. Measured Results As discussed i n e a r l i e r chapters the wave height and vessel motion were recorded simultaneously for the vessel subjected to head and beam seas. "Maximum motion" (also denoted A) i n the subsequent discussion i s taken as the excursion between the maximum po s i t i v e and negative displacements recorded during the ten c y c l e s measured f o r each t r i a l . In cases of good response uniformity, t h i s simply means twice the amplitude of o s c i l l a t i o n . For more i r r e g u l a r behavior, i t i s related to the maximum forces in the l i n e s and maximum displacements which can be expected. R.A.O. i s defined as i n chapter 8. 71 9.2.1. Comparison of Mathematical and Experimental Models As a test of the v a l i d i t y of the computer model, the r e s u l t s , as generated mathematically, f o r s p e c i f i c wave and moorage conditions were compared to the actual measured response obtained from experimentation. This comparison i s i l l u s t r a t e d f o r surge i n figure 33 and sway i n figure 34. The t y p i c a l examples are presented to roughly correspond to the generalized response as discussed i n as s o c i a t i o n with figures 31 and 32. See appendix II for the actual test conditions corresponding to each output. For surge agreement i s good. The main objective of determining maximum o s c i l l a t i o n i s s a t i s f i e d and, although the computer model exaggerates some of the d i s c o n t i n u i t i e s , the basic shape of the response curves agree. The differences between the predicted and measured performance of fig u r e 33a and 33b can be explained as r e s u l t i n g from the moorage l i n e s i m p l i f i c a t i o n s introduced into the computer model which f a i l to damp out secondary, high frequency motion. For the actual experiments, i t i s expected that such high frequency damping does occur. For more sinusoidal behaviour, the agreement between the measured and predicted time h i s t o r y r e s u l t s i s better. For sway, because of the complex influence of r o l l , p a r t i c u l a r l y at the natural frequency of r o l l , the r e s u l t s are less accurate. Figure 34a presents reasonably good agreement between predicted and measured response as the r e l a t i v e influence of r o l l was less s i g n i f i c a n t than at higher frequencies. Figure 34b i s a l i t t l e l e s s encouraging. Although the peak-to-peak magnitude i s as expected, the intermediate o s c i l l a t i o n s appearing i n the s t r i p recording do not appear i n the computer simulation. The reason seems to be a combination of inadequate a c c o u n t a b i l i t y of r o l l at resonance and the v i o l e n t nature of the vessel motion which tends to invalidate such t h e o r e t i c a l assumptions as small motion, l i n e a r wave theory and equivalent viscous damping. Nevertheless the computer model has merit i n i t s a b i l i t y to predict maximum motion. Figure 34c presents a more accurate computer simulation than would o r i g i n a l l y appear evident. The behaviour i s e s s e n t i a l l y unrestrained and o s c i l l i t o r y . The s t r i p recording appears influenced by a secondary d r i f t e f f e c t which causes i t to behave at a secondary low frequency sinusoid. No such e f f e c t was included i n the computer a n a l y s i s . Nevertheless, the main frequency peak-to-peak amplitude of motion i s v a l i d . Based on these comparisons, i t can be concluded that the computer model, for the purposes to be discussed l a t e r o f f e r s a good simulation of a c t u a l observed behaviour. 9.2.2. Unrestrained o s c i l l a t i o n s To test the expectation that a vessel i n an unrestrained condition would tend to follow the motion of the water p a r t i c l e s , a series of t r i a l s were run with the vessel r e s t i n g unrestrained on a wavy surface. A l l t r i a l s were conducted at the node of the p a r t i a l l y standing wave. Figure 35 presents the r e s u l t s of these t r i a l s . The s o l i d l i n e indicates the l i n e a r theory p r e d i c t i o n of the maximum horizontal excursion of the water p a r t i c l e s . As expected, the tests for surge and sway f a l l somewhere below t h i s l i n e i n d i c a t i n g that while the vessel attempts to follow the p a r t i c l e motion, some slippage does occur. For surge, the motion s t a r t s out very low at a dimensionless period of 2 and r a p i d l y increases to follow the p a r t i c l e motion. The small motion at low periods r e f l e c t s the fact that the waves are of a length in the order of the waterplane length. Therefore, the forces of successive 73 waves on the body cancel each other and very l i t t l e net force i s applied. Sway on the other hand s t a r t s with high motion as the waterplane beam, being much l e s s than the length, experiences only one part of one wave at one time. It i s i n t e r e s t i n g to note that for higher periods, the behaviour i n surge and sway i s almost i d e n t i c a l . 9.2.3. Results of surge tests The f i r s t s e r i e s of tes t s for the ve s s e l i n surge was an e f f o r t to determine the e f f e c t of wave r e f l e c t i o n . As can be seen from figure 36, the e f f e c t i s dramatic. For r e f l e c t i o n of only 30 percent, there i s a clear difference of motion between the node and the anti-node of the standing wave. At the node the water p a r t i c l e s are expected to undergo th e i r maximum horizontal excursion and i t should come as no surprise that the vessel also experiences i t s maximum surge. Figure 37 i l l u s t r a t e s the r e s u l t s of t r i a l s to determine the influence of wave height on motion i n surge. P a r t i c l e excusion, and hence vessel motion, i s by theory d i r e c t l y dependent on the wave height. In the case of a vessel moored with slackness, i t w i l l o s c i l l a t e within the free movement for small wave heights. As the wave height i s increased, i t w i l l increase i t s motion u n t i l i t begins to extend the moorage l i n e s . Beyond t h i s point further increases i n height w i l l increase l i n e a r l y the elongation of the l i n e s . The r e s u l t s as presented i n the figure v e r i f y t h i s theory. The influence of slackness on motion i n surge can be very dramatic. Figure 38 i l l u s t r a t e s t h i s very well by considering the case of two d i f f e r e n t wave periods. When examining t h i s graph, i t i s emphasized that what i s of most concern i s the difference between the maximum 74 vessel motion and the free movement, as t h i s represents the forces i n the l i n e s . For high dimensionless periods, the flow can be considered uniform and pe r i o d i c . This being the case, the forces i n the l i n e s tend to be the same regardless of the slackness because i t becomes as pseudo-s t a t i c equilibrium problem as discussed e a r l i e r . The wave pushes the l i n e s to equilibrium and the wave force, being reasonably constant u n t i l i t reverses, i s evenly maintained. There i s nothing about tauter l i n e s to change t h i s r e a l i t y . For lesser periods, however, the wave frequency i s c l o s e r to the natural frequency of the system. An adjustment i n slackness w i l l a l t e r the natural period to such an extent that resonance at some point may occur. In t h i s case, resonance does occur r e s u l t i n g i n the high forces i n the l i n e s . The l a s t point on the curve i s very important. It indicates what the maximum vessel excursion would be i f the vessel was unrestrained, as i t i s since the motion i s less than the free movement. What t h i s says i s that the l a s t four points on the dotted curve would not na t u r a l l y be so high, were i t not for the dynamic a m p l i f i c a t i o n of the moorage l i n e s . Another of the parameters examined during the experimentation was the frequency dependence of the vessel response. Figure 39 i l l u s t r a t e s what has become recognized as the general form of the frequency dependence response curve. If the vessel were permitted to o s c i l l a t e unrestrained from the wave forces, i t would carry the form of the frequency dependent response i l l u s t r a t e d i n figure 35, i n t h i s case for surge. Generally, the peak to peak motion would begin at a zero response and increase with period. The influence of the moorage l i n e s , i n the absence of dynamic a m p l i f i c a t i o n would be to te^ther the motion. Such behaviour i s represented by the lower branch of the curve on figure 75 39. I f , however, dynamic a m p l i f i c a t i o n occurs, i t w i l l do so at the natural frequency of the system. Furthermore, i f , as i s the case for surge, the damping i s l i g h t , the resonant curve w i l l be narrow and peaked. As can be seen l n the figure, t h i s i s p r e c i s e l y what i s observed. Therefore, the general shape of the frequency response curve Is a sharply defined peak at the natural period, f a l l i n g to an intermediate value and r i s i n g again towards higher periods. There i s , however, a further complicating influence which can have a wide ranging impact on moorage procedure. As discussed e a r l i e r , waves of a frequency high enough to generate waves i n the order of the vessel's waterplane length w i l l introduce l i t t l e i f any force on the v e s s e l . The threshold for t h i s behaviour i s near the natural frequency of the system. The natural frequency, of course i s determined by the s t i f f n e s s of the moorage l i n e s . Should the s t i f f n e s s be high enough to increase the natural frequency above t h i s threshold value than the vessels frequency response would be dictated by the lower l i n e on the curve. This represents an obvious advantage over the upper l i n e . The r e s u l t s as presented i n figure 40 show th i s e f f e c t . Each l i n e on the graph represents a d i f f e r e n t s t i f f n e s s of moorage l i n e s . The s t i f f e s t of these i s the l i n e denoted "E = 16". The graph represents an attempt to test the influence of d i f f e r e n t moorage l i n e s by r e l a t i n g t h e i r s t i f f n e s s e s to the vessel's frequency response. As can be seen, the curve "E = 6", being the softest of the l i n e s e x h i b i t s well defined resonance behavior at the highest natural frequency. The intermediately s t i f f l i n e shows a lower frequency, l e s s defined resonant condition. F i n a l l y , the s t i f f l i n e with a natural period presumably le s s than the threshold l i m i t f o r motion, presents no dynamic a m p l i f i c a t i o n behaviour. 