( SLAMMING MOTIONS OF A RECTANGULAB-SECTIQN HAaHONIC WAVES BARGE MODEL IN by DOUGLAS NEIL WOEDEN B. A, S c , , U n i v e r s i t y of B r i t i s h Columbia, 1977 A THESIS SUBMITTED IN PARTIAL . FULFILLMENT THE REQUIREMENTS FOE THE DEGREE CF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE ( D e p a r t m e n t of M e c h a n i c a l STUDIES Engineering) We a c c e p t t h i s t h e s i s a s c o n f o r m i n g to the reguired standard., THE UNIVERSITY OF BRITISH COLUMBIA December, Jc) D o u g l a s N e i l 1979 Worden, 1979 OF In presenting this thesis in p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mechanical Engineering The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver,. Canada V6T 1W5 D a t e 21 A p r i l 1980 11 Abstract The work presented in this thesis concerns the theoretical analysis of the motion of f l o a t i n g rectangular cross-section bodies in s i n g l e frequency harmonic waves. When a conventional l a t e r a l l y symmetric ship is modelled by such a body, the computation of strip-theory c o e f f i c i e n t s (derived from the solution of Laplace's equation for the f l u i d surrounding the ship) is s i m p l i f i e d . This technique is used here to model a typical barge, with actual cross-sections rectangular shapes. very close to the assumed In p a r t i c u l a r , slamming motions are investigated using two conventional linear slamming c r i t e r i a . The rectangular section model is also applied to the investigation of slamming motions by use of 'quasi-harmonic' slamming c r i t e r i a , which are developed from an updating technique used with conventional s t r i p theory c o e f f i c i e n t s . are presented for an example. Results ill Table of Contents page Abstract ii L i s t of Tables v L i s t of Figures V 1 Acknowledgement 1 X Nomenclature x 1. INTRODUCTION 1 1.1 Problem Description 1 1.2 State of the Art 2 1.2.1 Two-Dimensional Strip Theory (Heave-Pitch) 2 1.2.2 P r o b a b i l i s t i c Ship Motion Theory 4 1.2.3 Slamming 6 1.3 Objectives and Scope 8 1.3.1 Conventional Linear Slamming C r i t e r i a 8 1.3.2 Quasi-Harmonic Motion Analysis 9 2. LINEAR MODEL (HARMONIC RESPONSE) 10; 2.1 Design of a Simplified Linear Model 10 2.1.1 Salvesen-Tuck-Faltinsen S t r i p Theory 10 2.1.2 Simplifying Assumptions 12 (Strip Theory) 2.1.3 Sectional Added Mass Coefficients 13 2.1.4 Sectional Damping Coefficients 14 2.1.5 Dynamic Simplifications 15 2.2 Equation of Motion 15 2.2.1 Coordinate System and Wave Parameters 15 2.2.2 Complex Amplitude Formulation 16 2.2.3 Measures of Response 17 3. NON-LINEAR MODEL (QUASI-HARMONIC RESPONSE) 3.1 Background 3.2 Method 3.2.1 Calculation of Motion Coefficients 3.2.2 Quasi-Harmonic O s c i l l a t i o n s 4. RESULTS 4.1 Purpose 4.2 Model Specifications 4.3 Linear Strip Theory Results 4.3.1 Motion S e n s i t i v i t y to Variation of Sectional Parameters 4.3.2 Linear Slamming C r i t e r i a 4.4 Quasi-Harmonic Results 4.4.1 Motion S e n s i t i v i t y to Variation of Sectional Parameters 4.4.2 Slamming C r i t e r i a 5. CONCLUDING: REMARKS 5.1 Summary 5.2 Recommendations for Further Work Bi bliography Appendix A Appendix B Appendix C L i s t of Tables Coefficients for Equation 2.1 Coefficients for a Single Section of the Rectangular Section Barge Model Rectangular Section Model of Seaspan 250: Specifications L i s t of Figures .page 1 Typical relationships between motion spectra and wave energy spectra 52 2 Slamming stresses superimposed on bending stress cycle 53 3 Conventional l i n e a r slamming c r i t e r i a 54 4 Rectangular section barge model 55 5 Method for f i t t i n g rectangular section to c u r v i l i n e a r section 56 6 Vertical o s c i l l a t i o n s of a rectangular section on a free surface 57 7 Added mass and damping c o e f f i c i e n t s for rectangular sections 58 8 Moment of i n e r t i a approximations for the rectangular section barge model 59 9 Fluid coordinates and ship coordinates 60 10 Harmonic wave notation 60 11a Absolute v e r t i c a l displacement and absolute v e r t i c a l velocity 61 lib Wave amplitude and v e r t i c a l v e l o c i t y 61 11c Relative v e r t i c a l displacement and r e l a t i v e v e r t i c a l v e l o c i t y 61 12 Phase relationships for linear slamming c r i t e r i a 62 13 Cases of sectional emergence considered 63 14a Wave p r o f i l e and s t i l l water p r o f i l e 64 14b Freeboard allowance for a section 64 15 Flowchart for calculation of motion c o e f f i c i e n t s 65 16 Flowchart for calculation of wave-surface intercept for a section 66 17 Graphical construction of quasi-harmonic response curve 67 18a Forced single-degree-of-freedom system, to i l l u s t r a t e quasi-harmonic vibration 68 vii page 18b Forced o s c i l l a t i o n at d i f f e r e n t values of spring stiffness 68 19 Quasi-harmonic motion cycle for single-degree-of freedom system 69 20 Tanker from Reference 21 and equivalent rectangular section model 70 21 Seaspan. 2-50 overall dimensions. Plate 1, p- lOa 22 Seaspan 250 cross-sections Plate 2, p. lot, 23 Seaspan 250 bow p r o f i l e Plate 3,p. 70c 24 Unit heave response, tanker, U=9 m/s 71 25 Absolute v e r t i c a l displacement at x=L/4, tanker, U=2.12 m/s 72 26 Absolute v e r t i c a l displacement at x=L/2, tanker, U=3 m/s 73 27 Frequency of encounter vs. r a t i o , L=107.2 m 74 28 Absolute v e r t i c a l displacement at f o r e f o o t , loaded barge, U=2.06, 4.11, 6.17 m/s 75 29 Absolute v e r t i c a l displacement at forefoot, l i g h t barge, 5 and 8 sections, U=2.06 m/s 76 30 Absolute v e r t i c a l displacement at forefoot, l i g h t barge, U=2.06 m/s 77 31 Absolute v e r t i c a l displacement at forefoot, loaded barge, U=2.06 m/s 78 32 Unit draft c r i t i c a l wave amplitude and absolute v e r t i c a l displacement, l i g h t barge, U=2.06 m/s 79 33 Relative v e r t i c a l displacement and r e l a t i v e v e r t i c a l v e l o c i t y , l i g h t barge, U=2.06 m/s, L/L =1.8 w Relative v e r t i c a l displacement and r e l a t i v e v e r t i c a l v e l o c i t y , l i g h t barge, U=2.06 m/s, L/L =2.9 w Harmonic and quasi-harmonic r e l a t i v e v e r t i c a l ;, displacements, l i g h t barge, U=2.06 m/s, L/L =2.9 w Quasi-harmonic r e l a t i v e v e r t i c a l displacement, f o r 3 motion c y c l e s , l i g h t barge, U=2.06 m/s, L/L =0.1 w 80 34 35 36 shiplength/wavelength 81 82 83 vi!i i page 37 38 Magnitudes of quasi-harmonic response c o e f f i c i e n t s , l i g h t barge, U=2.06 m/s, L/L =2.9 w Phase angle of quasi-harmonic response c o e f f i c i e n t s , l i g h t barge, U=2.06 m/s, L/L =2.9 84 Quasi-harmonic r e l a t i v e v e r t i c a l displacement, f o r 13 and 20 sections, l i g h t barge, U=2.06 m/s, L/L =2.9 w Quasi-harmonic r e l a t i v e v e r t i c a l displacement for three wave amplitudes, U=2.06 m/s, L/L =2.9. w Quasi-harmonic r e l a t i v e v e r t i c a l displacement for two values of t ' , l i g h t barge, U=2.06 m/s, L/L =2.9, =2.3 m *" 86 85 w 39 40 41 87 88 w 42 Quasi-harmonic r e l a t i v e v e r t i c a l displacement and sectional added mass for a section close to the \ forefoot, l i g h t barge, U=2.06 m/s, L/L =2.9 w 89 ix Acknowledgement The author wishes to thank Dr. H. Vaughan for his supervision of this thesis. Financial support was obtained through the Naval Architecture Fund, Account 65-6085, University of B r i t i s h Columbia, during the period January 1978 - December 1979. Mr. Tom Ward, of Seaspan International L t d . , provided drawings and specifications of the 'Seaspan 250' barge. whdch'were sincerely appreciated.' y The graduate students in the Mechanical Engineering department, p a r t i c u l a r l y Ken Lips and Mitchell Wawzonek, provided technical assistance and moral support whenever i t was required, for which I am most g r a t e f u l . My .wife, Eleanor, deserves praise for her patience. Nomenclature 2x2 matrices of c o e f f i c i e n t s (real) coupled motion equation for ship beam (m) beam for one section (m) typical dimension for calculation of v e r t i c a l slamming v e l o c i t y constants for forcing terms of coupled motion equation (complex) 2 moment of i n e r t i a about y-axis (tonne-m ) complex amplitudes of heave and pitch forcing functions respective heave and pitch forcing terms a r i s i n g from afterbody section heave unit response amplitude (m) pitch unit response amplitude (rad) ship length (m) wave length (m) ship mass (tonnes) f l u i d coordinates ship coordinates ship motion energy spectrum wave energy spectrum draft (m), period of motion cycle (sec) draft for one section (m) draft of afterbody section (m) forefoot draft (m) ship speed (m/sec) threshold v e r t i c a l slamming v e l o c i t y (m/sec) r e l a t i v e v e r t i c a l v e l o c i t y when forefoot impacts wave surface (m/sec) response amplitude operator (heave) average heave motion amplitude (m) average 1/3 highest heave-amplitudes (m) sectional added mass c o e f f i c i e n t (tonne/m) sectional damping c o e f f i c i e n t (tonne-sec/m) respective sectional c o e f f i c i e n t s for aftermost section freeboard allowance for a section gravitational constant; -1 wave number (m ) mass of one section (tonne) area under ship motion energy spectrum section number real time (sec) non-dimensional time absolute v e r t i c a l v e l o c i t y at forefoot (m/sec) r e l a t i v e v e r t i c a l v e l o c i t y at forefoot (m/sec) aftermost x-coordinate for a section (m) forwardmost x-coordinate for a section (m) X n+l" n x (x +x •V n+1 ±1 )/2 n' x-coordinate of center of gravity x-coordinate of afterbody section (m) x-coordinate of forefoot (m) x-coordinate of wave surface/section bottom intercept (m) absolute v e r t i c a l displacement at forefoot (m) r e l a t i v e v e r t i c a l displacement at forefoot (m) heave amplitude (m) pitch amplitude (rad) __wave amplitude (m) maximum wave amplitude (m) v e r t i c a l velocity of wave surface at forefoot (m/sec) .un.it draft c r i t i c a l wave amplitude (m) 3 water density (tonne/m ) r e l a t i v e angle at forefoot (rad) _wave frequency (rad/sec) frequency of encounter (rad/sec) 1 INTRODUCTION 1-. 