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Slamming motions of a rectangular-section barge model in harmonic waves Worden, Douglas Neil 1980

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( SLAMMING MOTIONS OF A RECTANGULAB-SECTIQN BARGE MODEL IN HAaHONIC WAVES by DOUGLAS NEIL WOEDEN B. A, Sc, , 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 OF THE REQUIREMENTS FOE THE DEGREE CF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Department of Mechanical Engineering) We accept t h i s t h e s i s as conforming to the r e g u i r e d s tandard., THE UNIVERSITY OF BRITISH COLUMBIA December, 1979 Jc) Douglas N e i l Worden, 1979 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Brit ish 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 Brit ish Columbia 2075 Wesbrook Place Vancouver,. Canada V6T 1W5 D a t e 21 Apri l 1980 11 Abstract 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 latera l ly symmetric ship is modelled by such a body, the computation of strip-theory coefficients (derived from the solution of Laplace's equation for the f lu id surrounding the ship) is s implif ied. 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 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 strip theory coeff ic ients. Results are presented for an example. ill Table of Contents page Abstract i i L ist of Tables v List 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 Probabil ist ic Ship Motion Theory 4 1.2.3 Slamming 6 1.3 Objectives and Scope 8 1.3.1 Conventional Linear Slamming Cr iter ia 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 Strip Theory 10 2.1.2 Simplifying Assumptions (Strip Theory) 12 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 Oscil lations 4. RESULTS 4.1 Purpose 4.2 Model Specifications 4.3 Linear Strip Theory Results 4.3.1 Motion Sensit ivity to Variation of Sectional Parameters 4.3.2 Linear Slamming Cr i ter ia 4.4 Quasi-Harmonic Results 4.4.1 Motion Sensit ivity to Variation of Sectional Parameters 4.4.2 Slamming Cr iter ia 5. CONCLUDING: REMARKS 5.1 Summary 5.2 Recommendations for Further Work Bi bliography Appendix A Appendix B Appendix C List 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 List of Figures .page 1 Typical relationships between motion spectra and wave 52 energy spectra 2 Slamming stresses superimposed on bending stress cycle 53 3 Conventional l inear slamming c r i ter ia 54 4 Rectangular section barge model 55 5 Method for f i t t ing rectangular section to curvil inear 56 section 6 Vertical osci l lat ions of a rectangular section on a 57 free surface 7 Added mass and damping coefficients for rectangular 58 sections 8 Moment of inert ia approximations for the rectangular 59 section barge model 9 Fluid coordinates and ship coordinates 60 10 Harmonic wave notation 60 11a Absolute vertical displacement and absolute vertical 61 velocity l ib Wave amplitude and vertical velocity 61 11c Relative vertical displacement and relative vertical velocity 61 12 Phase relationships for linear slamming c r i te r i a 62 13 Cases of sectional emergence considered 63 14a Wave prof i le and s t i l l water prof i le 64 14b Freeboard allowance for a section 64 15 Flowchart for calculation of motion coefficients 65 16 Flowchart for calculation of wave-surface intercept 66 for a section 17 Graphical construction of quasi-harmonic response 67 curve 18a Forced single-degree-of-freedom system, to 68 i l lus t rate quasi-harmonic vibration v i i 18b Forced osc i l lat ion at different values of spring stiffness 19 Quasi-harmonic motion cycle for single-degree-of freedom system 20 Tanker from Reference 21 and equivalent rectangular section model 21 Seaspan. 2-50 overall dimensions. 22 Seaspan 250 cross-sections 23 Seaspan 250 bow prof i le 24 Unit heave response, tanker, U=9 m/s 25 Absolute vertical displacement at x=L/4, tanker, U=2.12 m/s 26 Absolute vertical displacement at x=L/2, tanker, U=3 m/s 27 Frequency of encounter vs. shiplength/wavelength rat io, L=107.2 m 28 Absolute vertical displacement at forefoot, loaded barge, U=2.06, 4.11, 6.17 m/s 29 Absolute vertical displacement at forefoot, l ight barge, 5 and 8 sections, U=2.06 m/s 30 Absolute vertical displacement at forefoot, l ight barge, U=2.06 m/s 31 Absolute vertical displacement at forefoot, loaded barge, U=2.06 m/s 32 Unit draft c r i t i c a l wave amplitude and absolute vertical