76 One f i n a l observation concerning t h i s graph i s i n reference to the low frequency response. In t h i s region, i t appears that the maximum motion i s independent of the s t i f f n e s s of the l i n e s . Figure 41 i l l u s t r a t e s the r e s u l t s of an attempt to evaluate the performance of a number of moorage conditions. A l l curves i l l u s t r a t e the general frequency dependence as described above. It i s the r e l a t i v e magnitude of peak to peak motions and the degree of frequency dependence upon which comparisons can be made. Once again the frequency dependence of the response i s r e l a t e d to the natural frequency peculiar to each moorage condition. As the s t i f f n e s s depends on the length of the moorage l i n e s , i t follows that a moorage condition u t i l i z i n g short l i n e s o f f e r s the highest natural frequency and hence the best opportunity to avoid resonant a m p l i f i c a t i o n . As well, f o r low frequency conditions, the softer moorage arrangement would be expected to permit the highest pseudo-static d e f l e c t i o n s . In an e f f o r t to understand the meaning of figure 41, i t i s important to f i r s t consider the v e c t o r i a l r e s o l u t i o n of the moorage l i n e s to recognize t h e i r s t i f f n e s s against i n l i n e motion. To do so requires the consideration, for moorage l i n e s of the same mat e r i a l , of the length of each l i n e , i t s angle r e l a t i v e to the i n l i n e d i r e c t i o n , and the number of such l i n e s used to moor the v e s s e l . By doing so, i t becomes c l e a r that the s o f t e s t system i s represented by moorage condition "V" while the s t i f f e s t i s MC "VI". These two also e x h i b i t r e s p e c t i v e l y the greatest and l e a s t motion, implying the d i r e c t r e l a t i o n s h i p between system s t i f f n e s s and response. The other two conditions f a l l within these two extremes. Furthermore, the degree of dynamic a m p l i f i c a t i o n exhibited by MC "V" i s much higher than the others. This i n d i c a t e s that i t s natural period i s above the threshold period f o r wave motion and resonance occurs. MC X also exhibits t h i s behaviour but the other two presumably are s t i f f enough to possess natural periods below the threshold value. One f i n a l note i n reference to t h i s figure concerns asymmetry of the moorage condition r e l a t i v e to i n l i n e motion. MC X i s the only arrangement which has a d i f f e r e n t s t i f f n e s s for abow and astern motions. It also exhibits the l e a s t motion and the l e a s t dynamic a m p l i f i c a t i o n . By introducing asymmetry into the moorage arrangements the opportunity exists to r e s t r i c t dynamic a m p l i f i c a t i o n by avoiding a unique natural frequency f o r the system. The f i n a l series of tests to access vessel response i n surge involves the influence of protection fenders on surge motion. Only a moorage system possessing asymmetry about the l o n g i t u d i n a l axis w i l l a f f o r d the opportunity to examine fender behaviour. For t h i s reason, moorage condition " I I " was tested and the r e s u l t s presented in figure 42. It becomes immediately obvious that there i s l i t t l e d i f f e r e n c e between the two fender conditions. The fact that an included fender r e s u l t s i n s l i g h t l y higher response could possibly be the r e s u l t of two f a c t o r s . Fenders tend to smooth the vessels behaviour i n a way that a bare dock impacting the vessel cannot. The "fender excluded" condition causes jerking and bucking of the vessel creating more motion damping and hence l e s s motion. Secondly, the dynamic c h a r a c t e r i s t i c s of the fenders, although perpendicular to the surge motion, can introduce a degree of dynamic a m p l i f i c a t i o n into the surge r e s u l t s . This i s due to the fact that the moorage l i n e s are diagonal to the d i r e c t i o n of motion. This means that some component of the l a t e r a l motion affected by the fenders can be introduced into the diagonal moorage l i n e s and thereby surge. 78 9.2.4. Results of sway t e s t s The r e s u l t s of the sway t e s t s are f o r the most part very s i m i l a r to those of surge. To some extent the tendencies as discussed f o r surge are even more pronounced f o r sway. Tests on the v e s s e l i n sway were run at higher frequency than f o r surge. The waterplane beam being much l e s s than the waterplane l e n g t h means th a t s h o r t e r waves s t i l l i n f l u e n c e the v e s s e l i n sway and there i s no c a n c e l l a t i o n of forces as there was f o r surge. I t f o l l o w s t h a t i t i s more d i f f i c u l t to make the system s t i f f enough to keep the n a t u r a l period below the threshold l i m i t f o r sway. The d i s c u s s i o n of f i g u r e s 43, 44 and 45, examining the e f f e c t s of wave height and slackness on sway, and frequency dependent response r e s p e c t i v e l y , i s p r e c i s e l y the same as t h a t f o r surge. Figure 46 i l l u s t r a t e s the frequency dependent e f f e c t of s t i f f n e s s on sway. It i s important to note that f o r the motion of sway, the bulk of the t e s t s are performed with moorage c o n d i t i o n " I I " . This means that the moorage l i n e s represent the system s t i f f n e s s f o r only one h a l f of the c y c l e . The other h a l f i s provided by the dockside fenders. In p a r t , t h i s s i t u a t i o n e x p l a i n s the higher dynamic a m p l i f i c a t i o n which occurs for the case "E = 5". I t i s t h i s moorage l i n e s t i f f n e s s which i s the same as the e q u i v a l e n t s t i f f n e s s of the fenders. Therefore, i t i s f o r t h i s case only that the opportunity f o r balanced resonance occurs. For the other two cases, the moorage l i n e s and fenders each seek a d i f f e r e n t n a t u r a l frequency. I t i s the compromise of both that r e s u l t i n the somewhat reduced a m p l i f i c a t i o n s which are i l l u s t r a t e d . Because of the r e l a t i o n s h i p between the moorage l i n e s and fenders, there i s no c l e a r cut n a t u r a l period as was the case i n surge. However, as wi t h surge, the response at low frequencies i s reasonably s t i f f n e s s 79 independent. It would appear that one method of reducing sway dynamic response i s to ensure that s t i f f n e s s c h a r a c t e r i s t i c s of the fenders and moorage l i n e s are s i g n i f i c a n t l y d i f f e r e n t to avoid the opportunity f o r balanced resonance. As can be expected, the r e l a t i v e performance of the moorage conditions i n sway i s also dependent on the e f f e c t i v e s t i f f n e s s e s of the systems. A l l conditions of moorage ex h i b i t the now standard, frequency dependent behavior (see figure 47). Once again, i f the moorage l i n e s are resolved to evaluate t h e i r i n l i n e s t i f f n e s s , an understanding of the re s u l t s for sway w i l l be gained. In the case of sway, i t i s the side-to-side d i r e c t i o n which i s of concern. Moorage conditions II, VI, VII and X have approximately the same s t i f f n e s s against sway. This explains why, as i l l u s t r a t e d i n fig u r e 47, these four moorage conditions e x h i b i t frequency dependent response curves, s i m i l a r i n both form and magnitude. MC V, because of i t s long l i n e s i s the sof t e s t system and r e s u l t s i n the highest magnitude of motion. Several factors contribute to i t s rather dramatic resonant behaviour. F i r s t l y , i t s s o f t s t i f f n e s s r e s u l t s i n a higher natural period and hence an opportunity for more pronounced resonance. Also the s t i f f n e s s of the moorage l i n e s are more c l o s e l y related to that of the fenders so that the opportunity for balanced resonance i s more possible. F i n a l l y , the softness of l i n e s simply o f f e r s a greater magnitude of s t a t i c d e f l e c t i o n under a given force. The behaviour of MC I i s a l i t t l e more puzzling. Its system s t i f f n e s s i s not appreciably greater than the f i r s t four already discussed, but the motion i s c o n s i s t e n t l y l e s s . The explanation r e s t s i n the recognition that although the s t i f f n e s s away from the dock i s the same as f o r the other four moorage conditions, the s t i f f n e s s against dockward 80 motion i s increased i n MC I by the consideration of both the balloon fenders and the opposite pair of moorage l i n e s . In r e a l i t y , therefore, MC I i s a s t i f f e r system and enjoys the reduced motion r e s u l t i n g from th i s s t i f f n e s s . The f i n a l parameter investigated f o r Influence to sway i s the fender condition. Figure 48 i l l u s t r a t e s the r e s u l t s of these t r i a l s . As was the case with surge, the fender included s i t u a t i o n r e s u l t s i n greatest motion. In t h i s case, the cause Is c l e a r l y the cushion and rebound e f f e c t of the fenders on sway. A bare dock simply impacts the vessel and brings i t to a j o l t i n g halt without y i e l d i n g . The vessel r e s t s by the dock u n t i l the wave moves i t i n the opposite d i r e c t i o n . When a fender i s used the vessel rebounds as i f i t were a moorage l i n e and the a b i l i t y to y i e l d and rebound exaggerates the motion of the ve s s e l . However, although the actual t r a n s l a t i o n a l motion of the vessel i s reduced by a bare dock condition the forces from impact are undoubtedly greater. 81 CHAPTER 10 RECOMMENDATIONS 10.1. Reducing Vessel Damage Within the introduction to t h i s t h e s i s , i t was presented that damage to a moored vessel can be characterized as r e s u l t i n g from either excessive motion or excessive loads on the moorage l i n e s . It therefore remains to be established what can be done both during marina design and by v e s s e l owners to minimize damage to moored v e s s e l s . From a marina design standpoint, i t has become clear that the character of the incident wave t r a i n has a dramatic e f f e c t on vessel response. Wave height, wave r e f l e c t i o n , and, i n the absence of dynamic a m p l i f i c a t i o n , wave periods have a d i r e c t and proportional r e l a t i o n s h i p with vessel response i n surge and sway. To reduce vessel motion and, by as s o c i a t i o n , hawser forces, i t i s desi r a b l e to place l i m i t s on some of these parameters. In addition, water depth and i t s associated influence on water p a r t i c l e motion has an e f f e c t on vessel response. The interdependence of each of the factors also suggests that from a design standpoint, they should be examined together. That i s , i t i s not s u f f i c i e n t to merely prescribe a maximum wave height while taking no account of wave period, r e f l e c t i o n , or water depth. Mercer and Isaacson (1981) have discussed acceptable l i m i t s for v e s s e l motion by r e l a t i n g maximum response to expected wave action, f o r d i f f e r e n t p r o b a b i l i t i e s of occurrence. Three return periods are presented: the 50 year storm, the one year storm, and the weekly occurrence. Acceptable damage Is related to each return period and 82 j u s t i f i c a t i o n presented f o r e s t a b l i s h i n g l i m i t s on vessel motion which w i l l r e s u l t i n these l e v e l s of acceptable damage. For