1,1 Problem Barge Columbia Description towing are conditions, combined often with and motions of near bow, the Slamming between surface, extreme barge the as can for may hull Recent Columbia slamming by (near the "bow damage Bhen the his ship v i b r a t i o n ) . reported to their begins would weather to of any be noticed slamming h u l l plating involves impact and the been ships, the with studied barges are from the i s o l a t i o n problem water associated has slam, When v e l o c i t y , in to slamming s e l f - p r o p e l l e d unaware slamming damage slamming of B r i t i s h winter result barges f l a r e " because of (towing forepeak) bottom barge tow, cause towed conventional continue conventional with Although severe configurations This to coast common . f e a t u r e , . barge. opposed crew. a conditions extensively tug towing bottom susceptible daring such ships. tugboat oat west loading) conventional more the waveheights associated h u l l on carried with . c e r t a i n heading the operations master of the (whereas on a immediately, due to 1 cases of Seaspan towing described damage, International companies) in this l e d of the (the to thesis. , the type largest described of the investigations above, B r i t i s h of barge 2 1.2 State of the A r t 1.2,1 .Two-Dimensional S t r i p T h e o r y The theory analysis of of ship ship simplifying motion model. in condition frequency) general, probabilistic {generally terms. motions, motion problem into pitching formulation Coupled heaving ship responses s p e e d , ,,, symmetric will directly It appreciate to develop a so t h a t s h i p ships been motions by s h a p e , relative to that for reasonable, for six-degree-of-freedom motions omitted). and are discussed i n shown i t i s ship waveheight heading has be t h e b a s i s and "heave-pitch" The c o u p l e d heaving f o r a l l motion: and theory thesis. and p i t c h i n g m o t i o n s c a n be r e p r e s e n t e d to a f o r c i n g response by the general input case i s t h a t o f s i n g l e - f r e q u e n c y theories, to "non-linear" motions i n p a r t i c u l a r , i s generally i a this a quantity, "roll-sway-yaw*} (surge as parameterized uncouple motions considered well conventional of the background be i n t r o d u c e d moments o f i n e r t i a , and to of Ship c o n f i g u r a t i o n i s s p e c i f i e d laterally small basis necessary will i s a statistical direction, conventional is model, a s mass, and a p p r o p r i a t e wave that and slamming the An u n d e r s t a n d i n g theory assumptions b a r g e slamming wave . motion linear Sea motions i s o f s h i p slamming linear (Heave-Pitch) amplitudes proportional to wave ( s u r f a c e waves). harmonic for height, The s i m p l e s t waves. harmonic as For linear responses w i l l allowing use be of 3 superposition The two-dimensional conventional It techniques (heave-pitch) l i n e a r ships in long-crested i s e v a l u a t e d as a " s t r i p practical They d e t e r m i n e d a method between and heave presented their own coefficients. 8ang and Strip but for five known. first developed and for in Jacobs , 2 coupling a u t h o r s , most n o t a b l y of the with same slightly motion of as different t h e o r y has degrees form included freedom, by Sclavounos . 6 theory formulations a r i s e potential;theory. waves i s w e l l account Other theories, F u r t h e r study of s h i p L o u k a k i s and rigid S a l v e s e n , Tuck a n d . F a l t i n s e n * have Jacobs, o f c o u p l e d motions and s strip model by K o r v i n - K r o u k o v s k y motions., 3 for was rationally B e u k e l m a n , and Korvin-Kroukovsky analysis to pitch harmonic t h e o r y " , and c o m p u t a t i o n a l . form G e r r i t s m a and 1.2.2). (Section I t i s necessary to £ = from hydrodynamic velocity satisfy: O (1.1) where: (j) =• (J) (J)^ = total velocity potential = incident velocity t o ocean <waves) (f) ° = diffracted velocity potential (waves r e f l e c t e d from s o l i d s u r f a c e of ship) = motion i n d u c e d potential with the boundary potential(due conditions: (1) a r i s i n g velocity from the e g u a t i o n s of motion of surface, it i s the body a n d (2) on f l u i d on s h i p s i d e s , and a t i n f i n i t y . assumed that linear, the r e s u l t motion amplitudes equations are direction, of hydrodynamic the resulting c a n be e x p r e s s e d as unknowns. obtained ship by properties. problem to velocity Salvesen, Tuck and F a l t i n s e n * . equations, in an effect of with these appropriate i s t o reduce the vibration potential problem. A formulation i s Hehausen^ and t h e a p p e n d i x t o t h e p a p e r by 7 and atthe i s done, and coefficients The the by Newman , detail, The i.e. a r e harmonic and as coupled mechanical presented exclusively motions a treatment used of When t h i s integration, detailed is boundaries, The s t r i p i n this thesis, compared with other and t h e o r y from i s strip the l a t t e r presented in more theories, i n Section 2. 1.1. 1.2.2 Probabilistic Ship Hotion Theory The t h e o r y o f m o t i o n s i n r e g u l a r waves no l o n g e r c a n be c o n s i d e r e d by i t s e l f a l o n e . , I t i s o n l y a p a r t o f the picture, the hydromechanical phase which e s t a b l i s h e s t h e d e p e n d e n c e o f a s h i p s m o t i o n on i t s form and mass d i s t r i b u t i o n . > , The r e s u l t s o b t a i n e d (from s t r i p t h e o r y ) , t h e s h i p r e s p o n s e s to regular waves o r "response factors", a r e t h e n t r e a t e d by methods o f m a t h e m a t i c a l statistics i n conjunction with a measured o r assumed s p e c t r u m o f a r e a l i s t i c i r r e g u l a r sea t o g i v e t h e r e a l i s t i c s h i p motions. Korvin-Kroukovsky Although methods u s e d probabilistic of this thesis t o convert terms will single and J a c o b s . 2 not frequency consider harmonic (that i s , s h i p response waveheight-wavelength combinations), in d e t a i l the inputs into to a s t a t i s t i c a l s e t i t i s useful to outline 5 the principles and probabilistic It involved, waveheight vs. heading and s p e e d . for ship a using strip wave t h e development (as (proportional amplitude motions, a theories of sea c o n d i t i o n . unidirectional wave f r e q u e n c y i s r e g u i r e d , Response a amplitude function is adjusted operators expressed in Section f o r ship are in terms 1.2. 1. of For spectrum computed o f wave f r e q u e n c y and s h i p speed) When wave the energy t o t h e s g u a r e o f t h e w a v e h e i g h t ) , and t h e r e s p o n s e o p e r a t o r s (commonly termed s h i p motion example, and p i t c h a spectrum theories as o u t l i n e d spectrum of s t r i p methods h a s n o t o c c u r r e d i n d e p e n d e n t l y . i s necessary to obtain the c a s e o f heave of since "S. A.O. *s") e n e r g y s p e c t r u m c a n be e x p r e s s e d f o r heave as a r e known, t h e follows, (for motions) : where: ship motion enerqy spectrum response amplitude operator (heave motion) heave amplitude/wave one f r e q u e n c y wave e n e r g y If is t h e wave s p e c t r u m the area under i s given the amplitude a t spectrum by a R a y l e i g h d i s t r i b u t i o n , ship m o t i o n a m p l i t u d e s c a n fee d e f i n e d and m motion energy spectrum, then t h e as f o l l o w s 6 2 average amplitude in example o f Figure 1 the {from -/E r e l a t i o n s h i p between Hawscn and Probabilistic the ship designer, ship range of sea first to apply linear theories, conditions. St. superposition w h i l e more r e c e n t l y to Probabilistic methods also s l a m m i n g ) , and described Section for probabilistic 1.2.3 1 has 2 the motion important i s shown D e n i s and Pierson technigues be to Bishop applied made t h e 1.2.3. 1 1 linear were ship a the motion have w r i t t e n a dynamics. t o extreme required for over 1 0 ship motions analysis for slamming c r i t e r i a , Bhattacharyya standard tool performance probabilistic model o f c o n v e n t i o n a l in reference lick an P r i c e and introduction a statistical is t o compare comprehensive (e.g. spectra 9 him can various Tupper ). dynamics allowing wide 0 average of o n e - t h i r d highest amplitudes 2.0 An L2&Jl^-n = 1 3 calculations is a as useful required for studies. Slamming Slamming has self-propelled Investigations First, concerned pressures been ships, are study widely as one broadly of when case classed the a of for conventional extreme s h i p i n t o two "hydrodynamic w i t h e x p e r i m e n t a l and created studied, response. areas. problem ' is t h e o r e t i c a l i n v e s t i g a t i o n of the ship section impact (perhaps modelled 1 by a 7 flat plate resulting wedge) response Hodelling where or of methods f o r applicable) classification thesis ship While as types can of (Figure 2). Local response to a single A by J o n e s * contained The review second directly results impact change of added in Section to of by N a g a i can and problem be bottom amidships, bending l s stress modelled as on a f u l l and C h u a n g 1 7 near the nor t h e this i s thesis i s bottom time of the slamming periods, due t o a h i g h r a t e o f 1 2 mass ,, predicts Unfortunately, variation t h e o r y model, s i n c e , the l i n e a r theory ship). theory Studies that forefoot added 1 6 . occurrence. have shown Lewis scale response w i t h which t h a t t h e slam i s p a r t l y mass h a s been , and HcLean and accelerations, f o r short a linear strip 2. 