the case of surge, a one year return period response of ±1 foot (300 mm) i s suggested. One week and f i f t y year occurrence l i m i t s of ±0.5 feet (150 ram) and ±2 feet (600 mm) re s p e c t i v e l y , are also offered with the l i m i t i n g c r i t e r i o n to govern. In sway, the acceptable vessel motion has been presented as one h a l f of those f o r surge except f o r the f i f t y year return period where ±0.75 feet (225 mm) i s suggested. It should be noted that these v e s s e l motion l i m i t s have been established assuming that the vessels are moored with s u f f i c i e n t slackness to permit heave, p i t c h and r o l l . Therefore, there i s some free movement p o t e n t i a l introduced into the proposed c r i t e r i a . F i n a l l y , Mercer and Isaacson o f f e r a concept of grading the marina environment as moderate, good, or excellent by applying m u l t i p l i c a t i o n factors to the proposed c r i t e r i a . The computer model, having been v e r i f i e d by the experiment program, was used to generate expected vessel response to a number of wave conditions. Typical, rather than the poorest or the best, examples of moorage conditions and moorage materials were used as input. A slackness of ±150 mm was taken as conforming to the intent of Mercer and Isaacson when they prescribed acceptable l i m i t s of motion. The wave conditions, characterized by wave height, wave r e f l e c t i o n , wave period and water depth, r e s u l t i n g i n the l i m i t s of vessel motion defined above were determined and form the recommended marina design c r i t e r i a as presented i n table 6. The d e f i n i t i o n of "5" r e f l e c t s the Influences of wave height, wave r e f l e c t i o n , and water depth on water p a r t i c l e motion i n the incident wave t r a i n , and hence the vessel's e x c i t i n g force. The values appearing 83 i n the table were determined by examining data generated by the computer program and r e l a t i n g i t to the formula for t,. In using the recommended c r i t e r i a , a marina designer would, f o r a p a r t i c u l a r frequency with known wave r e f l e c t i o n and for a s p e c i f i e d water depth and return period, assign a value for h and n from the table. The value of wave height which s a t i s f i e s the design equation would be the recommended maximum wave height f or a "good" wave climate. For "excellent" wave climate the value for wave height would be reduced. For "moderate", the opposite i s true. Over a number of frequencies, a family of design heights can be determined. The marina should then be protected such that at each frequency of incident waves, the recommended design height i s not exceeded during the s p e c i f i e d return period. Of the ten parameters suggested i n chapter 8 as a f f e c t i n g vessel motion, the f i r s t f i v e can be considered as being co n t r o l l e d at the marina design stage of concern. Wave height, degree of r e f l e c t i o n , wave period, and water depth have been outlined above. Po s i t i o n i n the standing wave i s treated by assuming the most severe condition, at the wave node, to p r e v a i l . Indeed, table 6 has been developed based on t h i s assumption. The remaining f i v e parameters can be considered as being under the control of the i n d i v i d u a l vessel owner. The matter of recommending d e t a i l s f o r mooring i n d i v i d u a l vessels requires a more s p e c i f i c treatment. In the absence of dynamic a m p l i f i c a t i o n , i t has become cl e a r that the s i t u a t i o n can be treated as a pseudo-static case of applying forces to the moorage l i n e s . This implys that to reduce vessel motion, a s t i f f system must be u t i l i z e d to reduce s t a t i c d e f l e c t i o n s . The e f f e c t of dynamic a m p l i f i c a t i o n near resonance presents 8 4 probably the most dramatic observations noted by t h i s research. It seems appropriate that t h i s matter should be dealt with by the vessel moorage system and not at the marina design stage. The e f f o r t s to reduce pseudo-static defl e c t i o n s become meaningless i f the system has the opportunity to achieve resonance with the associated excessive motion and force concerns. It therefore remains to consider steps to reduce the p o s s i b i l i t y of a resonant condition developing. Undoubtedly each moored vessel represents a unique dynamic system but c e r t a i n approaches to the problem are common to a l l . For the case of surge, i t i s desirable to increase the natural frequency of the system above the threshold l i m i t f o r vessel motion as described i n chapter 10. This threshold l i m i t can be considered as the wave frequency necessary to generate wavelengths of the order of the vessel length. To this end, i t i s once again desirable to moor the v e s s e l with the s t i f f e s t possible system. For sway, although the objective remains the same, the s i t u a t i o n has changed somewhat. The vessel i n sway, f o r reasons discussed i n chapter 9, presents no opportunity for threshold behaviour. The objective i s therefore not to eliminate resonance, but rather to minimize i t s e f f e c t . This can be accomplished by introducing an asymmetry into the dynamic behaviour for sway. That i s , steps should be taken to avoid a s i t u a t i o n where the protection fenders and the moorage l i n e s e x h i b i t the same s t i f f n e s s e s . Also i t i s d e s i r a b l e , once again to use a s t i f f moorage condition as t h i s w i l l increase the natural frequency to a l e v e l at which the wave e x c i t i n g function possesses le s s energy. Based on the above, the following s p e c i f i c recommendations can be 85 made. On the subject of moorage condition, i t i s desirable to u t i l i z e an arrangement which o f f e r s short moorage l i n e s to accommodate the four moorage posit i o n s discussed i n chapter 2. For head seas and a dockside moorage p o s i t i o n , moorage conditions VII, IX, X, I I , XI, V and III s a t i s f y the moorage objective i n the observed order of preference. If a header or finger system i s to be used, moorage condition VI can be used i n a d d i t i o n to the above. In t h i s case, VI would o f f e r the best arrangement with the others following i n order. A bow-in moorage p o s i t i o n can be accommodated by moorage conditions I, XII, VIII i n order of e f f e c t i v e n e s s . For beam seas, the order of preference i s primarily the same as discussed f o r surge. Line s t i f f n e s s i s also an important c r i t e r i o n f o r motion reduction. For vessels up to 1500 kg, i t i s recommended that a minimum moorage s t i f f n e s s be equivalent to 13 mm nominal diameter double braided nylon. For heavier vessels a 20 mm nominal diameter i s suggested as a recommended minimum. The f i n a l v a r i a b l e to consider for the moorage of vessels i s the fender condition. This parameter, s i g n i f i c a n t for beam seas only, i s d i f f i c u l t to characterize i n the absence of more comprehensive information on the r e l a t i v e performance of d i f f e r i n g fender systems, as t h i s study d e a l t with one type only. It has become c l e a r that i t i s an easier matter to control vessel response In surge than sway. This exists not only because of the threshold concept but a l s o because the large sway drag p r o f i l e r e s u l t s i n an increased force on the vessel for sway motions. It can therefore be concluded that i t i s more desi r a b l e to p o s i t i o n the vessel with i t s 86 end to the p r i n c i p a l incident waves rather than i t s side. This, of course, comes as no surprise. 10.2. Future Work The r e s u l t s and discussions presented herein were directed toward serving i n tandem with the report by Mercer and Isaacson (1981) for the remaining modes of motion. It therefore remains an obvious extension to both reports to piece the d e t a i l s together and further r e f i n e comprehensive guidelines for marina design and vessel moorage. Such work would draw upon the e f f o r t s of Mercer and Isaacson to define "unfavourable vessel response" i n a l l modes of motion and could r e s u l t i n s i g n i f i c a n t contribution to the industry. The computer program "YACHT" also presents i n t e r e s t i n g p o s s i b i l i t i e s for further research. It could be further developed to accommodate the remaining four modes of v e s s e l motion. Upon t h i s expansion, i t could be used to generate response behaviour i n a l l modes of motion f o r a greater v a r i e t y of vessel shapes and moorage conditions. Such a tool would be most he l p f u l i n s a t i s f y i n g the future objective stated above of developing and r e f i n i n g comprehensive guidelines for vessel moorage. A v i t a l piece of data to be input into the program "YACHT" involves the hydrodynamic c h a r a c t e r i s t i c s of the v e s s e l . For the present discussions, empirical values were used as determined from the model te s t s . In order to expand "YACHT" into a more useful t o o l , meaningful data on hydrodynamics of a number of vessel shapes i s required. Such a data base could be established, for the cases of the added mass and r a d i a t i o n damping c o e f f i c i e n t s and the e x c i t i n g function, by generating a program to solve equations 17, 18, 19 and 20 i n a manner sim i l a r to Kim (1965) for common h u l l types. S i m i l a r l y hydrostatic c h a r a c t e r i s t i c s could be obtained. For e m p i r i c a l l y established c o e f f i c i e n t s such as drag c o e f f i c i e n t s and viscous damping, model tests on a number of vessel shapes could be performed. F i n a l l y , the present discussion deals only with regular incident wave t r a i n s . The frequency dependent d e t a i l s of the r e s u l t s would make the extension to a spe c t r a l a n a l y s i s an obvious one. With such a transfer function within easy reach, i r r e g u l a r wave trains could be examined which could more e f f e c t i v e l y e s t a b l i s h c r i t e r i a f o r marina design. CHAPTER 11 CONCLUSION Concern among the owners of small vessels over damage to t h e i r boats may take several forms. One of the most pronounced of these r e l a t e s to damage sustained while the vessels are moored i n a marina or yacht club. Impact damage, as well as damage r e s u l t i n g from excessive loads i n the moorage l i n e s can be attributed to a vessel's motion i n surge and sway. Re s p o n s i b i l i t y f o r both forms of damage l i e s i n part with the marina designer and i n part with the i n d i v i d u a l vessel owner: the former, i n h i s e f f o r t s to r e s t r i c t the nature of waves within the marina, and the l a t t e r as he moors the c r a f t to minimize vessel motion. Based upon experimental model tests and a s i m p l i f i e d computer model, recommendations have been made which w i l l aid both designer and ve s s e l owner i n t h e i r e f f o r t s . A recommended marina design c r i t e r i a i s presented which relates the character of waves within a protected harbour to vessel action i n surge and sway. Acceptable l i m i t s of vessel motion for three return periods have been defined and wave conditions r e s u l t i n g i n those l i m i t s comprise the recommended c r i t e r i a . In addition, to aid the i n d i v i d u a l vessel owner as he moors his boat, suggestions are forwarded concerning the choice of moorage material, conditions and techniques, and fender systems. 