1.1, "bottom led to this responses structural i s slamming neither the acceleration admitted as stresses response The (and s i m i l a r l y case t h a t bending ,Ochi and H o t t o r i n high v e r t i c a l have t h e o r i z e d l e d to the known wave area of study, that concerned, hydrodynamic and is the pressures (1977) i n a paper damage, slamming . b e h a v i o r . experimental observation of stresses current forces. c o l l a p s e n e a r t h e bow, signifigant of 1 have the pressure pulse. Investigation (an and o i l tankers) structural and structural w i t h towed b a r g e s w i t h m a g n i t u d e s o f t h e same o r d e r a s undertaken (and damage i n t h e p a r t i c u l a r generate surface, to transmitted developed, was c o n f i n e d t o l o c a l p l a t e slamming water response have been such the ship girder most f r e q u e n t l y vessels, slamming". the of different type a s s o c i a t e d shaped impacts can be a s we w i l l s e e harmonic motions 8 {at the frequency are a l l o w e d , and t h e once, in the o f wave e n c o u n t e r ) notion are substituted f a c t o r s present importance - and illustrated The in amidships. i n various of narrowed occurs, and in (2) r e l a t i v e angle the criteria order ship bottom between t h e forefoot, , ship These are the tankers are subject R e c e n t slamming and 2 slamming Ochi* , 9 interest to occurrence Price, large models Bishop m o t i o n s o f an " e l a s t i c " and ship, and Scope Slamming Criteria p r o j e c t , as o r i g i n a l l y stated, was a m a t h e m a t i c a l model f o r p r e d i c t i n g t h e slamming As of h a r m o n i c modes. purpose o f t h i s barges. only observation, o i l t a n k e r s has i n c r e a s e d motions, since 1.3.1 C o n v e n t i o n a l L i n e a r develop near large considered 1,3 O b j e c t i v e s The These a r e , from v e l o c i t y a n d (3) p r e s e n t e d by T i c k * vibrating evaluated 3, i n Figure stresses have z models, surface advent o f very h a v e been Tam ° vertical are frequencies more s u i t a b l e slamming when b o t t o m slamming wave extreme s h i p bending Instead, i n linear no h i g h r e s t p o s i t i o n , s o t h a t no c h a n g e i n a r e : {1) f o r e f o o t emergence wave s u r f a c e hottom coefficients still-water added mass i s a l l o w e d , so t h a t the project progressed, the to motion objectives were to the following: { 1 ) d e v e l o p m e n t o f a model t o p r e d i c t t h e l i n e a r heave and pitch barge {an response of a rectangular section 9 approximation to t y p i c a l waves, u t i l i z i n g (2) barge shapes) i n r e g u l a r conventional investigation of the strip theory slamming Extension either 1.3.2 to p r o b a b i l i s t i c of the directly In response Hotion s h i p motion i s o l a t e the order to not a criteria. considered in do m o d e l s , as e x p l a i n e d p r e v i o u s l y , c a n n o t this, motion a coefficients f o r m u l a t i o n , while Chapter was model m e n t i o n e d developed above. method quasi-harmonic non-rlinear 3, was such Analysis factors associated on a s t e p w i s e evaluating motion r e s u l t s slamming of above c a s e s . , Quasi-Harmonic Linear techniques. occurrence model, i n terms o f c o n v e n t i o n a l l i n e a r harmonic and was with bottom developed b a s i s over on each which p r e d i c t s a motion f o r the cycle, interval.,. subject to r e s t r i c t i o n s refined slamming. noted This in rectangular section 10 L I N E AB 2. 2. 1 D e s i g n of a The used two-dimensional in this (henceforth widely employed results theory from was theory to Linear from as S-T-F chosen s t r i p for and Theory s t r i p s t r i p because Tuck ship it is one of has forms, and r e s u l t s , because The i t s the w i l l been shows and with S-T-F s t r i p use more be Faltinsen* It formulations. thesis that and theory). experimental theory this theory Salvesen, conventional with BESPONSE) Model Strip is for other (HARMONIC (heave-pitch) c o r r e l a t i o n documented, is well recent s t r i p formulations. The thesis S-T-F w i l l assumed). This is a reponse given frequency and of Forcing the actual Bishop wave and 2 1 , for amplitude an which only are harmonic small once, are each to evaluated for unit the obtained, heave pitch motions, is that the ship ship freguency, thus so was p o s i t i o n . , responses and this motion rest for in d i r e c t l y (relative s t i l l - w a t e r amplitude, in that functions ship theory, responses assuming some c o e f f i c i e n t s " by l i n e a r ship evaluated allowing multiplied Price are a (recall about position. amplitude, wave result motions rest is predicted waveheight c o e f f i c i e n t s that theory that to dimensions) Motion s t r i p mean proportional "unit thesis referred reasonable wave Simplified Salvesen-Tuck-Faltinsen 2.1.1 at MODEL to response be for result. investigation of the motions of 11 an o i l tanker and between r e s u l t s theory "(a) a d e s t r o y e r , note o b t a i n e d f o r S-T-F of G e r r i t s m a and there i s l i t t l e v a l u e s o f L/L effects of f l u i d the strip following theory differences and the strip Beukelman : 3 t o c h o o s e between t h e two t h e o r i e s a t h i g h (shiplength/wavelength ratio) since the damping a r e t h e n s m a l l . w (b) their p r e d i c t i o n s a r e somewhat d i f f e r e n t a t low v a l u e s o f L/L,y where wave r e s p o n s e is the more serious problem, s i n c e f l u i d damping i s t h e n dominant. (c) the d i f f e r e n c e s are i n the magnitudes of the {of t h e s h i p r e s p o n s e s ) r a t h e r t h a n i n the t h o s e peaks i n terms of L/L^/" Eguation equations laterally respective ship, the 2. 1 is 2.1 {in i s the the of the coupled encounter" domain) s h i p i n harmonic coefficients. integral form "frequency symmettric S-T-F linear mechanical vibration specified frequency, amplitudes by of system, and a s t a n d a r d methods Table motion for I lists t o summations. constant the Equation coupled coefficients be s o l v e d f o r t h e h a r m o n i c {Appendix at / B t C = 2x2 A): matrices of c o e f f i c i e n t s frequency amplitudes {complex) of a motion (2. A a multi-section two-degree-of-freedom with can waves. In p r a c t i c e , f o r a signs are converted representative of r e s o n a n t peaks locations of 1) {real) encounter o f h e a v e and pitch motions amplitudes o f heave and f u n c t i o n s (complex) pitch forcing 12 2.1.2 Simplifying Barge shapes conventional section to ships. sections the the h u l l i s of length, bow). that the ship strip form. and in and The signifigant reduction f o r the S-T-F useful sectional mass as the and correct for in a reguired time for (on "first computing the a digital sections approximation" time becomes model o f C h a p t e r section waterlines. cross-sectional approximation. fact damping substituting rectangular which a r e c t a n g u l a r reasonably for reasonable (unit added i n s i m p l i f i e d form. s e c t i o n , a t two section (except Figure is fitted This areas 5 ship are describes to a technique f o r the more 3. coefficients for a rectangular-section curvilinear stern complex f o r a g e n e r a l c u r v i l i n e a r quasi-harmonic i n Table II method by unity cross and a r i s e s from calculation by bow of 4). coefficients, quite to rectangular sectional that i s obtained technique. presented are reduction signifigant, close those constant I t would t h u s a p p e a r (Figure of nearly that with short this simplification inertial theory, The computer) The to calculations coefficients, S-T-F sections advantage much s i m p l e r c o e f f i c i e n t s are near cross Theory) general model a b a r g e w i t h a s e r i e s o f The the in Often Sectional coefficient) is are (Strip s h a p e o v e r most of t h e sections. cross Assumptions typical gives rectangular 13 2.1.3 Sectional Sectional described an on that infinitely the long on ship non-linear as effects vertical sectional of is by a of suitable analytic conformal mapping o f of requiring cross-sections, the at for graph. variety that in Chapter and acceleration as 1.1). ship where sections, Lewis 2 2 was at the ignored, a and particular sectional the More e x a c t sections, be added mass methods first to (from present methods, i n c l u d i n g have been used and added 7a)., They a r e breadth/depth were o b t a i n e d by (Appendix C ) . mass, by a the Macagno 2 3 disadvantage for used i n t h i s t h e s i s (Figure frequency modelling on motion position, velocity potential sectional will 7 and computation. of that its quasi-harmonic are constant moves shape, instant 3, instantaneous by mass d e p e n d s oscillating at length) when i t added an of technique. i n Newman values f o r values unit encounter. extensive values reproduced The physically C o n f o r m a l mapping t e c h n i q u e s have t h e 2 The of discussed velocity Equation Vugts *. (per l i n e a r m o d e l s , and general curvilinear ship determined the be section 6)• of added mass i s d e f i n e d solutions and (Figure For evaluated freguency of For cross acceleration parameters. are value constant and coefficients the of water e n t r a i n e d (at some i n s t a n t ) models of ( f o r heave motion) c a n surface velocity c e r t a i n shape Coefficients mass o f a free position transverse Bass added mass as vertically Added The are defined ratio. The rectangular from for a Vugts, range numerical a polynomial curve f i t to sensitivity of motion responses 14 to v a r i a t i o n s i n s e c t i o n a l added mass i s d i s c u s s e d For the sectional p u r p o s e o f most l i n e a r added mass when m u l t i p l i e d by 2, 1,4 S e c t i o n a l Damping C o e f f i c i e n t s per a unit damping length) free surface. sectional an appropriate a an In mathematical terras, the infinitely is result similar of viscous damping i s n o t noted, however, change of that a sectional to Architecture* , damping same s o u r c e 7 numerically the damping (see, pp. 8 as by also frequencies, the computing of computing sectional the ship o c c u r s due ship potential flow motions, since to the be rate moves f o r w a r d , equal t o zero motion a m p l i t u d e s . example, on of s o l u t i o n , , I t should the as force vertically problem by coefficient for so does not is not This Principles of of Naval 6 36)., added mass c o e f f i c i e n t s , a f i t to the graph (Figure damping c o e f f i c i e n t s a r e so damping moving in i n the 4.. arm, c o e f f i c i e n t s used i n t h i s t h e s i s a r e the f i g u r e that that occurs permitted damping 0=0. ship waves g e n e r a t e d cause unrestrained where The long added mass a t a s e c t i o n as necessarily true that theories, i s the on damping as moment ( f o r h e a v e motion) added mass. .,• S e c t i o n a l damping model motion f o r heave m o t i o n i s used f o r p i t c h i n g well, Sectional ship i n Chapter that damping, as stated more d e t a i l i n Chapter 7b). small ship responses are earlier. 4. and not Motion s e n s i t i v i t y are from evaluated It i s clear at high very the from encounter dependent i s discussed on in 15 Sectional h e a v e damping 2.1.5 damping f o r pitch coefficients by an is obtained appropriate by multiplying moment arm. Dynamic S i m p l i f i c a t i o n s The above model may be properties treated extended of as s i m p l i f i c a t i o n s f o r a rectangular the by seme a p p r o x i m a t i o n s b a r g e . , The rest position waterline. and the x=0 p o s i t i o n as: moment o f inertia (Figure Small changes i n the will about vertical p o s i t i o n s of the 2.2 of Coordinate Figure mass mass c a n transverse barge w i l l be centers be written a x i s through on as the p o s i t i o n of the on CG center motion r e s u l t s . will not be of gravity Variations in considered. notion System and Wave P a r a m e t e r s 9 shows s h i p c o o r d i n a t e s center of on undisturbed the the with inertia! 8) longitudinal 2.2.1 density, Thus, the have a n e g l i g i b l e e f f e c t Equation the rectangular-section s e c t i o n s of c o n s t a n t the for cross-section gravity at 0', level and fluid surface of 0'x*z* f i x e d coordinates the fluid, Oxz, and to the where 0 ship lies translates in 16 the positive defined x-direction i n the considered The positive here. The following with speed x-direction. wave p r o f i l e wave and , wave f r e q u e n c y W shiplength/wavelength Note that CUe = CO * = L/L„ ratio freguency of encounter shiplength/wavelength r a t i o as head parameters c i s also seas will be 10. will be used: 2-fT k .= — . — of encounter Have s p e e d i s shown i n F i g u r e wave number freguency Only freguency , U. kO can be e x p r e s s e d i n terms of ( f o r head s e a s ) : 2.4 2.2.2 Complex Amplitude For the harmonic is a useful are easily addition of only The motions t h a t mathematical preserved pitch (displacements part Formulation and and device, a r e assumed, complex n o t a t i o n Magnitudes in complex heave is velocities) and phase n o t a t i o n , so t h a t possible. are Motion understood angles vectorial amplitudes t o be the o f complex e x p r e s s i o n s . wave d i s p l a c e m e n t (in the vertical direction) is real 17 Ship motions are that h e a v e and h a r m o n i c , a t t h e same f r e q u e n c y pitch where K, , K a r e t h e 2 solution Section presented. (following (1) Tick 1 2 Relative when (1) ). These so x=z=0: o b t a i n e d from here, the that within the to simplify length scale i t is of rectangular section forefoot must not neccessary f o r the physical boundaries of of the forefoot. satisfied correct V , d and phase surface) from analysis the model. is velocity the were wave s u r f a c e ) to o c c u r , t h e the velocity wave criteria are: F o r a slam section arises used slamming ( w i t h r e s p e c t t o the Note be has threshold 2.9 be linear w h i c h i s d e f i n e d f o r our above approaching a typical will vertical velocity Equation two emerged. to amplitudes" at wave, % = 1. three (X/«,T^) . rectangular (2) for (respectively) the Response forefoot, forefoot are response displacement model by be 2,1 1.2*3 Only Relative the "unit of Equation -2.2*3 M e a s u r e s o f In amplitudes as g i v e n by a impact of the occurs vertical (ship bottom magnitude e x c e e d i n g Bhattacharya hydrodynamic slam the r e l a t i v e relationship with f o r movement a 1 3 theory, a as: where B free surface., 1 is 18 The above c o n d i t i o n s response amplitudes Vertical c a n be s p e c i f i e d i n t e r m s o f t h e u n i t f o r t h e barge displacement of as f o l l o w s ship at ( f o r some forefoot ): (rest positon waterline) Vertical v e l o c i t y of ship Haveheight at forefoot a t bow Vertical v e l o c i t y o f wave s u r f a c e Relative vertical displacement a t t h e bow at f o r e f o o t 2.13 \*,Cx T t)=z-y f> Relative vertical +K fa)x -e *)e' #. = Kfafa) r !kx x -? ra f v e l o c i t y at forefoot 2.if- * 2 „ = -CiAJ V, (pc t) f) It i s also useful e to define («, wave a m p l i t u d e , unity, causes the harmonic motion +K z fa)X ~JL **) ik f draft c r i t i c a l waveheight" f o r d r a f t at the forefoot equal to the f o r e f o o t cycle e the "unit which i s t h a t that Cco ) (derived t o j u s t emerge a t one p o i n t i n Appendix B) . , on 19 Equations Figure likely Figure 11. 2.3 The .through 2.14 relationship are between t o s a t i s f y t h e l i n e a r slamming 12. presented graphically equation criteria 2.13 and in 2.14- i s illustrated in 20 3. 3.1 NGN-LINEAB HODEL (QOASI-HARMONIC RESPONSE) Background What i s desired i s not a rigourous solution for a n o n - l i n e a r problem, b u t r a t h e r a s o l u t i o n of a substitute l i n e a r p r o b l e m which would a p p r o x i m a t e a t r u e s o l u t i o n Korvin-Kroukovsky The linear non-linear ship S-T-F rectangular strip section motion and J a c o b s theory presented here model), evaluated cycle. at a I t i s termed number over a motion the amplitudes obtained are not d i r e c t l y the uses the theory c o e f f i c i e n t s from Chapter 2 ( f o r the intervals motion 2 of non-^linear time because proportional to waveheight. Host motion authors have analysis, determination damping of (for concentrated f o r coupled sectional example (pp. non-linear technique, Korvin-Kroakovsky problem 614) and heaving refers but observed. is apparent The " r e s t coefficients ). to an (above), at non-linear pitching, f o r added mass Principles of analogue the a on t h e and Naval computer suggestion substitute of linear ignored. The i m p o r t a n c e o f i m p r o v e d coefficients 2 5 despite Jacobs seems t o have been and coefficients Golovato Architecture attempts methods f o r e v a l u a t i o n when position" the wave p r o f i l e assumption i s not a p a r t i c u l a r l y valid f o r the o f motion on a s h i p i s linear model one when v a r i a t i o n s i n 21 wave h e i g h t , relative Even i f a b s o l u t e the relative noted that to the design ship amplitudes waterline, (particularly this to for absolute angles is not rolling of coefficients slamming, vertical motions roll of reguire ships, where with 1.2.3. the Slamming allows neither coefficients added mass. of tihile these discussion Figure are evaluated This i s discussed with respect f u r t h e r i n Chapter disadvantages of a 14a of motion of bottom i s characterized both large position. high the conventional features, be analysis the consideration). a c c e l e r a t i o n s n e a r t h e bow, c a u s e d by a of The associated Section change ( I t should advantage of i n s t a n t a n e o u s e v a l u a t i o n is in are small, t h e same s o r t o f n o n - l i n e a r shows v a r i a t i o n s i n wave p r o f i l e f r o m t h e r e s t further significant. pitch) v e r t i c a l d i s p l a c e m e n t s c a n be l a r g e . referred A are are by high rate linear theory possible to actual of when wave p r o f i l e . 