89 BIBLIOGRAPHY Abramson, H.N. and Wilson, B.W., 1955, "A Further Analysis of the l o n g i t u d i n a l Response of Moored Vessels to Sea O s c i l a t i o n s " , Proc.  of the J o i n t Midwestern Conf. on S o l i d and F l u i d Mechanics, Purdue U n i v e r s i t y , Indiana. Biggs, J.M., 1964, Introduction to S t r u c t u r a l Dynamics, McGraw H i l l , New York. Browne, A.D., Moullen, E.B. and Perkins, A.J., 1929-1930, "The Added Mass of Prisms Floating i n Water", Proc. of the Cambridge  P h i l l s o p h i c a l Society, Vol. 26, pp. 258-272. Clough, R.W. and Penzien, J., 1975, Dynamics of Structures, McGraw H i l l , San Francisco. Craig, R.R. J r . , 1981, S t r u c t u r a l Dynamics: An Introduction to Computer  Methods, Wiley, Toronto. Dunham, J.W. and Finn, A.A., 1974, "Small Crafts Harbours: Design Construction and Operation", Special Report No. 2, prepared for the U.S. Army Corps of Engineers, Coastal Engineering Research Center. Hermans, A.J. and Remery, G.F.M., 1970, "Resonance of Moored Objects i n Wave Trains", Proc. of the 12th Conf. on Coastal Engineering, Washington, D.C. pp. 1685-1700. Ki l n e r , F.A., 1960, "Model Tests on the Motion of Moored Ships Placed i n Long Waves", Proc. of the 7th Conf. on Coastal Engineering, The Hague, pp. 723-745. Kim, W.D., 1965, "On the Harmonic O s c i l l a t i o n s of a Rigid Body on a Free Surface", Journal of F l u i d Mechanics, Vol. 24, pp. 293-301. Knapp, R.T., 1951, "Wave Produced Motion of Moored Ships", Proc. of the  2nd Conf. on Coastal Engineering, Houston. Le Mehaute, B., 1976, "Wave Agitation C r i t e r i a for Harbours", Proc. of  the 16th Conf. on Coastal Engineering, Lee, C.E., 1964, "On the Design of Small Craft Harbours", Proc. of the  9th Conf. on Coastal Engineering, Lisbon, pp. 713-725. Mercer, A.G., Isaacson, M., 1981, Study to Determine Acceptable Wave  Climate In Small Craft Harbours, Small Crafts Harbours Branch, Fis h e r i e s and Oceans Canada, Prepared by Northwest Hydraulics Consultants Ltd. Muga, B.J. and Wilson, J.F., 1970, Dynamic Analysis of Ocean Structures, Plenum Press, New York. Newman, J.N., 1977, Marine Hydrodynamics, M.I.T. Press, Boston. Raichlen, F., 1966a, "Wave Induced O s c i l l a t i o n s of Small Moored Vessels", Report No. KH-R-10, W.M. Keck Laboratory of Hydraulics and Water Resources, C a l i f o r n i a I n s t i t u t e of Technology, Monterey, C a l i f . Raichlen, F., 1966, "Wave Induced O s c i l l a t i o n s of Small Moored Vessels", Proc. of the 10th Conf. on Coastal Engineering, Tokyo, Vol. I I , pp. 1249-1273. Raichlen, F., 1968a, "The Motions of Small Boats i n Standing Waves", Report No. KH-R-17 , W.M. Keck Laboratory of Hydraulics and Water Resources, C a l i f o r n i a I n s t i t u t e of Technology, Monterey, C a l i f . Raichlen, F., 1968, "The Motions of Small Boats i n Standing Waves", Proc. of the 11th Conf. on Coastal Engineering, London, pp. 1531-1554. Russel, R.C.H., 1959, "A Study of the Movement of Moored Ships Subjected to Wave Action", Proc. of the I n s t i t u t e of C i v i l Engineers, Vol. 12, London. Sarpkaya, T. and Isaacson, M., 1981, Mechanics of Wave Forces on  Offshore Structures, Van Nostrand Reinhold Co., Toronto. 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Wendal, K., 1956, "Hydrodynamic Masses and Hydrodynamic Moments of In e r t i a " , The David Taylor Model Basin, T r a n s l a t i o n 260, C a l i f o r n i a . Wilson, B.W., 1967, " E l a s t i c C h a r a c t e r i s t i c s of Moorings", Jour, of  Waterways and Harbours Div., ASCE, Nov. pp. 27-56. Wong, R., 1981, "Strength, Elongation and Energy Absorption of Synthetic Fiber Rope", Report 8-81, Sampson Ocean Systems Inc., Boston. 91 Marina Location V i s i t e d Owners Interviewed Kingston Yacht Club Kingston, Ont. X X Partsmouth Olympic Harbour Kingston, Ont. X Sea Cove Marina Deep Cove, B.C. X Deep Cove Yacht Club Deep Cove, B.C. X Mosquito Creek Marine Basin N. Vancouver, B.C. X Eagle Harbour Yacht Club W. Vancouver, B.C. X West Vancouver Yacht Club W. Vancouver, B.C. X Thunderbird Yacht Club W. Vancouver, B.C. X Royal Vancouver Yacht Club Vancouver, B.C. X Point Roberts Yacht Club Point Roberts, Wash. X TABLE 1. MARINAS AND YACHT CLUBS EXAMINED. Model F i n Keel Prototype S a i l b o a t Swiftsure 24 Basic S p e c i f i c a t i o n s L Q Length o v e r a l l (mm) 762 7220 L w Waterplane length (mm) 680 6330 B Beam (mm) 244 2320 D D r a f t (mm) 149 1270 W^  Displacement (N) 22.1 12900 W, B a l l a s t i n k e e l (N) D 12.5 6500 Dimensionless R a t i o s B/L Beam/Waterplane length w 0.365 0.367 D/L Draft/Waterplane length w 0.224 0.200 W Displacement 0.00755 0.0052 YL 3 w W b — Ballast/Displacement W t 0.56 0.50 * TABLE 2. PHYSICAL COMPARISON BETWEEN MODEL AND PROTOTYPE . * Source: Mercer and Isaacson (1981) 93 Spring 1 Spring 2 Spring 3 mass (g) 2311 2311 2311 spring s t i f f n e s s (kg/s 2) 190.5 94.8 54.1 l i n e s t i f f n e s s (kg/s 2) 22.7 22.7 22.7 t o t a l s t i f f n e s s (kg/s 2) 213.2 117.5 76.8 natural frequency ( s - 1 ) 9.605 7.132 5.76 natural period (s) 0.66 0.88 1.10 TEST I damped natural period (s) 0.66 0.88 1.10 ( C / C c r i t > l 0.0031 0.0026 0.0022 TEST II damped natural period (sec) 0.66 0.88 1.10 < C / C c r i t > l + 2 0.013 8 0.0128 0.0110 ( C / C c r i t > 2 0.012 7 0.0112 0.0088 TABLE 3. RESULTS OF TESTS TO DETERMINE MEASUREMENT SYSTEM DAMPING. Spring 1 Spring 2 Spring 3 mass (g) 2311 2311 2311 s p r i n g s t i f f n e s s ( k g/s 2) 190.5 94.8 54.1 damped n a t u r a l period (s) 0.76 1.08 1.44 damped n a t u r a l frequency(s) 8.22 5.82 4.36 v i r t u a l mass (g) 2787 2801 2842 added mass (g) 0.476 0.490 0.531 ( C / C c r i t ) 1 + 2 + 3 0.0480 0.0460 0.0430 < C / C c r i t > 3 0.0342 0.0337 0.0320 TABLE 4. RESULTS FROM TESTS TO DETERMINE ADDED MASS AND DAMPING COEFFICIENTS IN SURGE. R o l l Only - Natural period = 1.15 sec. - Damping — ) = 0.0276 c r i t - V i r t u a l mass + = 70.4 " ^ " N * ™ rad * „.„„ N-mm - S t i f f n e s s = 2100 - — j Sway Only Spring s Td( s) C _ (%) u c r i t T n ( s ) K s " 1 ) Vmass<kS> Amass^g) 1 190.5 1.5 24.1 1.46 4.32 10.2 7.90 2 94.8 1.9 23.7 1.85 3.40 8.2 5.90 3 54.1 2.5 24.9 2.42 2.60 8.0 5.69 R o l l and Sway Spring 1 190.5 1.5 20.0 1.47 4.27 10.45 8.14 2 94.8 1.8 21.0 1.76 3.57 7.44 5.13 3 54.1 2.6 22.4 2.53 2.48 8.8 6.49 TABLE 5. MODEL VESSEL'S DYNAMIC BEHAVIOR TO BEAM SEAS. + at natural frequency of r o l l * Source: Mercer and Isaacson (1981) 96 TABLE 6. PROVISIONAL RECOMMENDED CRITERIA FOR "GOOD"* WAVE CLIMATE IN SMALL CRAFT HARBOURS FOR RESISTING VESSEL SURGE AND SWAY. Wave Period (sec) Wave Directio n Values of £ for each Return Period 1 week 1 year 50 years 0-2 2-6 6+ Head Beam Head Beam Head Beam 1.2 1 1.5 1 1.7 + 300 400 100 400 100 + 300 500 200 500 200 + 450 600 300 600 300 Figures represent the combined e f f e c t of wave height, wave r e f l e c t i o n and water depth such that: R± (1 + K R) C, = i n mm tanh n(kd) + I n s i g n i f i c a n t vessel response to incident waves. * For c r i t e r i a f o r "excellent" wave climate m u l t i p l y by 0.75 and for "moderate" wave climate multiply by 1.25. F I G U R E I M O D E S OF V E S S E L MOTION (c) FINGER (d) BOW- IN FIGURE 2 COMMON MOORAGE POSITIONS FIGURE 3 COMMON MOORAGE CONDITIONS 100 FIGURE 4 GENERAL MOORAGE STIFFNESS CURVE 101 S E A DIRECTION /// / DOCK L L L N L \ ^ ' ^ ^ - ^ ^ FIGURE 5 MODEL VESSEL SIMPLIFIED S H A P E FIGURE 6 VERIFICATION OF TIME STEP SOLUT ION FOR SURGE FIGURE 7 VERIFICATION OF TIME STEP SOLUTION FOR SWAY 0 10 20 30 ELONGATION (PER CENT) FIGURE 8 NYLON ROPE FORCE-ELONGATION CURVE (a) SPECIMEN (b) TEST LOADING FIGURE 9 PROTOTYPE BALLOON FENDER 800 + H 1 Ii 1 , 1. S 10 15 20 25 3 0 DEFLECTION ( m m ) FIGURE 10 FENDER FORCE-DEFLECTION CURVE 106 F I G U R E 1 1 W A V E B A S I N L A Y O U T 107 F I G U R E 1 2 W A V E B A S I N 108 FIGURE .13 MODEL MOORAGE FRAME 110 F I G U R E 1 5 M O D E L V E S S E L T E S T E D SURGE SWAY F IGURE 1 6 WAVE P R O B E S Y S T E M I l l SWAY FIGURE 17 VESSEL MOTION MEASUREMENT SYSTEM STEEL NYLON RUBBER r r ^ >i i n m BOAT DOCK RGURE 2 0 MODEL MOORAGE LINES 112 FIGURE 1 8 MEASUREMENT SYSTEM VESSEL ATTACHMENT 113 FIGURE 1 9 MEASUREMENT SYSTEM L.V.D.T. 0 . 0 1.0 2.0 3.0 4 .0 S.O 6 . 0 7.0 8 . 0 9 . 0 10.0 FL O N G R T I O N (%) I G U R E 2 I MOORAGE LINES FORCE-ELONGATION BEHAVIOUR I—• I—• • p -FIGURE 2 2 BALLOON FENDER STIFFNESS 116 FIGURE23 PARTIAL STANDING WAVE <D BEFORE EXPERinENTS • AFTER EXPERIMENTS - BASIN CHARACTER FIGURE24 WAVE BASIN REFLECTION CHARACTERISTICS FIGURE 2 5 TYPICAL LVDT CALIBRATION CURVE 119 mum tllMMI % 3 HAAH MASS i — (a) SYSTEM EXCLUDED *fHtr-g 3 c K 1 #WV\A-[MASS *2 C2 (b) SYSTEM INCLUDED FIGURE 2 6 MEASUREMENT SYSTEM DAMPING TESTS c K 1 i H ^ - w v -I Hi I f— —111 *"' • tl MASS A. MASS -^ 1 f Co FIGURE 2 7 ADDED MASS AND DAMPING TESTS 0_ r— CX \ CO o CO cr: o - | CO cr o LU CN Q_ o _ CD 0.0 0.5 1.0 1.5 — i 1 1 r 2.0 2.5 SLflCK/URTPLL (X100) ~ r ~ 3.0 3.5 4.0 4.5 FIGURE 2 8 EFFECT OF SLACKNESS ON NATURAL PERIOD IN SURGE FIGURE 2 9 VESSEL MOORED FOR SURGE TESTS (3) 122 N - P (b) l - l (c) t 3 ) (d) u (e) fa u y (•F) L f i f i r u i r i ^ ^ L D L.O < LXJ o >-o z UJ Z> O UJ LEGEND V _ . SLACK FIGURE30 GENERALIZED SURGE RESPONSE FIGURE 3 1 V E S S E L MOORED FOR SWAY T E S T S a) TEST I EXPERIMENTAL 'YACHT' FIGURE 3 3 TYPICAL SURGE TIME HISTORY CURVES 126 EXPERIMENTAL V YACHT FIGURE 3 4 TYPICAL SWAY TIME HISTORY CURVES © VESSEL MOTION Y /WAVE HEIGHT 0.2 x 0.3 X 0.5 FIGURE 3 6 EFFECT OF BASIN REFLECTION ON SURGE 128 129 3 + 0 6 7 8 9 FIGURE 4 0 EFFECT OF'LINE STIFFNESS ON SURGE M.C. 0 8 l a FIGURE 4 1 EFFECT OF MOORAGE CONDITION ON SURG — o FENDER INCL. X X FENDER EXCL. 0 3 4 5 6 7 8 9 F IGURE42 EFFECT OF FENDER CONDITION ON SURGE o o X 12-10-8 -6-4 -2-YACHT x-© 4.3 0.015 x 7.6 0.015 -+-3 H; -jjCx IOO) FIGURE 4 3 EFFECT OF WAVE HEIGHT ON SWAY FREE MOVEMENT Cx ioo) FIGURE 4 4 EFFECT OF SLACKNESS ON SWAY 132 1 1 1 1 1 r 2 3 4 5 6 7 8 FIGURE4.6 EFFECT OF LINE STIFFNESS ON SWAY 133 34 ©-x-® F E N D E R E X C L . F E N D E R INCL . Q 2-< CC 0 1 1 1 1 1 1 r 2 3 4 5 6 ? 8 FIGURE 4 8 E F F E C T OF F E N D E R CONDIT ION ON SWAY APPENDIX I COMPUTER LISTING OF PROGRAM "YACHT" 135 ^ c ********************************************************************** 2 C * * 3 C * PROGRAM Y A C H T * 4 C * * 5 c ********************************************************************** 6 C T h e f o l l o w i n g p r o g r a m ' Y A C H T ' e v a l u a t e s t h e p r e d i c t e d b e h a v i o r o f 7 C a s m a l l m o o r e d v e s s e l i n m o t i o n s o f s u r g e a n d s w a y . I t u t i l i z e s 8 C l i n e a r w a v e t h e o r y a n d 1s a p p l i c a b l e f o r p r o g r e s s i v e , p a r t i a l l y 9 C r e f l e c t e d a n d s t a n d i n g w a v e s . T h e p r o g r a m 1s s e t u p t o o f f e r a t i m e 1 0 C s t e p p e d s o l u t i o n t o t h e s t a n d a r d SDOF s p r i n g - m a s s - d a s h p o t d y n a m i c 11 C e q u a t i o n o f m o t i o n f o r s u r g e , a n d a c o u p l e d 2 D 0 F , i n c l u d i n g r o l l , f o r 12 C s w a y . 