4. non-linear approach such as t h i s are: (1) superposition techniques motions a r e not p u r e l y motion response complicated (2) computational are computed than data are not d i r e c t l y harmonic. in a probabilistic with c o n v e n t i o n a l time Thus, i s increased, a t many t i m e linear applicable, presentation of f o r m i s much more theories. s i n c e motion intervals. since coefficients 22 3.2 Method 3.2.1 Calculation The o f Motion method t h a t will Coefficients be u s e d h e r e response c o e f f i c i e n t s , f o r a ship of encounter, (1) f i x the with (2) i s as ship speed f o r each draft, solve for that using and be calculate suitable draft b a s e d on s i m p l e the e f f e c t i v e length approximations, t o f i n d and mean the s e c t i o n a l response c o e f f i c i e n t s , a t that Figure evaluation of the s e c t i o n a l 13 i l l u s t r a t e s considered, The c o r r e s p o n d i n g mean and f r e q u e n c y t o t h e motion c o e f f i c i e n t s and f o r c i n g requires each s e c t i o n . section. unit i n some r e l a t i v e p o s i t i o n t o t h e wave, moving section, (2) will a t some p o s i t i o n the follows; f o r the u n i t Step calculate 0 contribution (3) to for values are, referring the the four position. , properties possible relative position of e f f e c t i v e terms o f each section t o the notation cases length o f Figure l i n e a r approximation... These a r e , 13, respectively: CAStT I Z Z,(Xr) 3 4- The 2 o l i n e a r approximation o f o r mean d r a f t i s r e a s o n a b l e when 23 the wavelength d i m e n s i o n (AX„) the , so an that b o u n d a r i e s of one here f o r the the i s of two or larger An the associated f r e e b o a r d , the (see Figure 14b) . Figure 15 d e t a i l s the modified that (x*) the point wave 3.2.2 on for at so that water of reguired section (case 2 or c a s e 3) i f draft maximum a fixed position. routine a considered calculation bisection within value motion Figure 16 to calculate that intercepts surface. Quasi-Harmonic O s c i l l a t i o n s In the preceding coefficients could (relative to the of response unit section be directly wave). predict to use coefficients the For as the an the to unit response at i n s t a n t a n e o u s ship positions manner. response a motion the relative positions, not, however, the ship, since relative of I t i s not values time-dependent technique series would cycle. differential t o compute a This purpose of t h i s s e c t i o n , The how I t i s thus p o s s i b l e instantaneous obtain ordinary coefficients. shown c o e f f i c i e n t s for various p o s i t i o n changes over how i t was computed c h o s e n i n some s y s t e m a t i c 2.1 been to that section not original still i s set flowchart response c o e f f i c i e n t s for a ship are with each s e c t i o n , mean d r a f t shows t h e smallest a l l o w a n c e has mean d r a f t e x c e e d s t h e plus than the more wave c r e s t s section.. freeboard calculated order we entirely of unit clear response response. will equation for i t s solution consider with that equation time-varying will be used 24 here can be discussed oscillator required (two in terms coupled simplifications the ship,, (1) A ship o s c i l l a t i n g steady-state (relative of equivalent mechanical d e g r e e s of freedom)., when we i n terms o f in wave the harmonic oscillation to the of hydrodynamic waves given motion) will the model of return amplitude regardless make of to and its a phase initial position (2) As t h e relative increments, for those position the of the corresponding p o s i t i o n s vary in ship changes u n i t response small in small coefficients increments (phase and amplitude) . expresses small The first the steady-state variations assumption), small in unit When Chapter The is intuitively s o l u t i o n of the response the actual s i z e of coupled the equations (the of the for second above a s s u m p t i o n s coefficients these and imply motion variations i s discussed in 4. During a c y c l e of s h i p motion/ over a s m a l l time p o s i t i o n of the amount. , Ie will ship changes apply this by fact here "quasi-harmonic" o s c i l l a t i o n s , oscillations position. useful reasonable, coefficients taken together, periodic variations in equation. the assumption due to While t h i s information Figure 17 the unit i s not about s h i p shows the a correspondingly to introduce which response increment, «e what a r e consider coefficients, small termed as at r i g o r o u s l y t r u e , i t appears to the one give motions. graphical construction of a 25 quasi-harmonic vertical response curve, d i s p l a c e m e n t a t t h e bow. variation in unit correct since, implies that response ship for smaller carefully, i t cumulative solution has i t has i s only the r e s u l t i n g are the s h i f t oscillating beem relative motions not from with cycle, small quite t, to the which oscillating the motion c a n be seen period). i s not with t h a t e a c h new which Shen positon makes t h e tz unit those a p p r o x i m a t e l y e q u a l t o A t (where p r e v i o u s motion, is interesting illustration, shown single-degree-of-freedcm variation in stiffness spring variation with the 2.1. The r e s u l t i n g Although shows At is considered depends on t h e rigorously in Figure spring a simpler 18a correct case as and. mass s y s t e m , coefficients the as a with a small in Figure constructed quasi-harmonic i s not r i q o r o u s l y correct, The associated of the equations of shown an forced o v e r one p a r t o f t h e c y c l e , reponses a r e motions T8b, while motion cycle. some useful i s obtained, Interestingly, technique consider corresponds t o t h e s m a l l changes the procedure information to stiffness hydrodynamic 19 described dynamics, a field Hughes , in 2 6 been for very c o m p l i c a t e d . It Figure there However, t h e m o t i o n s a period than case o f t, f o r a c o m p l e t e Instead, parameters Shen seen on t h e f i g u r e , coefficients case. much as the this response c o e f f i c i e n t s , are very n e a r l y harmonic. the in a parallel the quasi-harmonic above c a n be f o u n d removed f r o m a to the consideration i n a paper marine of on response satellite environment. quasi-modal analysis P.C. of 26 v i b r a t i o n of a deployment deploying rate characteristics. harmonic to satellite boom, consider cycle as a that is sufficiently results". to "mode" c o n s i d e r a t i o n of "instantaneous" motions states the "instantaneous" While i t i s i n c o r r e c t motion neglects consider of boom modal a forced vibration, i s similiar. the Hughes "... i f the e x t e n s i o n r a t e ( i . e . , time dependence) gradual, a modal viewpoint will give useful 27 4. 4.1 Purpose The the development previous usefulness of of of the experimental a loaded a barge) The First, which the 4.2 for f o r an require (i.e. with a v e s s e l with 21, and loaded amounts a v a i l a b l e . .,, In scale Beference motion 21, a shape s i m i l a r of motion are "critical" slamming results may be are here to so t h a t an i s obtained. criteria motions) presented (for where to that presented i s twofold. variations indication Secondly, both f o r an of of numerical harmonic and example, so that considered.. The compared.... Specifications T h r e e r e c t a n g u l a r - s e c t i o n models w i l l first the motions large full example from f o c u s o f the r e s u l t s techniques Model comparisons detail (guantitatively) T h i s would parameters i s i n v e s t i g a t e d , quasi-harmonic t h e two only sensitivity ( i f any) results in of i s considered. primary sectional considered which i s u n f o r t u n a t e l y n o t qualitative, o i l tanker not barges, data the has r e c t a n g u l a r - s e c t i o n b a r g e model model i n p r e d i c t i n g actual full-scale results are of the chapters this section, of BESOMS is the and an approximation o t h e r two light t o the are based on be loaded tanker, the from "Seaspan 250" condition, respectively. Small Beference barge, variations in of 28 sectional p r o p e r t i e s are c o n s i d e r e d For the overall model, displacement rationally section and tanker f i t length, developed, motion results results with calculation sectional S-T-F with those The theory. from typical of c o a s t a l damage (plate bottom were was barges, The on not Instead, typical with damping it bow and chosen and was model the actual section to as fitting models f o r over h Table section example tanker entire I I I and models conditions, an the of IV of using shape one rectangular constant up" 2, of the barge with achieved linear strip remains i t is slamming the method o f C h a p t e r For the to specify because "setting whereas t h e n o n - l i n e a r model r e q u i r e s s m a l l (AX ) model, specifications subjected to severe rectangular section sections. i s only necessary 21. ship section) f o r modelling referred using the stern three Reference coefficients, compares t h e to dimensions, 50-section number o f s e c t i o n s c h o s e n s o t h a t more d e t a i l i s the a simple from a by i t s possible tanker those from only 21. collapse, plating). specified F i g u r e 20 250 was (for a curvilinear sectional Reference Seaspan based were o b t a i n e d added mass and strip i t sections. compared Note t h a t t h e s e accurate since i t i s specified rectangular model was the and f o r each m o d e l , as r e q u i r e d . the near theory model, section where over some