13 C 14 C O M M O N / P 1 / H E I G H T , P E R O I D , R E F C O . P O S I T , D E P T H , A L E N T H 15 C 0 M M 0 N / P 2 / D I S P , F O R C E 16 C 0 M M 0 N / P 3 / 5 1 , S 2 , V M A S S 1 , V M A S S 2 , D A M P 1 , D A M P 2 . D A M P 3 , Y M 2 1 , Y C 2 1 , V M A S S 3 17 C 0 M M 0 N / P 4 / W A T P L B , W A T P L L , D R A F T , W K E E L . H K E E L . X L K E E L , S L 18 C 0 M M 0 N / P 5 / A , B , C , D , I X . N X 19 C 0 M M 0 N / P 6 / X M A S S 2 . Z B A R , E C C , S T F R Y S , C D R A G 1 , C D R A G 2 , Z C E N , P A R E A 2 0 C 0 M M 0 N / P 7 / B D I S P , B F O R C E , N B A L , N P O I T 21 C 0 M M 0 N / P 9 / Y M 1 2 , Y C 1 2 , Y K 1 2 2 2 D I M E N S I O N D I S P ( 5 0 ) , F O R C E ( 5 0 ) . A ( 1 0 ) . B ( 1 0 ) , C ( 1 0 ) , D ( 1 0 ) , B D I S P ( 5 0 ) , B F O 2 3 $ R C E ( 5 0 ) 24 C  2 5 C E n t e r i n f o r m a t i o n c o n c e r n i n g t h e w a v e t r a i n : H e i g h t , P e r o i d , 2 6 C r e f l e c t i o n c o e f f i c i e n t , p o s i t i o n i n s t a n d i n g w a v e , w a t e r d e p t h 2 7 C 2 8 R E A D ( 5 . 6 0 ) H E I G H T , P E R O I D , R E F C O , P O S I T , D E P T H 2 9 W R I T E ( 6 , 6 0 ) H E I G H T , P E R O I D , R E F C O , P O S I T . D E P T H 3 0 C 31 C E n t e r a d e s c r i p t i o n o f t h e f o r c e e l o n g a t i o n c h a r a c t e r i s t i c s o f 3 2 C t h e m o o r a g e l i n e s . W h e n t h e m a t e r i a l b e h a v e s d i f f e r e n t l y u n d e r 3 3 C s t a t i c a n d c y c l i c l o a d i n g t h e l a t t e r c u r v e s h o u l d b e u s e d . I n 34 C a n y c a s e t h e d i s p l a c e m e n t ! 1 n d i m e n s i o n l e s s p e r c e n t ) s h o u l d 3 5 C b e f o l l o w e d b y t h e c o r r e s p o n d i n g f o r c e ( i n n e w t o n s ) FOR ONE 3 6 C MOORAGE L I N E O N L Y . 3 7 C 3 8 R E A D ( 5 , 6 1 ) N X 3 9 W R I T E ( 6 . 6 1 ) N X 4 0 DO 2 0 O 1 1 = 1 . N X 41 R E A D ( 5 , 5 9 ) D I S P ( I I ) , F 0 R C E ( I I ) 4 2 W R I T E ( 6 , 5 9 ) D I S P ( I I ) , F 0 R C E ( I I ) 4 3 2 0 O C O N T I N U E 4 4 C 4 5 C E n t e r a d e s c r i p t i o n o f t h e f o r c e c o m p r e s s i o n c h a r a c t e r i s t i c s o f 4 6 C t h e b a l l o o n f e n d e r s . T h e n u m b e r o f f e n d e r s a r e i n p u t , f o l l o w e d b y 4 7 C t h e n u m b e r o f p o i n t s u s e d t o d e s c r i b e t h e c o m p r e s s i o n c u r v e . T h e 4 8 C p o i n t s a r e t h e n e n t e r e d , o n e p e r l i n e , a s p e r a s t a n d a r d s t r e s s 4 9 C s t r a i n p l o t w i t h d e f 1 e c t l o n ( m m ) o n t h e a b s c i s s a a n d f o r c e ( N ) o n 5 0 C t h e o r d i n a t e . 51 C 5 2 R E A D ( 5 . 6 1 ) N B A L 5 3 W R I T E ( 6 , 6 1 ) N B A L 54 R E A D ( 5 , 6 1 ) N P 0 I T 5 5 W R I T E ( 6 , 6 1 ) N P 0 I T 5 6 DO 2 0 4 1 1 = 1 . N P O I T 5 7 R E A D ( 5 , 5 9 ) B D I S P ( I I ) . B F O R C E ( I I ) 5 8 W R I T E ( 6 . 5 9 ) B D I S P ( I I ) . B F O R C E ( I I ) 5 9 2 0 4 C O N T I N U E 6 0 C 136 61 C E n t e r a d e s c r i p t i o n o f t h e m o o r a g e c o n d i t i o n b y n o t i n g t h e p o s t 1 o n 6 2 C o f t h e e n d s o f e a c h m o o r a g e 1 l n e . S p e c i f l e a l l y t h e X a n d Y c o o r d i n a t e s 6 3 C f o r t h e d o c k a n d v e s s e l c o n t a c t p o i n t s a r e t o b e t h e i n p u t f o r e a c h 6 4 C l i n e . A l l d i m e n s i o n s a r e 1n m i l l i m e t e r s . 6 5 C 6 6 R E A D ( 5 . 6 1 ) I X 6 7 W R I T E ( 6 . 6 1 ) I X 6 8 DO 2 0 1 1 1 = 1 , I X 6 9 R E A D ( 5 , 7 4 ) A ( I I ) , B ( I I ) , C ( I I ) , D ( I I ) 7 0 W R I T E ( 6 , 7 4 ) A ( I I ) . B ( I I ) , C ( I I ) , D ( I I ) 71 2 0 1 C O N T I N U E 7 2 C - -7 3 C T h i s l i n e o f i n p u t d e s c r i b e s t h e a m o u n t o f s l a c k n e s s i n t h e v e s s e l ' s 7 4 C m o o r a g e c o n f I g u r a t 1 o n . S I a c k n e s s I s d e f i n e d a s t h e d i s t a n c e t h e 7 5 C v e s s e l c a n m o v e a b o w a n d a s t e r n f r o m t h e c e n t e r i n g p o s i t i o n w i t h o u t 7 6 C e n c o u n t e r i n g r e s i s t a n c e f r o m t h e m o o r i n g l i n e s . T h a t i s : o n e h a l f o f 7 7 C t h e f r e e p l a y i n t h e s u r g e d l r e c t i o n . F o r s w a y t h e d e f i n i t i o n i s s i m i l a r 7 8 C e x e p t i t i s s t a r b o a r d / p o r t m o t i o n w h i c h i s c o n s i d e r e d . 7 9 C 8 0 R E A D ( 5 . 7 2 ) S L 1 , S L 2 81 W R I T E ( 6 , 7 2 ) S L 1 . S L 2 8 2 C 8 3 C E n t e r t h e d y n a m i c c h a r a c t e r i s t i c s o f t h e v e s s e l a s p e r t h e f o l l o w i n g 8 4 C s c h e d u l e : 8 5 C 8 6 C V A R I A B L E D E S C R I P T I O N U N I T S 8 7 C 8 8 C 8 9 C VMASS1 V i r t u a l m a s s f o r s u r g e K g 9 0 C DAMP 1 T o t a l s y s t e m d a m p i n g f o r s u r g e K g 91 C V M A S S 2 V i r t u a l m a s s f o r s w a y K g 9 2 C DAMP2 T o t a l s y s t e m d a m p i n g f o r s w a y K g / s 9 3 C V M A S S 3 V i r t u a l m a s s f o r r o l l K g . m . m m / r a d 9 4 C X M A S S 2 B o d y m a s s K g 9 5 C DAMP3 T o t a l s y s t e m d a m p i n g f o r r o l l K g . m . m m / r a d . s 9 6 C CDRAG1 S u r g e d r a g c o e f f i c i e n t D/L 9 7 C CDRAG2 S w a y d r a g c o e f f i c i e n t D/L 9 8 C S T F R Y S R o l l s t i f f n e s s K g . m . m m / r a d . s . s 9 9 C 1 0 0 C 101 R E A D ( 5 , 5 8 ) V M A S S 1 , D A M P 1 , V M A S S 2 , D A M P 2 , V M A S S 3 , X M A S S 2 , D A M P 3 , C D R A G 1 . C D R 1 0 2 $ A G 2 , S T F R Y S 1 0 3 W R I T E ( 6 , 5 8 ) V M A S S 1 , D A M P 1 , V M A S S 2 , D A M P 2 . V M A S S 3 , X M A S S 2 , D A M P 3 , C O R A G 1 . C D 104 $ R A G 2 , S T F R Y S 1 0 5 R E A D ( 5 . 6 3 ) Y M 1 2 , Y C 1 2 , Y M 2 1 . Y C 2 1 , Y K 1 2 106 W R I T E ( 6 . 6 3 ) Y M 1 2 , Y C 1 2 , Y M 2 1 , Y C 2 1 . Y K 1 2 107 C 108 C E n t e r a d e s c r i p t i o n o f t h e s h a p e o f t h e v e s s e l u n d e r c o n s i d e r a t i o n 1 0 9 C b y n o t i n g a n u m b e r o f k e y d i m e n s i o n s . ' W A T P L B ' i s t h e w a t e r p l a n e 1 1 0 C b e a m . ' W A T P L L ' i s t h e w a t e r p l a n e l e n g t h , ' D R A F T ' i s s i m p l y t h a t , 111 C ' W K E E L ' l s t h e k e e l w i d t h , ' H K E E L ' i s t h e k e e l h e i g h t , ' X L K E E L ' i s t h e 112 C k e e l l e n g t h , a n d ' S L ' l s t h e s l o p e o f t h e h u l l l e n g t h a n d c a n b e 113 C c o n s i d e r e d a s t h e d i s t a n c e u n d e r b o a t f r o m t h e b o w ( o r s t e r n ) 114 C w a t e r l l n e t o t h e l e a d i n g e d g e o f t h e k e e l . ' Z B A R ' i s t h e v e r t i c a l 1 1 5 C d i s t a n c e f r o m t h e c e n t e r o f f l o a t a t i o n t o t h e c e n t e r o f b o u y a n c y . 1 1 6 C ' E C C i s t h e h e i g h t a b o v e t h e s t i l l w a t e r l e v e l ( c e n t e r o f f l o a t a t 117 C i o n ) a t w h i c h t h e v e s s e l i s m o o r e d . ' Z C E N ' i s t h e d e p t h b e l o w t h e 1 1 8 C s t i l l w a t e r l e v e l o f t h e c e n t r o l d o f t h e p r o f i l e a r e a . ' P A R E A ' i s 1 1 9 C t h e v e s s e l ' s p r o f i l e a r e a . 1 2 0 C -137 121 R E A D ( 5 , 6 2 ) W A T P L B . W A T P L L , D R A F T , W K E E L , H K E E L . X L K E E L . S L , Z B A R , E C C , Z C E N . 122 $ P A R E A 1 2 3 W R I T E ( 6 , 6 2 ) W A T P L B , W A T P L L , D R A F T , W K E E L , H K E E L , X L K E E L , S L , Z B A R . E C C , Z C E N 124 $ , P A R E A 1 2 5 c-126 C T h e v a r i a b l e ' I D N ' , I n p u t a t t h i s p o i n t I s a n o p e r a t o r p e r m i t t i n g 127 C t h e u s e r t o c h o o s e t h e t y p e o f m o t i o n t o b e a n a l y z e d . ' 1 ' i n s t r u c t s 128 c t h e p r o g r a m t o e x e c u t e s u r g e o n l y . ' 2 ' c a l l s f o r s w a y , w h i l e ' 3 ' 129 c e x a m i n e s b o t h s u r g e a n d s w a y . 1 3 0 131 R E A D ( 5 , 7 3 ) I D N 132 C A T = 0 . 0 0 2 * W A T P L L 133 C A R = 0 . 0 0 0 * W A T P L L 134 I F ( S L 1 . L E . C A T ) S 1 = 0 . 0 135 I F ( S L 2 . L E . C A R ) S 2 = 0 . 0 1 3 6 W R I T E ( 6 , 7 3 ) I D N 137 I F ( S L 1 . G T . C A T ) S 1 = S L 1 - C A T 138 I F ( S L 2 . G T . C A R ) S 2 = S L 2 - C A R 139 C A L L WAVEL 1 4 0 I F ( P O S I T . L T . O . O ) P 0 S I T = A L E N T H / 4 . 141 I F ( I D N . E O . 1 . 0 R . I D N . E Q . 3 ) C A L L SURGE 142 I F ( I D N . E 0 . 2 . 0 R . I D N . E 0 . 3 ) C A L L SWAY 1 4 3 5 8 F 0 R M A T ( 9 F 1 0 . 3 , F 1 5 . 3 ) 144 5 9 F 0 R M A T ( 2 F 1 5 . 5 ) 145 6 0 F O R M A T ( 5 F 1 2 . 5 ) 146 61 F 0 R M A T ( I 1 5 ) 147 6 2 F O R M A T ( 1 1 F 1 0 . 3 ) 148 6 3 F 0 R M A T ( 5 F 1 2 . 3 ) 149 7 2 F 0 R M A T ( 2 F 1 5 . 5 ) 1 5 0 7 3 F O R M A T ( 1 2 0 ) 151 74 F 0 R M A T ( 4 F 1 5 . 5 ) 152 S T O P 153 END 154 c 155 c * * 156 c * S U B R O U T I N E WAVEL * 157 c * 158 c 1 5 9 c 1 6 0 c C a l c u l a t e s t h e i n c i d e n t t r a i n w a v e l e n g t h . 161 c 162 r* L> -163 S U B R O U T I N E WAVEL 164 C O M M O N / P 1 / H E I G H T , P E R O I D , R E F C O , P O S I T , D E P T H , A L E N T H 165 W L E N T H = 5 0 O 0 . 166 DO 9 6 11 = 1 . 15 167 B L E N T H = ( ( 9 8 1 0 . * P E R O I D * * 2 ) / 6 . 2 8 3 2 ) * T A N H ( 6 . 2 8 3 2 * D E P T H / W L E N T H ) 168 W L E N T H = B L E N T H 169 9 6 C O N T I N U E 1 7 0 A L E N T H ' B L E N T H 171 W R I T E ( 6 , 9 7 ) A L E N T H 172 9 7 F O R M A T ( / / , 1 5 X , ' L I N E A R WAVE L E N G T H = ' , F 6 . 0 , / / ) 173 R E T U R N 174 END 175 c 176 c * * 177 c * S U B R O U T I N E SURGE * 178 c * * 179 c 1 8 0 c 138 181 C P e r f o r m s t h e t i m e s t e p c a l c u l a t i o n f o r t h e s u r g e p o r t i o n o f t h e 182 C p r o g r a m . 183 C 1 8 4 C - -1 8 5 S U B R O U T I N E S U R G E 1 8 6 C O M M O N / P 1 / H E I G H T . P E R O I D , R E F C O . P O S I T . D E P T H , A L E N T H 187 C 0 M M 0 N / P 3 / S 1 , S 2 , V M A S S 1 , V M A S S 2 , D A M P 1 , D A M P 2 , D A M P 3 , Y M 2 1 , Y C 2 1 , V M A S S 3 188 D I M E N S I O N X T R U E ( 5 0 0 ) , X X X ( 5 0 0 ) 189 W R I T E ( 6 . 7 5 O ) P 0 S I T 1 9 0 7 5 0 F 0 R M A T ( 2 0 X . F 1 5 . 3 ) 191 W R I T E ( 6 , 5 0 3 ) 192 W R I T E ( 6 . 5 0 2 ) 193 W R I T E ( 6 . 5 0 1 ) 194 X 0 = 0 . 195 X 1 0 = 0 . 196 D T = P E R 0 I D / 3 O . 197 DO 1 0 0 1 = 1 . 4 0 0 198 T = F L 0 A T ( I ) * D T 199 T 1 = ( ( F L 0 A T ( I ) + F L 0 A T ( I - 1 ) ) / 2 . ) * D T 2 0 0 C A L L F R O K R Y ( T 1 , F K F O R ) 2 0 1 X X X ( I ) = F K F 0 R 2 0 2 C A L L S T I F F ( X 0 , S T E F . S 1 ) 2 0 3 H A P = X 0 - 4 0 0 . * D T 2 0 4 H A P P = 4 0 . * D T 2 0 5 DO 1 0 4 J d = 1 , 6 2 0 6 R F M I N = 1 0 0 O O 0 . 2 0 7 DO 101 J = 1 , 2 1 2 0 8 X = H A P + F L 0 A T ( d - 1 ) * H A P P 2 0 9 C A L L S T I F F ( X , S T I F . S 1 ) 2 1 0 R E S F 0 R = A B S ( S T I F / 2 . - F K F 0 R + S T E F / 2 . + ( V M A S S 1 * 0 . 