l e n g t h , section dimensions length. show the the respectively. dimensions Seapan 250 The of motion response in typical loaded the the rectangular light model p r i m a r i l y f o r a form discussed p r e v i o u s l y , while for similar light model and serves t o the is loaded of as loaded more 29 interest of f o r slamming t h e 25 0, model, compared Figure cross-section respective light Motion that 22 vertical Sensitivity here {4) u n i t results 26 the with displacement become very equal draft linear Tf to in and strip tanker vertical shown responses and s h i f t e d velocity at forefoot. , in from theory, are shifted i s not e n t i r e l y a (2) r e l a t i v e theory f o r the rectangular are the 2.2.3, displacement zero) wave a m p l i t u d e response i n t h e same d i r e c t i o n at The models f o r Section vertical (3) r e l a t i v e critical to reproduce different, 23. of S e c t i o n a l Parameters form 45 d e g r e e h e a d i n g 24 2 1. those {near a t 1>/Lw f o r absolute relative magnitudes {Note t h a t f o r since angle. as somewhat "peak" oppositely. valid, Figures Reference i s observed a t x=L/2, w h i l e f o r x=I/4 t h e travelling Figure w h i l e o f t h e same q u a l i t a t i v e x=L/4, t h e c o m p a r i s o n was in a r e : (1) a b s o l u t e more p r e c i s e s t r i p shift section p l a n o f t h e 250, s h o w i n g as d i s c u s s e d the corresponding =1), and f a i l A rectangular i n T a b l e I I I and IV. to Variation at forefoot from U n i t heave r e s u l t s , = 1.5. the l i n e s profile Results model o f t h e l o a d e d through the f o r the rectangular section drafts displacement The w data Theory be u s e d at f o r e f o o t L/L from of condition i s given {at t h e bow, f o r from those measures o f r e s p o n s e , will section i s inertial 4,3 L i n e a r S t r i p The with F i g u r e 21 shows t h e how shapes a t s t a t i o n s s p e c i f i e d and l o a d e d 4.3,1 results. the The more tanker exact 30 version of s t r i p theory incorporates rectangular-section from Reference x=L/2 case response 21 are isolated In was 3 m/s)• probably due for the value of U for the t o v a r i a t i o n s i n values of the unit these i n Reference 21,, addition errors to The t o the t a n k e r feature, 2 does n o t * large pitching, Unfortunately, approximation while model o f C h a p t e r coefficient distance. this differences magnified pitching by the coefficients i n t r o d u c e d by model, The the rough large were not 3-section the f o l l o w i n g should also be considered: (1) v a r i a t i o n s to the i n the assumed L/Lu; parameter are waterline length directly proportional f o r the r e c t a n g u l a r s e c t i o n model. „ |2) freguency limits, damping c o e f f i c i e n t s C). For the f o r the s e c t i o n a l o f F i g u r e 7, added mass and must be rectangular-section tanker are: sectional calculated model {Appendix (L = 384 m) . U = 3 »/s U -3 U-Z./Z mfc T=io 4.4 M 9.o T*lO 2.9 +8 S.4- m m I t i s p o s s i b l e to s h i f t the model closer model, t h r o u g h unless (i.e. these a more to those obtained careful f o r the manipulation rational more d e t a i l responses about actual the more e x a c t of s e c t i o n a l comparison the for o f t h e two tanker M/S 3-section strip theory parameters, but m o d e l s i s known dimensions), the 31 exercise would be of f u r t h e r comparative sources limited value. r e s u l t s between s h i p motion data from (experimental or otherwise) model were results not I t i s unfortunate that available, but other and the r e c t a n g u l a r s e c t i o n the indication (above) that are g u a l i t a t i v e l y reasonable i s enough j u s t i f i c a t i o n f u r t h e r c o n s i d e r the r e c t a n g u l a r s e c t i o n model. course, be necessary quantitative It would, to c o r r e l a t e r e s u l t s very c a r e f u l l y prediction for full-scale ships or to of before barges is considered. The the remainder of t h i s s e c t i o n c o n s i d e r s motion r e s u l t s rectangular section measures of response models of the Seaspan 250. (with the e x c e p t i o n of z ) r r e s p e c t t o wave amplitude, u n i t response for Since a l l are l i n e a r amplitudes with only s i l l be considered. Three typical values barge ("...the of U slamming incidence of have been speeds s e l e c t e d , on the b a s i s of mentioned in Reference n o t i c a b l e slamming decreases with and speed, and i s seldom a problem with . barges which 1. draft tow at speeds between 9 or 11 knots a t d r a f t s over 8 f e e t or s o " ) . The correspondence between L / L ^ , and frequency of encounter those is speeds shown in Figure 27, direction of save motion. Where JL/L W for opposite to Respective frequency l i m i t s are a l s o shewn, f o r the . l i g h t and loaded barge d e s c r i b e d i n 4.2. e noting that a l l cases c o n s i d e r e d are "head seas", with the s h i p advancing the (uJ ) values p l o t t e d f o r the motions exceed Section freguency l i m i t s , s e c t i o n a l added mass and damping c o e f f i c i e n t s are s e t to constant values (a =0.5, fa =0.0) respectively.,. 32 Figure bow, f o r the amplitudes shift shows t h e a b s o l u t e loaded occur towards L/I -3) w to 28 barge, i s the right near what we reach e q u i v a l e n t encounter Reference that 21 these independent linear states: 0 this theory". L/L near U. the Maximum values.. The ( f o r example, near f o r higher intuitively values appear values at speeds values. i t is resonance w "peaks" frequency d o e s not L/i.^, might expect "...while calculated of integral of response reverse of displacement as a f u n c t i o n o f L/L^, and i n a l l cases the vertical of to expected are be almost assured zero imply by the infinitely long waves. Figures the 29, 30, rectangular section displacement. shifts in 29), The values Chapter 2 variation 31, (and of theory because " A p p e n d i x C) of the barge. absolute variations reasonable. section afterbody when computed r e s u l t s these t o doubt their theory i s d e r i v e d from terras rigorous the 32 shows t h e are amplitude (particularly The terms to variation in Reference included". of Figure the agree made effect i s shown i n admissibility velocity unit seem 8 to 5 approximations model a r e i n c l u d e d , a c c o r d i n g t o The in small of vertical from mass and, damping that the afterbody parameters sections relatively are The of of with small a p p e a r t o be reason Figure number e f f e c t s o f added draft ... experiments the e f f e c t s of values c f L/Z.«i), s u g g e s t i n g f o r the loaded strip model on to the l e f t shown i n F i g u r e 30 low 31 i l l u s t r a t e Reducing the curve (Figure at and linear 4, better only with T h e r e i s some when the strip potential formulation. critical wave a m p l i t u d e (J£ ) for 33 the l i g h t The barge, values f o r might expect forefoot \j/ (relative reflects the not fact that since waves The we (that i s , shift in peak z i s an a b s o l u t e q u a n t i t y ~z K i s relative to the surface. Seaspan 2 50, (relative this at U=2.06 vertical threshold Criteria in are investigated sectcn are f o r the " l i g h t l y m/s. displacement vertical slamming linear and r e l a t i v e and velocity (Section m V* 0 V 0 time form are will = S.90 3.SS be beam dimensions Ws m/s „/s expressed in a where: T phase a n g l e s velocity) 2.2*3) , f o r 0 B' - IO non-dimensionalized vertical V V = 3,91 cycles, criteria variations. m m model slamming values, f o r t y p i c a l 0 loaded" wavelength, fi' = 5 B' =Z4.+ motion The f o r wave a m p l i t u d e 0 •= 2.06, h a s t h e f o l l o w i n g All occur). displacement. concern, long t o r e f e r e n c e s a x e s Oxz) w h i l e Results For infinitely should 4.3.2 L i n e a r Slamming The for co 0 —i** t o the absolute v e r t i c a l W/L^ - 0 a r e o f some near emergence amplitudes wave compared expressed ( c o s i n e wave a t t = 0 ) . . drr relative to the ocean wave 34 Figures cycle), light 33 and at the 34 show minimum the harmonic values of V relative impact, vertical slamming a r e shown. can t h e n (since ( f o r one F i g u r e 32, f o r t h e o c c u r s , and t h e velocity (magnitude), The wave a m p l i t u d e s be c a l c u l a t e d from c b a r g e . , The r e g i o n where slamming motions maximum at forefoot f o r slamming to r e s u l t s a r e l i n e a r with occur amplitude) as: L/Lui^l.8 L / L u ) 2.51 * 2 . $ Since waves that 3 ,» occur in nature is maximum wave a m p l i t u d e s maximum a m p l i t u d e - 3.7 L/c-tu - 2.9 maximum a m p l i t u d e - 2.3 figures difference velocity important curve indicate between i s always factor. that, vertical 90 d e g r e e s , ~ to ~> S.OZ 0.071 the ( f o r L =. 