0 0 1 / D T ) * ( 2 . * ( X - X 0 ) / D T - 2 2 1 1 * . * X 1 0 ) + D A M P 1 * 0 . 0 0 1 * ( X - X 0 ) / D T ) 2 1 2 I F ( R E S F O R . L E . R F M I N ) X T R U E ( I ) = X 2 1 3 I F ( R E S F O R . L E . R F M I N ) R F M I N = R E S F O R 2 1 4 101 C O N T I N U E 2 1 5 H A P P = H A P P / 1 0 . 2 1 6 H A P = X T R U E ( I ) - H A P P * 1 0 . 2 1 7 104 C O N T I N U E 2 1 8 B X 0 = X 0 2 1 9 B X 1 0 = X 1 0 2 2 0 X O = X T R U E ( I ) 2 2 1 X 1 0 = 2 . * ( ( X T R U E ( I ) - B X O ) / D T ) - B X 1 0 2 2 2 1 0 0 C O N T I N U E 2 2 3 DO 102 N J = 1 , 4 0 0 , 2 2 2 4 N I = N J + 1 2 2 5 X T = F L 0 A T ( N I ) * D T 2 2 6 W R I T E ( 6 , 5 0 0 ) X T . X T R U E ( N I ) , X X X ( N I ) 2 2 7 102 C O N T I N U E 2 2 8 5 0 0 F 0 R M A T ( 5 X , F 1 5 . 3 . 9 X , F 1 5 . 5 . 1 0 X . F 1 0 . 3 ) 2 2 9 5 0 1 F 0 R M A T ( 1 5 X , ' ( S E C ) ' . 1 5 X . ' ( M M ) ' . 1 5 X , ' ( N E W T O N S ) ' , / / ) 2 3 0 5 0 2 F 0 R M A T ( 1 5 X . ' T I M E ' , 1 1 X , ' D I S P L A C E M E N T ' . 6 X . ' F R O U D E - K R Y L O V F O R C E S ' . / ) 2 3 1 5 0 3 F O R M A T ( / / / , 1 0 X , ' T H E F O L L O W I N G I S T H E SURGE T I M E D E P E N D E N T R E S P O N S E 2 3 2 * : ' . / / / ) 2 3 3 R E T U R N 2 3 4 END 2 3 5 C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 2 3 6 C * * 2 3 7 C * S U B R O U T I N E SWAY * 2 3 8 C * * 2 3 9 C ********************************************************************** 2 4 0 C 139 2 4 1 C P e r f o r m s t h e t i m e s t e p c a l c u l a t i o n f o r t h e s w a y p o r t i o n o f t h e 2 4 2 C p r o g r a m . 2 4 3 C 2 4 4 C 2 4 5 S U B R O U T I N E SWAY 2 4 6 C O M M O N / P 1 / H E I G H T , P E R O I D . R E F C O . P O S I T , D E P T H . A L E N T H 2 4 7 C O M M 0 N / P 3 / S 1 , S 2 . V M A S S 1 , V M A S S 2 , D A M P 1 , D A M P 2 . D A M P 3 . Y M 2 1 , Y C 2 1 . V M A S S 3 2 4 8 C 0 M M 0 N / P 6 / X M A S S 2 . Z B A R , E C C . S T F R Y S , C D R A G 1 , C 0 R A G 2 , Z C E N , P A R E A 2 4 9 C 0 M M 0 N / P 9 / Y M 1 2 , Y C 1 2 , Y K 1 2 2 5 0 D I M E N S I O N Y Y Y ( 5 0 0 ) , R Y T R U E ( 5 0 0 ) , Y M O M ( 5 0 0 ) . Y T R U E ( 5 0 0 ) 2 5 1 W R I T E ( 6 . 4 0 3 ) 2 5 2 W R I T E ( 6 . 4 0 2 ) 2 5 3 W R I T E ( 6 , 4 0 1 ) 2 5 4 D T = P E R 0 I D / 3 0 . 2 5 5 Y 0 = 0 . 0 2 5 6 Y 1 0 = 0 . 0 2 5 7 Y 2 0 = 0 . 0 2 5 8 R Y O = 0 . 0 2 5 9 R Y 1 0 = 0 . 0 2 6 0 R Y 2 0 = 0 . 0 2 6 1 DO 4 5 0 1 = 1 , 4 0 0 2 6 2 T = F L O A T ( I ) * D T 2 6 3 T 1 = ( ( F L 0 A T ( I ) + F L 0 A T ( I - 1 ) ) / 2 . ) * D T 2 6 4 C A L L F K S W A A ( T 1 , F K S W A Y . F K M O M ) 2 6 5 Y Y Y ( I ) =FKSWAY 2 6 6 Y M O M ( I ) = F K M O M 2 6 7 C A L L Y S T I F F ( R Y 0 , Y 0 . R Y S T E F , S 2 , Y S T E F ) 2 6 8 R Y H A P = R Y 0 - 4 . * D T 2 6 9 R Y H A P P = 0 . 4 * D T 2 7 0 DO 4 5 7 K K = 1 , 5 2 7 1 R F M I N = 1 0 0 0 0 0 0 0 . 2 7 2 DO 4 5 1 J 1 = 1 . 2 1 2 7 3 d = J 1 - 1 2 7 4 R Y = R Y H A P + F L O A T ( J ) * R Y H A P P 2 7 5 C A L L Y S T I F F ( R Y . Y O , R Y S T I F , S 2 , Y S T E F ) 2 7 6 R E S F O R = ABS(-FKM0M+YM21»Y20+YC21 * Y 1 0 + R Y S T I F / 2 . + R Y S T E F / 2 . + ( V M A S S 3 / D T 2 7 7 $ ) * ( 2 . * ( R Y - R Y 0 ) / D T - 2 . * R Y 1 0 ) + D A M P 3 * ( R Y - R Y 0 ) / D T ) 2 7 8 I F ( R E S F O R . L E . R F M I N ) R Y T R U E ( I ) = R Y 2 7 9 I F ( R E S F O R . L E . R F M I N ) R F M I N = R E S F O R 2 8 0 4 5 1 C O N T I N U E 2 8 1 R Y H A P P = R Y H A P P / 1 0 . 2 8 2 R Y H A P = R Y T R U E ( I ) - R Y H A P P * 1 0 . 2 8 3 4 5 7 C O N T I N U E 2 8 4 B R Y 0 = R Y 0 2 8 5 B R Y 1 0 = R Y 1 0 2 8 6 B R Y 2 0 = R Y 2 0 2 8 7 R Y O = R Y T R U E ( I ) 2 8 8 R Y 1 0 = 2 . * ( ( R Y T R U E ( I ) - B R Y O ) / D T ) - B R Y 1 0 2 8 9 R Y 2 0 = 2 . * ( ( R Y 1 0 - B R Y 1 0 ) / D T ) - B R Y 2 0 2 9 0 C A L L Y S T I F F ( R Y 0 , Y 0 , R Y S T E F . S 2 . Y S T E F ) 2 9 1 Y H A P = Y 0 - 4 O O . * D T 2 9 2 Y H A P P = 4 0 . * D T 2 9 3 DO 4 5 2 J J = 1 , 6 2 9 4 Y R F M I N = 1 0 0 O 0 O . 2 9 5 DO 4 5 3 J = 1 , 2 1 2 9 6 Y = Y H A P + F L 0 A T ( J - 1 ) * Y H A P P 2 9 7 C A L L Y S T I F F ( R Y 0 , Y . R Y S T I F . S 2 , Y S T I F ) 2 9 8 R E S F 0 R = A B S ( + Y S T I F / 2 . - F K S W A Y + Y S T E F / 2 . + ( V M A S S 2 * 0 . 0 0 1 / D T ) * ( 2 . * ( Y - Y O ) / 2 9 9 $ D T - 2 . * Y 1 0 ) + D A M P 2 * 0 . 0 0 1 * ( Y - Y 0 ) / D T + Y M 1 2 * ( ( B R Y 2 0 + R Y 2 0 ) / 2 . ) + Y C 1 2 * ( ( R Y 1 3 0 0 $ 0 + B R Y 1 0 ) / 2 . + Y K 1 2 * R Y 0 ) ) 1 4 0 3 0 1 I F ( R E S F O R . L E . Y R F M I N ) Y T R U E ( I ) = Y 3 0 2 I F ( R E S F O R . L E . Y R F M I N ) Y R F M I N = R E S F O R 3 0 3 4 5 3 C O N T I N U E 3 0 4 Y H A P P = Y H A P P / 1 0 . 3 0 5 Y H A P = Y T R U E ( I ) - Y H A P P * 1 0 . 3 0 6 4 5 2 C O N T I N U E 3 0 7 BYO=YO 3 0 8 B Y 1 0 = Y 1 0 3 0 9 Y 0 = Y T R U E ( I ) 3 1 0 Y 1 0 = 2 . * ( ( Y T R U E ( I ) - B Y 0 ) / D T ) - B Y 1 0 3 1 1 Y 2 0 = 2 . * ( ( Y 1 0 - B Y 1 0 ) / D T ) - Y 2 0 3 1 2 4 5 0 C O N T I N U E 3 1 3 DO 4 5 6 N J = 1 . 4 0 0 3 1 4 N I = N J 3 1 5 X T = F L 0 A T ( N I ) * D T 3 1 6 W R I T E ( 6 , 4 5 5 ) X T , Y T R U E ( N I ) . R Y T R U E ( N I ) , Y Y Y ( N I ) , Y M 0 M ( N I ) 3 1 7 4 5 6 C O N T I N U E 3 1 8 4 5 5 F O R M A T ( 5 X . F 1 5 . 3 . 6 X . 2 F 1 5 . 5 . 2 F 1 5 . 3 ) 3 1 9 4 0 1 F O R M A T ( 1 5 X . ' ( S E C ) ' . 1 4 X , ' ( M M ) ' , 1 0 X , ' ( R A D I A N S ) ' , 8 X . ' ( N E W T O N S ) ' , 6 X . ' ( 3 2 0 $ N - m m ) ' , / / ) 3 2 1 4 0 2 F O R M A T ( 1 5 X , ' T I M E ' , 1 1 X , ' D I S P L A C E M E N T ' , 8 X , ' R O L L ' . 1 1 X , ' F - K F O R C E ' , 7 X , 3 2 2 $ ' F - K M O M ' , / ) 3 2 3 4 0 3 F O R M A T ( / / / , 1 0 X . ' T H E F O L L O W I N G I S THE SWAY T I M E D E P E N D E N T R E S P O N S E : 3 2 4 $'.///) 3 2 5 R E T U R N 3 2 6 END 3 2 7 c ********************************************************************** 3 2 8 c * * 3 2 9 c * S U B R O U T I N E S T I F F * 3 3 0 c * * 3 3 1 c ********************************************************************** 3 3 2 c 3 3 3 c C a l c u l a t e s t h e s t i f f n e s s m a t r i x f o r t h e s u r g e s y s t e m . 3 3 4 c 3 3 5 3 3 6 S U B R O U T I N E S T I F F ( X , S T I F , S 1 ) 3 3 7 C 0 M M 0 N / P 5 / A . B . C , D . I X , N X 3 3 8 D I M E N S I O N A ( 1 0 ) , B ( 1 0 ) , C ( 1 0 ) , D ( 1 0 ) 3 3 9 I F ( A B S ( X ) . G T . S 1 ) G 0 TO 6 0 0 3 4 0 S T I F = 0 . 0 3 4 1 GO TO 6 0 2 3 4 2 6 0 0 S T I F = 0 . 0 3 4 3 DO 6 0 1 N = 1 , I X 3 4 4 G T N = ( C ( N ) - A ( N ) ) / X 3 4 5 I F ( G T N . L T . 0 . O ) G 0 TO 6 0 1 3 4 6 SX1 =S1 3 4 7 I F ( C ( N ) . L T . A ( N ) ) S X 1 = - S 1 3 4 8 E X T = S O R T ( ( D ( N ) - B ( N ) ) * * 2 + ( C ( N ) - A ( N ) + X ) * * 2 ) - S Q R T ( ( D ( N ) - B ( N ) ) * * 2 + ( C ( N 3 4 9 * ) - A ( N ) + S X 1 ) * * 2 ) 3 5 0 C A L L S T I F F 0 ( E X T . F 0 R C E X . N , S 1 ) 3 5 1 S T I F = S T I F + F O R C E X » ( C ( N ) - A ( N ) + X ) / ( S Q R T ( ( D ( N ) - B ( N ) ) * * 2 + ( C ( N ) - A ( N ) + X ) * 3 5 2 * * 2 ) ) 3 5 3 6 0 1 C O N T I N U E 3 5 4 6 0 2 C O N T I N U E 3 5 5 R E T U R N 3 5 6 END 3 5 7 Q ********************************************************************** 3 5 8 c * * 3 5 9 c * S U B R O U T I N E Y S T I F F * 3 6 0 c * * 141 3 6 1 c »*•***»»»»*****»«*«*******»*«»*»*****»****»»***********»«*»*****»***** 3 6 2 c 3 6 3 c C a l c u l a t e s t h e s t i f f n e s s m a t r i x f o r t h e s w a y s y s t e m . 3 6 4 3 6 5 c 3 6 6 S U B R O U T I N E Y S T I F F ( R Y , Y . R Y S . S 2 , Y S ) 3 6 7 C 0 M M 0 N / P 5 / A , B . C , D . I X , N X 3 6 8 C 0 M M 0 N / P 6 / X M A S S 2 , Z B A R , E C C , S T F R Y S , C D R A G 1 , C D R A G 2 , Z C E N , P A R E A 3 6 9 D I M E N S I O N A ( 1 0 ) , B ( 1 0 ) , C ( 1 0 ) , D ( 1 0 ) 3 7 0 S T R T = Y - E C C * T A N ( R Y ) 3 7 1 Y S = 0 . 0 3 7 2 R Y S = R Y * S T F R Y S 3 7 3 I F ( Y . G T . S 2 ) C A L L B A L 0 0 N ( Y , S 2 , Y S 1 ) 3 7 4 I F ( Y . L E . S 2 ) Y S 1 » 0 . 0 3 7 5 I F ( S T R T . E O . O . O ) Y S = 0 . 0 3 7 6 I F ( S T R T . E 0 . O . O ) G 0 TO 8 5 1 3 7 7 I F ( S T R T . L E . S 2 . A N D . S T R T . G E . - S 2 ) G 0 TO 8 5 1 3 7 8 DO 8 5 2 N = 1 , I X 3 7 9 G T N = ( D ( N ) - B ( N ) ) / S T R T 3 8 0 I F ( G T N . L T . O . O ) G O TO 8 5 2 3 8 1 S X 2 = S 2 3 8 2 I F ( D ( N ) . L T . B ( N ) ) S X 2 = - S 2 3 8 3 E X T = S 0 R T ( ( C ( N ) - A ( N ) ) * * 2 + ( D ( N ) - B ( N ) + S T R T ) * * 2 ) - S 0 R T ( ( C ( N ) - A ( N ) ) * * 2 + ( 3 8 4 $ D ( N ) - B ( N ) + S X 2 ) * * 2 ) 3 8 5 C A L L S T I F F E ( E X T , F 0 R C E Y , N , S 2 ) 3 8 6 Y S = Y S + F 0 R C E Y * ( D ( N ) - B ( N ) + S T R T ) / ( S Q R T ( ( C ( N ) - A ( N ) ) * * 2 + ( D ( N ) - B ( N ) + S T R T 3 8 7 $ ) * * 2 ) ) 3 8 8 8 5 2 C O N T I N U E 3 8 9 R Y S = - Y S * E C C + R Y * S T F R Y S 3 9 0 8 5 1 C O N T I N U E 3 9 1 Y S = Y S + Y S 1 3 9 2 R E T U R N 3 9 3 END 3 9 4 3 9 5 C * * 3 9 6 C * S U B R O U T I N E S T I F F O * 3 9 7 C * * 3 9 8 3 9 9 c 4 0 0 c C a l c u l a t e s t h e f o r c e i n e a c h m o o r a g e l i n e f r o m t h e f o r c e - d e f l e c t i o n 4 0 1 c c u r v e f o r s u r g e m o t i o n s . 4 0 2 4 0 3 c 4 0 4 S U B R O U T I N E S T I F F O ( X , S T I F , N , S 1 ) 4 0 5 C 0 M M 0 N / P 2 / D I S P . F O R C E 4 0 6 C 0 M M 0 N / P 5 / A , B , C . D . I X , N X 4 0 7 D I M E N S I O N D I S P ( 5 0 ) , F O R C E ( 5 0 ) , A ( 1 0 ) , B ( 1 0 ) , C ( 1 0 ) , D ( 1 0 ) 4 0 8 S X 1 = S 1 4 0 9 I F ( C ( N ) . L T . A ( N ) ) S X 1 = - S 1 4 1 0 X = 1 0 0 . * X / S Q R T ( ( C ( N ) - A ( N ) + S X 1 ) * * 2 + ( D ( N ) - B ( N ) ) * * 2 ) 4 1 1 N X 1 = N X - 1 4 1 2 DO 4 0 0 I = 1 ,NX 1 4 1 3 11=1+1 4 1 4 I F ( X . L T . D I S P ( I 1 ) . A N D . X . G T . D I S P ( I ) ) S T I F = F 0 R C E ( I ) + ( F 0 R C E ( I 1 ) - F 0 R C E ( I 4 1 5 * ) ) * ( X - D I S P ( I ) ) / ( D I S P ( I 1 ) - D I S P ( I ) ) 4 1 6 4 0 0 C O N T I N U E 4 1 7 R E T U R N 4 1 8 END 4 1 9 4 2 0 C * * 141 4 2 1 C » S U B R O U T I N E S T I F F E * 4 2 2 C * « 4 2 3 C ***»»***«»*»*****»*•*********»****«*****«********»«*«»»•**»**»*«»»»»*» 4 2 4 C 4 2 5 C C a l c u l a t e s t h e f o r c e i n e a c h m o o r a g e l i n e f r o m t h e f o r c e - d e f l e c t i o n 4 2 6 C c u r v e f o r s w a y m o t i o n s . 4 2 7 C 4 2 8 C 4 2 9 S U B R O U T I N E S T I F F E ( E X T , F O R C E Y . N , S 2 ) 4 3 0 C 0 M M 0 N / P 2 / D I S P . F O R C E 4 3 1 C 0 M M 0 N / P 5 / A . B , C , D , I X . N X 4 3 2 D I M E N S I O N D I S P ( 5 0 ) . F O R C E ( 5 0 ) , A ( 1 0 ) , B ( 1 0 ) , C ( 1 0 ) . D ( 1 0 ) 4 3 3 S X 2 = S 2 4 3 4 I F ( D ( N ) . L T . B ( N ) ) S X 2 = - S 2 4 3 5 Y = 1 0 0 . * E X T / S Q R T ( ( D ( N ) - B ( N ) + S X 2 ) * * 2 + ( C ( N ) - A ( N ) ) * * 2 ) 4 3 6 N X 1 = N X - 1 4 3 7 DO 8 7 5 I = 1 , N X 1 4 3 8 11=1+1 4 3 9 I F ( Y . L T . D I S P ( I 1 ) . A N D . Y . G T . D I S P ( I ) ) F O R C E Y = F O R C E ( I ) + ( F O R C E ( I 1 ) - F O R C E 4 4 0 $ ( I ) ) * ( Y - D I S P ( I ) ) / ( D I S P ( I 1 ) - D I S P ( I ) ) 4 4 1 8 7 5 C O N T I N U E 4 4 2 R E T U R N 4 4 3 END 4 4 4 C * * * * * * » * * * * * * * * * * * * * * * * * * » * » * * * * * * * * * * * * * * * * * * * * * * * » * * * * * * » * . * « * * * * » « * 4 4 5 C * * 4 4 6 C * S U B R O U T I N E B A L O O N * 4 4 7 C * * 4 4 8 C ********************************************************************** 4 4 9 C 4 5 0 C C a l c u l a t e s t h e f o r c e s e x e r t e d o n t h e v e s s e l b y t h e b a l l o o n f e n d e r s . 4 5 1 C 4 5 2 C 4 5 3 S U B R O U T I N E B A L O O N ( Y 1 , S 2 , Y S ) 4 5 4 C 0 M M 0 N / P 7 / B D I S P , B F O R C E , N B A L . N P O I T 4 5 5 D I M E N S I O N B D I S P ( 5 0 ) , B F O R C E C 5 0 ) 4 5 6 Y = A B S ( Y 1 ) - S 2 4 5 7 N P 0 I T 1 = N P 0 I T - 1 4 5 8 DO 8 7 4 J I = 1 . N P 0 I T 1 4 5 9 J I 1 = J I + 1 4 6 0 I F ( Y . L T . B D I S P ( J I 1 ) . A N D . Y . G T . B D I S P ( d l ) ) Y S = + F L O A T ( N B A L ) * ( B F O R C E ( J I ) + 4 6 1 $ ( B F O R C E ( d I 1 ) - B F O R C E ( d l ) ) * ( Y - B D I S P ( J I ) ) / ( B D I S P ( J I 1 ) - B D I S P ( d l ) ) ) 4 6 2 8 7 4 C O N T I N U E 4 6 3 R E T U R N 4 6 4 END 4 6 5 C *»»*********»«»*«*********»»******«***».»***********************»»«.»» 4 6 6 C * * 4 6 7 C * S U B R O U T I N E FROKRY * 4 6 8 C * * 4 6 9 C »******»*****»«****»***»»****»«*»*******«***»*«*»»»**********»**.***»* 4 7 0 C 4 7 1 C C a l c u l a t e s t h e s u r g e e x c i t i n g f o r c e f r o m f r o u d e - k r y l o v f o r c e s . 4 7 2 C 4 7 3 C 4 7 4 S U B R O U T I N E F R O K R Y ( T , F K F O R ) 4 7 5 C O M M O N / P 1 / H E I G H T . P E R O I D , R E F C O , P O S I T , D E P T H . A L E N T H 4 7 6 C 0 M M 0 N / P 4 / W A T P L B , W A T P L L , D R A F T , W K E E L . H K E E L . X L K E E L . S L 4 7 7 C O M M O N / P 6 / X M A S S 2 . Z B A R . E C C . S T F R Y S , C D R A G 1 , C D R A G 2 , Z C E N , P A R E A 4 7 8 X L = 6 . 2 8 3 2 / A L E N T H 4 7 9 R H 0 G = 9 . 8 1 E - 6 4 8 0 0MEGA«6.2B32/PER0ID 143 4 8 1 N = 1 0 4 8 2 F K E E L = 0 . 0 4 8 3 F H U L L = 0 . 0 4 8 4 0 0 7 4 9 1 = 1 , N 4 8 5 A N = F L 0 A T ( N ) 4 8 6 A I » F L 0 A T ( I ) 4 8 7 A H U L L = W A T P L B * ( 1 . - ( 2 . * A I - 1 . ) / ( 2 . * A N ) ) * D R A F T / A N 4 8 8 Z E D = D R A F T * ( ( 2 . * A I - 1 . ) / ( 2 . * A N ) ) 4 8 9 XB = - ( ( W A T P L L / 2 . ) - ( Z E D / D R A F T ) * S L ) 4 9 0 7 5 1 X = X B + P 0 S I T 4 9 1 X 1 = - X B + P 0 S I T 4 9 2 P R E S S = O . 5 * R H 0 G * H E I G H T * ( C 0 S H ( X L * ( D E P T H - Z E D ) ) / C 0 S H ( X L * D E P T H ) ) * ( C 0 S ( X 4 9 3 * t _ * X - O M E G A * T ) + R E F C 0 * 0 . 0 1 * C O S ( X L * X + O M E G A * T ) - C O S ( X L * X 1 - O M E G A * T ) - R E F C O 4 9 4 »*0.01*COS(XL*X1+OMEGA*T)) 4 9 5 F H U L L = F H U L L + P R E S S * A H U L L 4 9 6 A K E E L = ( ( ( 2 . * A I - 1 . ) / ( 2 . * A N ) ) * W K E E L ) * H K E E L / A N 4 9 7 Z E D K = - ( D R A F T + ( ( 2 . * A I - 1 . ) / ( 2 . * A N ) ) * H K E E L ) 4 9 8 X K = X L K E E L / 2 . 4 9 9 7 5 2 X = X K + P O S I T 5 0 0 X 1 = - X K + P 0 S I T 5 0 1 PRESS=O.5«RHOG*HEIGHT*(C0SH(XL*(DEPTH+ZEDK))/C0SH(XL*DEPTH))*(COS( 5 0 2 * X L * X 1 - O M E G A * T ) + R E F C O * 0 . 0 1 * C O S ( X L * X 1 + O M E G A * T ) - C O S ( X L * X - O M E G A * T ) - R E F 5 0 3 * C 0 * O . O 1 * C 0 S ( X L * X + 0 M E G A * T ) ) 5 0 4 F K E E L = F K E E L + P R E S S * A K E E L 5 0 5 7 4 9 C O N T I N U E 5 0 6 X K S = X L * ( D E P T H - Z B A R ) 5 0 7 X K D = X L * D E P T H 5 0 8 V E L = ( 3 . 1 4 1 6 * H E I G H T / P E R 0 I D ) * ( C 0 S H ( X K S ) / S I N H ( X K D ) ) * ( C 0 S ( X L * P 0 S I T - 0 M E 5 0 9 $ G A * T ) - O . 0 1 * R E F C 0 * C 0 S ( X L * P 0 S I T + 0 M E G A * T ) ) 5 1 0 A R E A = 0 . 5 * ( D R A F T * W A T P L B + H K E E L * W K E E L ) 51 1 V F 0 R C E = 5 O O . E - 1 2 * C D R A G 1 * A R E A * A B S ( V E L ) * V E L 5 1 2 F K F O R = F K E E L + F H U L L + V F O R C E 5 1 3 R E T U R N 5 1 4 END 5 1 5 5 1 6 C * * 5 1 7 C * S U B R O U T I N E FKSWAA * 5 1 8 C * * 5 1 9 5 2 0 C 5 2 1 C C a l c u l a t e s t h e s w a y e x c i t i n g f o r c e f r o m f r o u d e - k r y l o v f o r c e s . 5 2 2 5 2 3 C 5 2 4 S U B R O U T I N E F K S W A A ( T , F K S W A Y , F K M O M ) 5 2 5 C O M M O N / P 1 / H E I G H T . P E R O I D , R E F C O , P O S I T . D E P T H , A L E N T H 5 2 6 C O M M O N / P 6 / X M A S S 2 , Z B A R , E C C . S T F R Y S . C D R A G 1 , C D R A G 2 . Z C E N , P A R E A 5 2 7 X K R = R E F C 0 * 0 . 0 1 5 2 8 0 M E G A = 6 . 2 8 3 2 / P E R O I D 5 2 9 X K = 6 . 2 8 3 2 / A L E N T H 5 3 0 X K S = X K * ( D E P T H - Z B A R ) 5 3 1 XKD=XK»DEPTH 5 3 2 V E L = ( 3 . 1 4 1 6 * H E I G H T / P E R 0 I D ) * ( C 0 S H ( X K S ) / S I N H ( X K D ) ) * ( C 0 S ( X K * P 0 S I T - 0 M E 5 3 3 $ G A * T ) - X K R * C O S ( X K * P O S I T + O M E G A * T ) ) 5 3 4 F K S D R A = 5 0 0 . E - 1 2 * C D R A G 2 * P A R E A * A B S ( V E L ) * V E L 5 3 5 F K S I N E = X M A S S 2 * ( ( 1 9 . 7 3 9 * H E I G H T / P E R 0 I D * * 2 ) * C O S H ( X K S ) / S I N H ( X K D ) ) * ( S I N 5 3 6 $ ( X K * P O S I T - O M E G A * T ) + X K R * S I N ( X K * P O S I T + O M E G A * T ) ) * 0 . 0 0 1 5 3 7 F K S W A Y = F K S I N E + F K S D R A 5 3 8 F K M O M = F K S I N E * Z B A R + F K S D R A * Z C E N 5 3 9 R E T U R N 5 4 0 END APPENDIX I I EXPERIMENTATION DATA 145 The f o l l o w i n g i s an explanation of the data which appears on the f o l l o w i n g t a b l e : Heading Explan a t i o n Coding Un i t s Sea D i r e c t D i r e c t i o n of wave propogation r e l a t i v e to the v e s s e l 1- head seas 2- beam seas N/A Moorage Condition Manner by which the v e s s e l i s secured See f i g . 3 N/A Fender C o n d i t i o n Whether or not a b a l l o o n fender i s i n s t a l l e d 1- not i n c l . 2- i n c l . N/A Slack Amount of f r e e movement seaward or leeward from a ce n t e r i n g p o s i t i o n N/A mm L i n e S t i f f n e s s R a t i o of s t i f f l i n e l e n g t h to e l a s t i c l i n e length i n the model moorage l i n e s N/A N/A Per i o d I n c i d e n t wave period N/A sec (T**2)G L Dimensionless period = —^ s- N/A D/L P o s i t i o n P o s i t i o n i n the wave b a s i n at which the t r i a l was performed 0.0 - not recorded f e e t R e f l e c t Coef R e f l e c t i o n c o e f f i c i e n t f o r the wave t r a i n N/A percent Wave He i g h t Wave height recorded at the p o s i t i o n of the v e s s e l N/A mm Incident Height Incident wave height of the wave t r a i n N/A mm Maximum Motion Maximum peak to peak h o r i z o n t a l motion of the v e s s e l i n the d i r e c t i o n of the i n c i d e n t waves N/A mm RAO Response Amplitude Operator maximum motion T i n . height N/A D/L 4 9 e TRIAL SEA MOORAGE FENDER SLACK LINE 7 DIRECT CONDITION CONDITION STIFFNESS 8 9 1 1 , , 9 25 9 00 10 2 1 1 1 9 25 9 00 11 3 1 1 1 9 25 9 00 12 4 1 t 1 9 25 9 00 13 5 1 f 1 9 25 9 00 14 6 1 1 1 9 25 9 00 15 7 1 1 1 9 25 9 00 16 a 1 1 1 25 50 9 00 17 9 1 1 1 25 50 9 00 IS 10 1 1 1 25 50 9 00 19 11 1 1 1 25 50 9 00 20 12 1 1 1 25 50 9 00 21 13 1 1 1 25 50 9 00 22 14 1 1 1 25 50 9 00 23 15 1 1 1 25 50 9 00 24 16 1 1 1 25 50 9 00 25 17 1 1 1 25 50 9 00 26 18 1 1 1 25 50 9 00 27 19 1 1 1 25 50 9 00 28 20 1 1 1 15 00 9 00 29 21 1 1 1 15 00 9 00 30 22 1 1 1 15 00 9 00 31 23 1 1 1 15 00 9 00 32 24 1 1 1 15 00 9 00 33 25 1 1 1 15 00 9 00 34 26 1 1 1 15 00 9 00 39 27 1 1 1 15 00 9 00 36 28 1 1 1 15 00 9 00 37 29 1 1 1 15 00 9 00 3B 30 1 1 1 15 OO 9 00 39 31 1 1 1 15 00 9 00 40 32 1 1 1 15 00 9 00 41 33 1 1 1 15 00 9 00 42 34 1 1 1 15 00 9 00 43 35 1 1 1 15 00 9 00 44 36 1 1 1 15 CO 9 00 45 37 1 1 1 15 00 9 00 46 38 1 1 1 10 oo 4 00 47 39 1 1 1 10 00 4 00 48 40 1 1 1 10 00 4 00 49 41 1 1 1 10 00 4 00 90 42 1 1 1 10 00 4 00 51 43 1 1 1 10 00 4 00 52 44 1 1 1 10 00 4 00 53 45 1 1 1 10 oo 4 oo 54 46 1 1 1 13 oo 9 00 55 47 1 1 1 13 00 9 00 56 48 1 1 1 13 00 9 00 57 49 1 1 1 13 00 9 00 58 50 1 1 1 13 00 9 00 59 51 1 1 1 13 oo 9 .00 60 52 1 1 1 1 1 00 9 .00 PER01DJ<T'*2)G' P O S I T I O N R E F L E C T WAVE 2.40 2.35 2.00 1 .64 1 . 33 0.92 0.62 2.50 2.50 2 . 35 2 .00 1 .64 1 . 20 00 00 00 00 00 00 40 20 00 00 00 00 00 80 2. 2. 2. 2. 2 . 2. 2 . 2. 2. 2. 2. 2. 2. 1 . 1 .64 1 .43 1 .43 1 .43 1 .43 1 .43 1 . 25 1 . 13 1 .02 0.94 2.40 1 .66 1 .40 1 .20 1 .03 1 .08 1 . 10 0.94 1 . 28 1 .25 1 .25 1 . 25 1 . 25 1 .25 1 .64 WATPLL 9.12 8.93 7 .60 6.23 5.05 3.49 2.35 9.50 9.50 8.93 7 6 4 7 7 7 7 COEF HEIGHT INCIDENT MAXIMUM HEIGHT MOTION 60 23 56 60 60 60 60 7 .60 7 .60 12 36 60 60 60 60 60 84 23 43 43 43 43 43 75 29 87 3.57 9. 12 31 32 56 91 10 18 57 86 75 75 75 75 75 23 83.50 84.00 82.50 83.00 83.00 83.00 83.00 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 81 00 83.00 80.00 80.00 80. OO 80.00 80.00 83.00 81 00 81 .00 81 .00 81 .00 81 .00 81 .00 82.50 82.50 82.50 82.50 0.0 O 0 O 0 O 0 O 81 0 O O O O O 0 50 43.67 42.50 35.80 20.71 B.44 0.0 0.0 46.00 46.00 42.50 35.80 20.7 1 5.55 35.80 35.80 35.80 35.80 35.80 35.80 43.67 39.00 35.80 35.80 35.80 35.80 35.80 29.00 20.7 1 11 .68 I 1 .68 II .68 11 .68 11 .68 43.67 22. 14 10.71 5.55 8 4.00 8 4 .00 8 4 O O 8 4 O O 8 4.00 8 4.50 20.00 16.50 27 .00 16 .00 32 .00 33 .00 19.00 13.00 15.00 19.00 15.00 14 .00 29.00 38.00 27.00 14 .00 19.00 1 1 .00 6.00 16.00 20.00 19.00 27 .00 13.00 7.00 5.00 20.00 24 .00 20.00 28 .00 33.00 15.00 10.00 24 .00 40.00 25.00 27 .00 16.00 24 .00 40.00 53.00 68.00 41 .00 26.00 37 O O 10.OO 16.00 29 O O 22 .00 37 00 45. O O 9.00 28.73 23.51 36.67 19.31 34.70 33.00 19.00 18 .98 21 .90 27.07 20.37 16.90 30.61 51 .60 36.67 19.01 25.80 14 .94 8. 15 22.99 27.80 25.80 36.67 17.65 9.51 6.79 25.80 28.97 22.34 31 .27 36.85 16.75 11 . 17 25.49 41 .62 25. 16 27.00 22.99 29.31 44.28 55.94 68.64 42.02 26.81 37 .00 10.68 17 .00 30.81 23 . 37 39 . 30 47 .80 10. 86 25. O O 31 .50 27.50 19.00 27.00 0.0 0.0 62.00 76.00 69.00 60.00 50.00 8 1 .00 64 .00 61 .00 50.00 57 .00 52 .00 50.00 38.00 39.00 37.00 41 .00 34.00 31 .50 28.50 34 .50 34.50 35.00 41 .50 42.50 33.50 30.00 39.50 40.50 0.0 0.0 20.00 20.00 20.00 27 . 50 40.00 25.50 22.00 25.OO 26.50 31 O O 33.OO 32.50 42 C O 45.OO 17 . 50 R.A.O. 0.870 1 340 0.750 0.984 O 778 0.0 0.0 3. 3. 2 . 2 . 2 . 2 . .267 .470 548 946 959 646 1 .240 1 .664 2.630 2.209 3.481 6. 136 1 .653 1 .403 1 .434 1 . 1 18 1 .926 3.314 4 . 197 1 . 337 1 . 191 1 .567 1 .327 1 . 153 2 C O O 2 .686 1 .549 0.973 0.0 0.0 0.870 0.682 0.452 0.492 0.583 0.607 O.B21 0.676 2 .481 1 .824 1 .07 1 1 .391 1 .069 0.94 1 1.611 o> 61 53 1 1 1 1 1 00 9 00 1 62 54 1 1 1 1 1 00 9 00 1 63 55 1 1 1 1 1 00 9 00 1 64 56 1 1 1 1 1 00 9 00 1 65 57 1 1 1 1 1 oo 9 00 1 66 58 1 1 1 1 1 00 9 oo 2 67 59 1 1 1 1 1 00 9 oo 2 68 60 1 1 1 1 1 00 9 00 2 69 61 1 1 1 12 50 9 oo 1 70 62 1 1 1 5 oo 9 00 1 71 63 1 1 1 30 00 9 00 1 72 64 1 1 1 26 oo 9 00 1 73 65 1 1 1 . 