92.7 m) a r e : m m because displacement the relative and r e l a t i v e the f o r e f o o t decreasing ~ r a t i o f o r the s t e e p e s t The e f f e c t o f i n c r e a s i n g down, t h e r e b y 6*18 approximately L/L.0—I.Q The P 3.78 t h e wave a m p l i t u d e / w a v e l e n g t h corresponding z 2.OS -» relative draft (T^) phase vertical is an T^ i s t o s h i f t t h e vertical slamming 35 v e l o c i t y at f o r e f o o t impact 4.4 Quasi-Harmonic 4.4.1 Motion It (V ) (see Appendix C) . c Motion R e s u l t s Sensitivity should f i r s t he v e r i f i e d that the assumptions 3.2.2 f o r the n o n - l i n e a r theory are v a l i d section model. of an approximate smoothing result r e l a t i v e v e r t i c a l displacement) s m a l l piecewise s l o p e changes. presented f o r the rectangular R e l a t i v e v e l o c i t y w i l l not be c o n s i d e r e d , s i n c e differentiation quasi-harmonic of S e c t i o n The (i.e. the w i l l only magnify quasi-harmcnic curves are here i n the same form as the computed r e s u l t s , and no r o u t i n e s have been employed. R e s u l t s i n t h i s s e c t i o n correspond to the minimum v a l u e s of Vc from Figure amplitude, the pure the 32. next s e c t i o n . reasonably form The respectively, of the number of quasi-harmonic Although (calculated while variation the p i t c h over one motion that f o r unit wave curve i s a good approximation to illustrates waveform, constant shows using l a r g e "jump" a t p t . F i g u r e 36 guasi-harmcnic 35 quasi-harmonic harmonic coefficients). Figure in A i s d i s c u s s e d i n the the Figure rest-position 37 periodicity and magnitudes 38 of the show the and phase, u n i t response c o e f f i c i e n t s f o r heave and cycle. sections Figure 39 i s an i s not indication a c r i t i c a l parameter that f o r the technique. not i l l u s t r a t e d here, i t can a l s o be v e r i f i e d that 36 the s t a r t i n g p o i n t , o r i n i t i a l l y assumed values o f u n i t response c o e f f i c i e n t s , have no e f f e c t on the motion c y c l e ( t r a n s i e n t s are removed a f t e r the f i r s t time increment)» , 4.4.2 Slamming Criteria The a b i l i t y predict of the slamming, guasi-harmonic for the technique rectangular dependent on two f e a t u r e s a s s o c i a t e d with mass was data fitted f o r the ratio computer model. variations in slope a motion the motion unsteady responses, response realistic limit restricts cycle. on shallow Secondly, of as noting that (w.r.t. ^ ) aspect s i m i l a r waveforms are (Figure 41) produces an when the f o r e f o o t impacts the water s u r f a c e . those responses are too large to be a representation. r a p i d i n c r e a s e i n added mass that i s p a r t l y slamming the bow the , varies. The decreasing time i n t e r v a l The amplitudes Figure added f i t may l e a d t o e r r o r s i n added mass d e t e r m i n a t i o n , observed. for is (from F i g u r e 7a) i n the F i g u r e 40 demonstrates the n o n l i n e a r The model, sectional First, of the curve the d r a f t , and t h e r e f o r e <jJpf~ of the (2 < B/T < 8) a r t i f i c i a l l y d r a f t s e c t i o n s t h a t "occur" during numerical section directly from Reference 7, Chapter 2, and the way i n which i t breadth/draft the to (as d e s c r i b e d i n S e c t i o n 1.2.3) i s i l l u s t r a t e d i n 42, where the added mass f o r a r e c t a n g u l a r {N = 18 from Table of f o r e f o o t submergence. responsible section near I I I ) i n c r e a s e s r a p i d l y near the p o i n t 37 5. 5.1 CONCLUDING REMARKS Summary This thesis theory. Hhile analysis easier, investigates simplifications making use the of displacements, have been shows Although to theory to s h i p motions i s a simple make problem. r e c t a n g u l a r s e c t i o n barge model i s designed, complex and notation f o r the a p p l i e d t o two examples. loaded t a n k e r , as t e s t e d with the s t r i p 21, applied i t i s not intended t o imply t h a t the g e n e r a l a p p l i c a t i o n of s t r i p Firstly, two p a r t i c u l a r a s p e c t s of s t r i p qualitative results ship wave Comparison with a theory that and from appear Beference satisfactory. no comparison i s p o s s i b l e , a method f o r l a y i n g out the equivalent rectangular section model i s presented f o r the Seaspan 250 barge, Secondly, the quasi-harmonic motion formulation is presented, and r e s u l t s compared with the pure harmonic case f o r a rectangular section agreement with harmonic guasi-harmonic model model, , As w e l l as demonstrating results at moderate appears to indicate, l e a s t , a d i r e c t method f o r p r e d i c t i n g slamming p o i n t o f f o r e f o o t submergence. amplitudes, qualitatively motions, good the at at the 38 5.2 Recommendations f o r F u r t h e r Work Although this thesis has limited case, t h a t of r e c t a n g u l a r s e c t i o n work presented directions here comparison with experiments or s t r i p First, tested or s h i p form. could with general ship should be Two Both the particular would from actual approximation towing a strip the q u a s i - h a r m o n i c preferably obtained require motion to other degrees as an needs t o full-scale for a curvilinear f o r m u l a t i o n of coefficients for technique may which prove i s also a Chapter f o r a more results have Despite itself approximate n o n - l i n e a r c f freedom be or f o r slamming b e h a v i o u r . r e g u i r e d , the developed tank theory one time Extension results with section shape, computation specific more e x t e n s i v e s h i p theory been and evident. other very i n h a r m o n i c waves, to expansion. a p p l i e d using s t r i p previously to a theories. either Secondly, be from rectangular more f u l l y , results 3 results the bodies lends i t s e l f f o r f u r t h e r work a r e itself the useful solution. possibility. 39 Bibliography 1. Ward, T . , Private communication to author, 1979. 2. Korvin-Kroukovsky, B.V. and Jacobs, W.R., "Pitching and Heaving Motions of a Ship in Regular Waves", Transactions of the Society of Naval Architects and Marine Engineers, Volume 65, 1957, pp. 590-632. 3. Gerritsma, J . and Beukelman, W., "Analysis of the Modified Strip Theory for the Calculation of Ship Motions and Wave Loads", International Shipbuilding Progress, Vol. 14, 1967. 4. Salvesen, N., Tuck, E.O., and Faltinsen, 0., "Ship Motions and Sea Loads", Trans. SNAME, V o l . 81, 1971, pp. 250-287. 5. Wang, S. "Dynamical Theory of Potential Flows With a Free Surface: A Classical Approach t o - S t r i p Theory of Ship Motions", Journal of Ship Research, Vol. 20, No. 3, September 1976, pp. 137-144. 6. Loukakis, T.A. and Sclavounos, P.D., "Some Extensions of the Classical Approach to S t r i p Theory of Ship Motions, Including the Calculation of Mean Added Forces and Moments", Journal of ShiD Research, V o l . 22, 1978. 7. Newman, J . N . , "Marine Hydrodynamics", MIT Press, Cambridge, 1977. 8. Wehausen, J . V . , "The Motion of Floating Bodies", Annual Review of Fluid Mechanics, 1971. 9. Rawson, K.J. and Tupper, E.C., "Basic Ship Theory", V o l . 2, Longman, London, 1968. 10. St. Denis, M. and Pierson, W.J., "On the Motions of Ships in Confused Seas", Trans. SNAME, Vol. 61, pp. 280-357. 11. P r i c e , W.G. and Bishop, R.E.D., " P r o b a b i l i s t i c Theory of Ship Dynamics"; Chapman and H a l l , London, 1974. 12. Tick, L . J . , "Certain P r o b a b i l i t i e s Associated with Bow Submergence and Ship Slamming in Irregular Seas", Journal of Ship Research, No. 1, Vol. 2, June 1958, pp.30-36. 13. Bha^tachar.yaV R.v- "Dynamics of Marine V e h i c l e s " , Wiley and Sons, Toronto, 1978. 14. Jones, N., "Slamming Damage", Journal of Ship Research, V o l . 17, No. 2, June 1973, pp. 80-86. 40 15. Ochi, M.K. and Mottor, L.E., "Prediction of Slamming Characteristics and Hull Responses for Ship Design", Trans. SNAME, Vol. 81, 1973, pp. 144-176. 16. McLean, W. and Lewis, E.V., "Analysis of Slamming Stresses on S.S. Wolverine State', Marine Technology, January, 1973. 1 17. Nagai, T. and Chuang, S., "Review of Structural Response Aspects of Slamming", Journal of Ship Research, Vol. 21, No. 3, September 1977, pp. 182-190. 18. Comstock, J.P. ( e d i t o r ) , " P r i n c i p l e s of Naval A r c h i t e c t u r e " , SNAME, New York, 1967. 19. Ochi, M.K., "Extreme Behaviour of a ShiD in Rough Seas - Slamming and Shipping of Green Water", Trans. SNAME, Vol. 72, 1964, pp. 143-202. 20. P r i c e , W.G., Bishop, R.E.D. and Tarn, P.K.Y., "On the Dynamics of Slamming", Transactions of the Royal Institute of Naval A r c h i t e c t s , Vol.)2©, 1978. '•• O 21. P r i c e , W.G. and Bishop, R.E.D., "A Unified Dynamic Analysis for Ship Response to Waves", Trans. RINA, V o l . n S , 1977^' 22. Lewis, F.M., "The Inertia of Water Surrounding a Vibrating Ship", Trans. SNAME, Vol. 37, 1929, pp. 1-20. 23. Macagno, M., "A Comparison of Three Methods for Computing the Added Mass of Ship Sections", Journal of Ship Research, Vol. 12, No. 4, December 1968, pp. 279-284. 24. Vugts, J . H . , "The Hydrodynamic Coefficients for Swaying, Heaving and Roiling Cylinders in a Free Surface", International Shipbuilding Progress, Vol. 15, 1968r 25. Golovato, P., "A Study of the Forces and Moments on a Surface Ship Performing Heaving O s c i l l a t i o n s " , David Taylor Model Basin Report No. 1074, 1956. 26. Hughes, P.C., "Deployment Dynamics of the Communications Technology S a t e l l i t e - A Progress Report", Transactions of the Canadian Aeronautics and Space I n s t i t u t e , March 1974, pp. 10-18. Appendix A Solution, of equation 2 . 1 : \_-wlA * - iuf B e c] Solve'for unit response c o e f f i c i e n t s K, (cu^) . K ( ^e) l 2 where: or Z/XcOe) - T CWe~)~ 2 let B -ccu [ D ] e e which can be expressed c] X,W) -D>] (ou ) - as: K = &.