23 00 9 00 1 74 66 1 1 1 19 00 9 00 1 75 67 1 1 1 19 00 9 00 1 76 68 1 1 1 1 1 50 9 oo 1 77 69 1 1 1 7 50 9 00 1 78 70 1 1 1 0 0 9 00 1 79 71 1 1 1 21 50 9 00 1 80 72 1 1 1 33 00 9 00 1 81 73 1 1 1 / 26 OO 9 00 1 82 74 1 1 l' 30 50 9 00 1 83 75 1 1 1 17 00 9 00 1 84 76 1 1 1 17 00 9 00 0 as 77 1 1 1 17 00 9 00 1 86 78 1 1 1 17 00 9 00 0 87 79 1 1 1 17 00 9 00 0 88 80 1 1 1 17 00 9 00 0 89 81 1 1 1 17 00 9 00 0 SO 82 1 1 1 17 00 9 00 1 SI 83 1 1 1 17 00 9 00 1 92 84 1 1 1 17 00 9 00 1 83 85 1 1 1 17 00 9 00 1 84 86 1 1 1 17 00 9 00 2 95 87 1 1 1 17 00 9 00 2 96 88 1 1 1 17 00 9 00 2 97 89 1 1 1 17 00 9 oo 2 98 90 1 1 1 17 00 9 00 2 99 91 1 1 1 17 00 9 00 2 100 92 1 1 1 17 00 9 00 2 101 93 1 1 1 17 00 9 00 2 102 94 1 1 1 17 00 9 00 2 103 95 1 1 1 17 00 9 00 2 104 96 1 1 1 17 00 9 00 2 105 97 1 1 1 17 00 9 00 2 106 98 1 1 1 17 00 9 00 2 107 99 1 1 1 21 00 6 00 2 108 100 1 1 1 21 00 6 00 1 109 101 1 1 1 21 00 6 00 1 1 10 102 1 1 1 21 00 6 00 1 1 1 1 103 1 1 1 21 00 6 00 1 112 104 1 1 1 23 00 16 00 1 113 105 1 1 1 23 00 16 00 1 114 106 1 1 1 23 00 16 00 1 1 15 107 1 1 1 23 00 16 00 2 1 16 108 1 1 1 23 00 16 00 0 1 17 109 1 2 2 7 50 9 00 2 1 18 1 10 1 2 2 7 50 9 oo 2 1 19 111 1 2 2 7 so 9 00 1 120 1 12 1 2 2 7 50 9 00 1 64 6 23 84 50 20 71 15 00 18 1 1 23 00 1 270 64 6 23 84 50 20 7 1 21 00 25 35 24 50 0 966 64 6 23 84 50 20 7 1 30 CO 36 21 33 50 0 925 64 6 23 84 50 20 7 1 4 1 00 49 49 38 00 0 768 64 6 23 84 50 20 71 50 CO 60 36 42 OO 0 696 40 9 12 83 00 43 67 15 00 2 1 55 30 50 1 415 40 9 12 83 00 43 67 10 00 14 37 27 50 1 914 40 9 12 83 00 43 67 7 50 10 77 23 50 2 181 64 6 23 83 00 20 71 24 00 28 97 32 50 1 122 64 6 23 83 OO 20 71 24 00 28 97 7 00 0 242 64 6 23 83 OO 20 71 24 00 28 97 69 OO 2 382 64 6 23 83 00 20 71 24 OO 28 97 57 00 1 967 64 6 23 83 00 20 71 24 OO 28 97 54 00 1 864 64 6 23 83 00 20 7 1 24 OO 28 97 46 50 1 605 13 4 29 83 00 4 06 27 OO 28 10 71 00 2 527 13 4 29 83 00 4 06 27 00 28 10 37 50 1 335 13 4 29 83 00 4 06 27 00 28 10 24 50 0 872 13 4 29 83 00 4 06 27 oo 28 10 2 00 0 071 13 4 29 83 00 4 06 27 00 28 10 77 50 2 7S8 13 4 29 83 00 4 06 27 00 28 10 37 50 1 335 13 4 29 83 00 4 06 27 oo 28 10 84 50 3 007 13 4 29 83 00 4 06 27 oo 28 10 81 OO 2 883 13 4 29 83 00 4 06 31 oo 32 26 56 50 1 751 91 3 46 83 00 0 0 30 00 30 00 33 00 1 100 02 3 87 83 00 0 63 33 oo 33 21 56 50 1 701 98 3 72 83 00 0 0 30 oo 30 OO 58 00 1 933 95 3 61 83 00 0 0 29 oo 29 00 64 50 2 224 89 3 38 83 00 0 0 32 00 32 00 65 50 2 047 84 3 19 83 00 0 0 32 oo 32 00 41 50 1 297 29 4 90 83 00 7 15 29 oo 31 07 56 50 1 818 43 5 43 83 00 1 1 68 28 00 31 27 56 50 1 807 63 6 19 83 00 20 00 24 00 28 80 48 40 1 681 88 7 14 83 00 32 20 23 00 30 4 1 51 50 1 694 08 7 90 83 00 37 08 20 00 27 42 53 00 1 933 44 9 27 83 00 44 60 16 oo 23 14 47 OO 2 031 30 8 74 84 50 41 33 21 00 29 68 52 00 1 752 00 7 60 81 50 35 80 29 oo 40 00 50 00 1 250 00 7 60 82 so 35 80 28 oo 40 OO 55 00 1 375 00 7 60 83 50 35 80 39 00 40 00 57 00 1 425 00 7 60 84 so 35 80 46 00 40 oo 50 00 1 250 00 7 60 85 50 35 80 52 oo 40 00 41 00 1 025 00 7 60 86 SO 35 80 46 oo 40 00 44 00 1 100 00 7 60 87 50 35 80 36 oo 40 00 52 50 1 313 00 7 60 88 50 35 80 30 00 40 00 59 OO 1 475 00 7 60 89 50 35 80 37 oo 40 oo 50 50 1 262 00 7 60 83 00 35 80 30 oo 40 00 57 00 1 425 12 8 05 86 00 37 72 22 00 30 30 66 OO 2 178 64 6 23 86 00 20 71 27 00 32 59 64 00 1 964 47 5 58 84 00 13 26 25 00 28 32 82 00 2 896 27 4 82 84 00 6 50 27 oo 28 75 86 00 2 991 14 4 33 84 00 4 38 30 00 31 31 24 00 0 766 27 4 82 84 OO 6 50 27 oo 28 75 7 1 00 2 469 47 5 58 84 00 13 26 25 00 28 32 73 00 2 578 64 6 23 83 00 20 71 22 00 26 56 61 50 2 316 12 8 05 86 00 37 72 22 00 30 30 68 50 2 261 92 3 49 82 50 0 0 27 00 27 00 23 00 0 852 35 8 93 91 00 42 50 18 00 25 65 41 00 1 598 OO 7 60 8B 00 35 80 23 00 31 23 4 1 00 1 313 68 6 38 87 oo 23 57 25 00 30 89 27 00 0 874 43 5 43 87 00 1 1 68 26 00 29 04 48 00 1 653 121 1 13 1 2 2 7 50 9 00 1 122 1 14 1 2 2 7 50 9 00 0 123 1 15 1 2 2 7 50 9 00 1 124 1 16 1 2 1 7 50 9 00 0 125 1 17 1 2 1 7 50 9 00 1 126 1 18 1 2 1 7 50 9 00 1 127 1 19 1 2 1 7 50 9 OO 1 126 120 1 2 1 7 50 9 00 2 129 121 1 2 1 7 50 9 00 2 130 122 1 5 2 5 00 9 00 2 131 123 1 5 2 5 00 9 00 2 132 124 1 5 2 5 00 9 00 1 133 125 1 5 2 5 00 9 00 1 134 126 1 5 2 5 00 9 00 1 135 127 1 5 2 5 00 9 00 o 136 128 1 7 2 5 00 9 00 0 137 129 1 7 2 5 00 9 00 1 138 130 1 7 2 5 00 9 00 1 139 131 1 7 2 5 00 9 00 1 140 132 1 7 2 5 00 9 00 2 141 133 1 7 2 5 00 9 00 2 142 134 1 10 2 5 00 9 00 2 143 135 1 10 2 5 00 9 00 2 144 n 6 1 10 2 5 00 9 00 1 145 137 1 10 2 5 00 9 00 1 146 138 1 10 2 5 00 9 00 1 147 139 1 10 2 5 00 9 00 0 148 140 1 6 2 5 00 9 00 0 149 141 1 6 2 5 00 9 00 1 1SO 142 1 6 2 5 00 9 00 1 151 143 1 6 2 5 00 9 00 1 152 144 1 6 2 5 00 9 00 2 153 145 1 6 2 5 00 9 00 2 154 146 1 1 1 22 00 9 00 2 155 147 1 1 1 22 00 9 00 1 196 148 1 1 1 22 00 9 00 1 157 149 1 1 1 22 00 9 00 1 158 150 1 1 1 22 00 9 00 1 159 151 2 2 2 10 00 9 00 1 160 152 2 2 2 10 00 9 00 1 161 153 2 2 2 10 00 9 00 1 162 154 2 2 2 10 00 9 00 1 163 155 2 2 2 10 00 9 00 1 164 156 2 2 2 10 00 9 00 1 169 157 2 2 2 10 00 9 00 1 166 158 2 2 2 10 00 9 00 2 167 159 2 2 2 10 00 9 00 2 168 160 2 2 2 10 00 9 00 2 169 161 2 2 2 10 00 9 00 2 170 162 2 2 2 10 00 9 00 2 171 163 2 2 2 10 00 9 00 2 172 164 2 2 2 3 50 9 00 2 173 165 2 2 2 18 00 9 00 2 174 166 2 2 2 28 00 9 00 2 179 168 2 2 2 3 50 9 00 1 176 169 2 2 2 18 oo 9 00 1 177 170 2 2 2 28 00 9 00 1 178 172 2 2 2 14 oo 9 00 1 179 173 2 2 2 14 00 9 00 1 180 174 2 2 2 14 00 9 00 1 13 4 29 87 00 4 06 28 00 29 14 60 00 2 059 92 3 49 87 00 0 0 32 00 32 00 37 50 1 172 03 3 91 87 00 0 94 37 00 37 35 86 00 2 303 92 3 49 87 00 0 0 32 00 32 00 30 00 0 938 13 4 29 87 00 4 06 32 00 33 30 56 00 1 682 43 5 43 87 00 1 1 68 25 00 27 92 37 OO 1 325 68 6 38 87 00 23 57 25 00 30 89 30 00 0 97 1 00 7 60 88 00 35 80 24 00 32 59 34 00 1 043 35 8 93 91 00 42 50 18 00 25 65 37 00 1 442 35 8 93 91 00 42 50 18 00 25 65 12 00 0 468 00 7 60 88 00 35 80 24 00 32 59 12 00 0 368 68 6 38 87 00 23 57 25 00 30 89 15 00 0 486 43 5 43 87 00 1 1 68 25 00 27 92 15 OO 0 537 13 4 29 87 00 4 06 32 00 33 30 27 00 0 811 92 3 49 87 00 0 0 32 00 32 00 30 00 0 938 92 3 49 87 oo 0 0 32 00 32 OO 2 1 OO 0 656 13 4 29 87 00 4 06 32 00 33 30 22 00 0 661 43 5 43 87 00 1 1 68 25 00 27 92 19 00 0 681 68 6 38 87 00 23 57 25 00 30 89 17 00 0 550 OO 7 60 88 00 35 80 24 00 32 59 17 00 0 522 35 8 93 91 oo 42 50 18 00 25 65 19 00 0 74 1 35 8 93 91 00 42 50 18 00 25 65 12 00 0 468 00 7 60 88 00 35 80 24 00 32 59 11 00 0 338 68 6 38 87 00 23 57 25 00 30 89 12 00 0 388 43 5 43 87 00 1 1 68 25 00 27 92 16 00 0 573 13 4 29 87 00 4 06 32 00 33 30 16 00 0 480 92 3 49 87 00 0 0 32 00 32 00 24 00 0 750 92 3 49 87 00 0 0 32 00 32 00 13 00 0 406 13 4 29 87 00 4 06 32 00 33 30 10 00 0 300 43 5 43 87 00 1 1 68 25 00 27 92 8 00 0 287 68 6 38 87 00 23 57 25 00 30 89 7 00 0 227 00 7 60 88 00 35 80 24 00 32 59 7 00 0 215 35 8 93 91 00 42 50 18 00 25 65 6 00 0 234 07 7 86 86 50 36 92 23 00 31 49 64 50 2 048 64 6 23 82 50 20 7 1 24 00 28 97 54 50 1 881 43 5 43 84 00 11 68 28 00 31 27 70 50 2 255 30 4 94 84 00 7 47 28 00 30 09 73 50 2 443 13 4 29 84 00 4 06 29 00 30 18 80 00 2 651 13 4 29 86 00 4 06 5 00 5 20 0 0 0 0 13 4 29 86 00 4 06 10 00 10 41 17 50 1 682 13 4 29 86 00 4 06 16 00 16 65 21 00 1 261 13 4 29 86 00 4 06 22 00 22 89 24 00 1 048 13 4 29 86 00 4 06 25 00 26 02 32 00 1 230 13 4 29 86 00 4 06 30 00 31 22 19 00 0 609 13 4 29 86 00 4 06 35 00 36 42 32 00 0 879 00 7 60 87 80 35 80 7 00 9 51 32 00 3 366 00 7 60 87 80 35 80 10 00 13 58 40 00 2 946 00 7 60 87 60 35 80 15 00 20 37 48 50 2 381 OO 7 60 87 80 35 BO 19 00 25 80 57 00 2 209 00 7 60 87 80 35 80 23 00 31 23 63 00 2 017 00 7 60 87 80 35 80 30 00 40 74 73 50 1 804 00 7 60 87 80 35 80 23 00 31 23 23 OO 0 736 OO 7 60 87 80 35 80 23 00 31 23 67 00 2 145 00 7 60 87 80 35 80 23 00 31 23 78 oo 2 497 13 4 29 87 80 4 06 30 00 31 22 23 50 0 753 13 4 29 87 80 4 06 30 00 31 22 53 50 1 7 14 13 4 29 87 80 4 06 30 00 31 22 87 50 2 803 93 7 33 87 80 34 20 23 00 30 87 77 00 2 495 68 6 38 86 00 23 57 26 00 32 13 65 00 2 023 43 5 43 85 50 1 1 68 28 oo 3 1 27 63 00 2 015 181 175 2 2 2 14 00 9 00 1 15 4 37 87 80 4 69 28 00 29 31 66 00 2 252 182 176 2 2 2 14 OO 9 00 0 86 3 27 85 50 0 0 30 00 30 00 83 00 2 767 183 177 2 2 2 14 00 9 OO 0 63 2 39 85 SO 0 0 30 00 30 00 53 00 1 767 184 178 2 2 2 14 00 16 00 0 63 2 39 85 50 0 0 30 00 30 OO 56 OO 1 867 IBS 179 2 2 2 14 00 16 00 0 88 3 34 85 50 0 0 30 00 30 00 92 00 3 067 186 180 2 2 2 14 00 16 00 1 15 4 37 B7 80 4 69 28 00 29 31 59 OO 2 013 187 181 2 2 2 14 00 16 00 1 44 5 47 85 50 12 OO 28 00 31 36 45 00 1 435 188 182 2 2 2 14 00 16 oo 1 68 6 38 86 00 23 57 26 OO 32 13 66 OO 2 054 189 183 2 2 2 14 00 16 00 1 93 7 33 B7 80 34 20 23 OO 30 87 7 1 OO 2 300 190 184 2 2 2 15 00 5 00 1 93 7 33 87 80 34 20 23 00 30 87 74 OO 2 397 191 1B5 2 2 2 15 00 5 00 1 68 6 38 86 00 23 S7 26 00 32 13 75 00 2 334 192 186 2 2 2 15 00 5 00 1 44 5 47 85 50 12 00 28 00 31 36 53 00 1 690 193 187 2 2 2 15 00 5 00 1 15 4 37 87 80 4 69 28 00 29 31 66 00 2 252 194 188 2 2 2 15 00 5 oo 0 88 3 34 85 50 0 0 30 00 30 00 108 00 3 600 199 189 2 2 2 15 00 5 00 0 63 2 39 85 50 0 0 30 00 30 00 53 oo 1 767 196 190 2 1 2 15 00 9 oo 0 63 2 39 85 SO 0 0 30 00 30 00 33 oo 1 100 197 191 2 1 2 15 00 9 00 0 88 3 34 as 50 0 0 30 oo 30 00 70 oo 2 333 198 192 2 t 2 15 00 9 00 1 15 4 37 87 80 4 69 28 oo 29 31 25 00 0 853 199 193 2 1 2 15 00 9 00 1 44 5 47 85 50 12 00 28 00 31 36 25 00 0 797 200 194 2 1 2 15 00 9 00 1 68 6 38 86 00 23 57 26 00 32 13 54 00 1 681 201 195 2 1 2 15 00 9 00 1 93 7 33 87 80 34 20 23 00 30 87 64 00 2 073 202 196 2 5 2 14 00 9 00 1 93 7 33 87 80 n 20 23 00 30 87 64 oo 2 073 203 197 2 5 2 14 00 9 00 1 68 6 38 86 00 •j7 26 00 32 13 69 00 2 148 204 198 2 5 2 14 oo 9 00 1 44 5 47 85 50 12 00 28 oo 31 36 43 00 1 371 205 199 2 5 2 14 00 9 00 1 15 4 37 87 80 4 69 28 00 29 31 48 00 1 638 206 200 2 5 2 14 00 9 00 0 88 3 34 85 50 0 0 30 00 30 00 135 00 4 5O0 207 201 2 5 2 14 00 9 00 0 63 2 39 85 50 0 0 30 00 30 00 20 oo 0 667 208 202 2 6 2 14 00 8 00 0 63 2 39 85 50 0 0 30 oo 30 00 36 00 1 200 209 203 2 6 2 14 oo 9 00 0 88 3 34 85 50 0 0 30 00 30 OO 91 oo 3 033 210 204 2 6 2 14 00 9 00 1 15 4 37 87 80 4 69 28 00 29 31 73 oo 2 490 211 205 2 6 2 14 00 9 00 1 44 5 47 83 50 12 OO 28 00 31 36 54 00 1 722 212 206 2 6 2 14 00 9 00 1 68 6 38 86 00 23 57 26 00 32 13 74 oo 2 303 213 207 2 6 2 14 00 9 00 1 93 7 33 87 80 34 20 23 00 30 87 84 00 2 721 214 208 2 10 2 15 00 9 00 1 93 7 33 87 80 34 20 23 00 30 87 71 oo 2 300 215 209 2 10 2 15 00 9 00 1 68 6 38 86 00 23 57 26 00 32 13 70 00 2 179 216 210 2 10 2 15 00 9 00 1 44 5 47 85 50 12 00 28 00 31 36 47 00 1 499 217 211 2 10 2 15 00 9 00 1 15 4 37 87 80 4 69 28 00 29 31 51 00 1 740 218 212 2 10 2 15 00 9 00 0 88 3 34 85 50 0 0 30 oo 30 00 72 oo 2 400 219 213 2 10 2 15 00 9 00 0 63 2 39 as 50 0 0 30 oo 30 00 13 50 0 450 220 214 2 7 2 14 00 9 00 0 63 2 39 85 50 0 0 30 oo 30 OO 35 oo 1 167 221 115 2 7 2 14 00 9 00 0 8B 3 34 85 50 0 0 30 00 30 00 64 oo 2 133 222 216 2 7 2 14 00 9 00 1 15 4 37 87 80 4 69 28 oo 29 31 46 00 1 569 223 217 2 7 2 14 00 9 00 1 44 5 47 85 SO 12 00 28 00 31 36 57 oo 1 818 224 218 2 7 2 14 00 9 00 1 68 6 38 86 00 23 57 26 oo 32 13 70 00 2 179 225 219 2 7 2 14 00 9 oo 1 93 7 33 87 80 34 20 23 00 30 87 76 00 2 462 226 220 2 2 2 • 14 00 9 oo 1 93 7 33 87 80 34 20 23 00 30 87 53 00 1 717 227 221 2 2 1 14 00 9 00 1 68 6 38 86 00 23 57 26 00 32 13 43 00 1 338 228 222 2 2 1 14 oo 9 00 1 44 5 47 85 50 12 OO 28 00 31 36 43 00 1 371 229 223 2 2 1 14 oo 9 00 1 15 4 37 B7 80 4 69 28 00 29 31 54 00 1 842 230 224 2 2 1 14 00 9 00 0 88 3 34 85 50 0 0 30 00 30 00 BO 00 2 667 231 225 2 2 1 14 00 9 00 0 63 2 39 85 50 0 0 30 00 30 00 12 00 0 400 End of fll» 

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