-T, ;~' GPS where: Or + Cu B e 3=- <»l AIM. - ^A 2 L - CUJ^B,^ - i LU B 9 2 / + Cz, Appendix B Unit draft c r i t i c a l wave amplitude y c : D e f i n i t i o n : For harmonic motions, unit draft c r i t i c a l wave amplitude is that wave amplitude that w i l l cause the forefoot r e l a t i v e vertical displacement to equal zero at one point on a motion c y c l e , where the forefoot position is defined at x=x^., _z=-l, i . e . unit d r a f t . #-# at some t*t* so that % ~ , t t*)*0 z,(x , t f / The minimum value .o-CJs£_that w i l l satisfy equation B l (above) w i l l be designated ^ - Recalling that only the real part is considered, we have Thus. Appendix C Numerical curve f i t for sectional added mass and damping c o e f f i c i e n t s : Sectional added mass and sectional damping c o e f f i c i e n t s are calculated (see FORTRAN routines, Appendix D) by use of the following polynomial, f i t t e d to the curves of Figure 7. y = p7 *2 +• p9 p,60 ?*(*,)* M +"'• Pd 6</)*i + ,+ Z : f> (*J+ p* 3 ?io (Xt)**, fx!+Pf(**) p< 3+ + Pa X, X + p z /£ . where: V % - ~ = — (respectively, for added mass and damoing) The respective c o e f f i c i e n t s P ' . , ^ a r e given in Appendix D. For both sectional added mass and sectional damping c o e f f i c i e n t s , the range of frequencies considered w i l l be limited to those shown in Figure C-l and Figure C-2. In addition, the values of breadth/draft r a t i o are r e s t r i c t e d toT2^:B^T"<8. Figure C - l - Numerical curve f i t to sectional damping curves for rectangular sections Figure C-2 - Numerical curve f i t to sectional added mass curves for rectangular sections \ \ reference 7 Figure C-3a - Comparison of sect.ional added mass curves (B/T=4) 47 Table I - Coefficients for Equation v f r . l , 4 (from Salvesen, Tuck and Faltinsen ) ft,, A B/2 B = 3' fa c/x ' C= - jfe 6g + m n /2 £\ 2I — / *w* / 6 B A J x * * ^YY Ua* = /2 B, = 2 /x%Jx C'//„ r g <V 4 vfc'faf. %lx*b * n = ^sfxBJx. r C/2 where: sectional added mass c o e f f i c i e n t sectional damping c o e f f i c i e n t CC^ ^ } N - respective sectional c o e f f i c i e n t s for aftermost position of aftermost section 0 - 0 A" ) ft ii is f\ B„ Ui when U=0 (respectively) section Table I - Coefficients for Equation 2.1 (continued) Forcing terms: T = % fe'**e"" {*[ B -cu6*„a.-U>,)] T ?9 z where: wave amplitude k = wave number / = draft at aftermost section Note 1 A l l integrals are over the length of the ship. 2 Coefficients are v a l i d for head seas only. Table II - Coefficients for a Single Section of the Barge Model (derived from Table I) Main-body c o e f f i c i e n t s (Integral terms) Rectangular Afterbody c o e f f i c i e n t s O f L/ C4 / A\Z/*, tJ -r sr* A -DA -DA 6 22 = C „ = 9 9 ^« ^ O O Q/ - C/2 r - U o - o 2 2 where: D, 2 D 3 Table II - Coefficients for a Single Section of the Rectangul Barge Model (derived from Table I) (continued) Forcing terms (for section n) Forcing terms:(for afterbody section) wherer C/ = £g B h = ; — UJUJ <Zn - icUh^ e 4- c Uu)cc* .n B 1 LIGHT LOADED LIGHT LOADED T. n n B„ 11 24.4 6.0 1.6 24.4 1.9 5.7 12 24.4 6.0 1.6 24.4 1.9 5.7 13 24.4 6.0 1.6 24.4 1.9 5.7 14. 24.4 6.0 1.6 24.4 1.9 5.7 4 24.4 6.0 1.6 24.4 1.9 5.7 16 23.8 5.8 1.6 23.2 1.8 5.7 17 22.9 5.5 1.6 21.2 1.7 6.7 18 21.2 5.2 3.6 16.7 1.6 7.7 19 16.2 4.1 4.4 8.3 0.9 8.5 20 7.8 1.6 4.8 7.8 -2.5 8.9 6.0 -53.6 24.4 1.2 2.1 24.4 -2.9 6.2 - 2 24.4 2.7 1.8 24.4 -1.4 5.9 3 24.4 4.2 1.6 24.4 0.1 5.7 4 24.4 5.7 1.6 24.4 1.6 5.7 5 24.4 6.0 1.6 24.4 1.9 5.7 - •43.9 6 24.4 6.0 1.6 24.4 1.9 5.7 7 24.4 6.0 1.6 24.4 1.9 5.7 - 8 24.4 6.0 1.6 24.4 1.9 5.7 9 24.4 6.0 1.6 24, 4 1.9 5.7 10 24.4 6.0 1.6 24.4 1.9 5.7 24.4 5 -29.3 •23.0 -5.0 6.0 29.3 34.1 39.0 -17.0 •11.0 18.0 21.0 •34.1 Table III 12.0 -48.8 -9.0 ; - 43.9 48.8 53.6 - Rectangular Section Model of Seaspan 250 Specifications (20 sections) cn 52 Figure 1 - Typical relationships between motion spectra and wave energy spectra Figure 2 - Slamming stresses superimposed on bending stress cycle (3) Relative angle at forefoot Figure 3- Conventional l i n e a r slamming c r i t e r i a 56 Figure 5 - Method for f i t t i n g rectangular to c u r v i l i n e a r section section Figure 6 - V e r t i c a l O s c i l l a t i o n s of a Rectangular Section on a Free Surface 58 59 eg © n A n+1 The center of gravity of section n i s assumed to be located at x=x , 0 . z = n jTD ) (T~ y-Y/'n h — , ~>B J£,, m a s °f section n s n n m f 12 [ (x»>o)~ y 2 '» * x a ? j_ momoment r of i n e r t i a of sectionabout transverse axis through " J ab( <v°> thus: — f * r > n and: YY Figure 8 - Moment-! of i n e r t i a approximations f o r the rectangular section barge model Figure J.Q; - Harmonic wave notation z,V t z'(t) Figure 11a - Absolute v e r t i c a l displacement and absolute vertical-veloci-ty Figure 11c - Relative v e r t i c a l displacement and r e l a t i v e vertical velocity 'A'denotes the region where slamming may occur, i . e . where v^ 0, z r 0 at the f o r e f o o t . zj r i s the amplitude of z ( t ) about z - T f r r = Jv j i s the amplitude of v ( t ) about v =0 v'*' i s the value of r e l a t i v e v e r t i c a l v e l o c i t y v^(t) at the point of '.' r r r i n i t i a l forefoot submergence, given by: Figure 12 - Phase r e l a t i o n s h i p s f o r l i n e a r slamming c r i t e r i a 63 z (x^)'<0 \ r CASE" 1 z ( - r t z (x )< 0 r j,' X l . J ) x t r 1 v (\i)<0 2 CASE Jt\ X 2 r (x. )>0 N. S t: i x r X r z (x .)>.0 r T s CASE <3X 1 z (x 4<0 r x" p R z (x )>0 r v CASE-.4 1 VV > 0 X 1 Figure 13 - Cases of sectional emergence considered Xr 64 freeboard wave still water Figure 14a - Wave p r o f i l e ~ a n d ~ s t i l l , water p r o f i l e Vl T = maximum draft m Figure-14b - Freeboard allowance for a section FROM CALLING ROUTINE: Barge and wave parameters *-j»f.j> ^ i ' ^ n ' ^ ' ^ l 2 x -**-— r X* X l * X* T %(x )/2 r T i + T r r: x 2.1 r RETURN T. x arrays Figure 15 - Flowchart for Calculation of Motion Coefficients 66 FROM CALLING ROUTINE Barge parameters: K^K^, U Section parameters: x-| ,x ,T Wave parameters: w,t I x = x,= (x^j/2 (x-,+x )/2 x = r x l (x +x )/2 1 •• r X RETURN x Figure 16 - Flowchart for calculation of wave-surface intercept for a section Figure 17 - Graphical construction of quasi-harmonic response curve 68 F(t)=F Sin wt Q ^ x=0 Figure 18a - Forced single-degree-of-freedom system, to i l l u s t r a t e quasi-harmonic v i b r a t i o n Figure 18b - Forced o s c i l l a t i o n at d i f f e r e n t values of spring s t i f f n e s s Figure 19 - Quasi-harmonic motion cycle for single-degree-of-freedom system <JD B=35 m 30 nr B=45-m B=45 m •162 rai- 162 m 384 m Rectangular section model: L=384 m Total weight = 2834 Tanker specifications (from Ref. 21): L=384 m Total weight = 2792 MN Figure 20 - Tanker from Reference 21 and equivalent rectangular section model B=35 tn 30 m 1 PLATE 1 SEASPAN 250 - OVERALL DIMENSIONS PLATE 2 SEASPAN 250 - CROSS-SECTIONS Figure 22 PLATE 3 SEASPAN 250 - BOW Figure 23 PROFILE Figure 24 - Unit .heave response, tanker, U=9 m/s 72 rectangular section model 0 1 2 3 4 L 5 6 7 / w L Figure 25 - Absolute v e r t i c a l displacement at x=L/4, tanker, U=2.12 m/s 73 Figure 26 - Absolute v e r t i c a l displacement at x L / 2 , tanker, U=3 m/s = 1.0 Y Figure 28 - Absolute v e r t i c a l displacement at forefoot, loaded barge, 11=2.06,4.11, 6.17 m/s L/L w Figure 29 - Absolute v e r t i c a l displacement at forefoot, light barge, 5 and 8 sections. 11=2.06 m/s Figure 32 - Unit draft c r i t i c a l wave amplitude and absolute v e r t i c a l displacement, l i g h t barge, U=2.06 m/s Fiqure 33 - Relative v e r t i c a l displacement and r e l a t i v e v e r t i c a l v e l o c i t y , l i g h t barge, U=2.06 m/s, L/U=l:8 ,3.0 Figure 34 - Relative v e r t i c a l displacement and r e l a t i v e vertical v e l o c i t y , light barge, U=2.06 m/s, L/L =2.9 OO 00 Figure 39 - Quasi-harmonic r e l a t i v e v e r t i c a l displacement, for 13 and 20 sections, l i g h t U=2.06 m/s, L/L =2.,9 barge, IO.O r 8.0 • -8.0 • - i o : o *• Figure 40 - Ouasi-harmonic r e l a t i v e v e r t i c a l displacement for three wave amplitudes, U=2.06 m/s, L/L =2.9 w oo 00 CO 00 to
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Slamming motions of a rectangular-section barge model in harmonic waves Worden, Douglas Neil 1980
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Title | Slamming motions of a rectangular-section barge model in harmonic waves |
Creator |
Worden, Douglas Neil |
Date Issued | 1980 |
Description | The work presented in this thesis concerns the theoretical analysis of the motion of floating rectangular cross-section bodies in single-frequency harmonic waves. When a conventional laterally symmetric ship is modelled by such a body, the computation of strip-theory coefficients (derived from the solution of Laplace's equation for the fluid surrounding the ship) is simplified. This technique is used here to model a typical barge, with actual cross-sections very close to the assumed rectangular shapes. In particular, slamming motions are investigated using two conventional linear slamming criteria. The rectangular section model is also applied to the investigation of slamming motions by use of 'quasi-harmonic' slamming criteria, which are developed from an updating technique used with conventional strip theory coefficients. Results are presented for an example. |
Subject |
Ships --Hydrodynamic impact --Mathematical models Wave-motion, Theory of |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-03-22 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080801 |
URI | http://hdl.handle.net/2429/22226 |
Degree |
Master of Applied Science - MASc |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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