displacement, l ight barge, U=2.06 m/s 33 Relative vertical displacement and relative vertical velocity, l ight barge, U=2.06 m/s, L/L =1.8 w 34 Relative vertical displacement and relative vertical velocity, l ight barge, U=2.06 m/s, L/L =2.9 w 35 Harmonic and quasi-harmonic relative vertical ;, displacements, l ight barge, U=2.06 m/s, L/L =2.9 w 36 Quasi-harmonic relative vertical displacement, for 3 motion cycles, l ight barge, U=2.06 m/s, L/L =0.1 w page 68 69 70 Plate 1, p- lOa Plate 2, p. lot, Plate 3,p. 70c 71 72 73 74 75 76 77 78 79 80 81 82 83 vi!i i page 37 Magnitudes of quasi-harmonic response coeff ic ients, 84 l ight barge, U=2.06 m/s, L/L =2.9 w 38 Phase angle of quasi-harmonic response coeff ic ients, 85 l ight barge, U=2.06 m/s, L/Lw=2.9 39 Quasi-harmonic relative vertical displacement, for 86 13 and 20 sections, l ight barge, U=2.06 m/s, L/L =2.9 w 40 Quasi-harmonic relative vertical displacement for 87 three wave amplitudes, U=2.06 m/s, L/L =2.9. w 41 Quasi-harmonic relative vertical displacement for 88 two values of t ' , l ight barge, U=2.06 m/s, L/L =2.9, =2.3 m *" w 42 Quasi-harmonic relative vertical displacement and 89 sectional added mass for a section close to the \ forefoot, l ight barge, U=2.06 m/s, L/Lw=2.9 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 Brit ish Columbia, during the period January 1978 - December 1979. Mr. Tom Ward, of Seaspan International Ltd., provided drawings and specifications of the 'Seaspan 250' barge.ywhdch'were sincerely appreciated.' The graduate students in the Mechanical Engineering department, particularly Ken Lips and Mitchell Wawzonek, provided technical assistance and moral support whenever i t was required, for which I am most grateful. My .wife, Eleanor, deserves praise for her patience. Nomenclature 2x2 matrices of coefficients (real) for coupled motion equation ship beam (m) beam for one section (m) typical dimension for calculation of vertical slamming velocity constants for forcing terms of coupled motion equation (complex) 2 moment of inert ia about y-axis (tonne-m ) complex amplitudes of heave and pitch forcing functions respective heave and pitch forcing terms arising from afterbody section heave unit response amplitude (m) pitch unit response amplitude (rad) ship length (m) wave length (m) ship mass (tonnes) f lu id 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 vertical slamming velocity (m/sec) relative vertical velocity 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 coeff icient (tonne/m) sectional damping coeff ic ient (tonne-sec/m) respective sectional coefficients for after-most 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 vertical velocity at forefoot (m/sec) relative vertical velocity at forefoot (m/sec) aftermost x-coordinate for a section (m) forwardmost x-coordinate for a section (m) Xn+l" xn (x ± 1+x )/2 •V n+1 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 vertical displacement at forefoot (m) relative vertical displacement at forefoot (m) heave amplitude (m) pitch amplitude (rad) __wave amplitude (m) maximum wave amplitude (m) vertical 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 ) relative angle at forefoot (rad) _wave frequency (rad/sec) frequency of encounter (rad/sec) 1 1-. I N T R O D U C T I O N 1,1 P r o b l e m D e s c r i p t i o n B a r g e t o w i n g o p e r a t i o n s o n t h e w e s t c o a s t o f B r i t i s h C o l u m b i a a r e o f t e n c a r r i e d o a t d a r i n g s e v e r e w i n t e r w e a t h e r c o n d i t i o n s , w i t h e x t r e m e w a v e h e i g h t s a common . f e a t u r e , . When c o m b i n e d w i t h . c e r t a i n t o w i n g c o n f i g u r a t i o n s ( t o w i n g v e l o c i t y , h e a d i n g a n d b a r g e l o a d i n g ) s u c h c o n d i t i o n s r e s u l t i n s l a m m i n g m o t i o n s o f t h e b a r g e . T h i s c a n c a u s e d a m a g e t o h u l l p l a t i n g n e a r t h e b o w , S l a m m i n g a s s o c i a t e d w i t h t o w e d b a r g e s i n v o l v e s i m p a c t b e t w e e n t h e h u l l b o t t o m ( n e a r t h e f o r e p e a k ) a n d t h e w a t e r s u r f a c e , a s o p p o s e d t o " b o w f l a r e " s l a m m i n g a s s o c i a t e d w i t h c o n v e n t i o n a l s h i p s . A l t h o u g h b o t t o m s l a m m i n g h a s b e e n s t u d i e d e x t e n s i v e l y f o r c o n v e n t i o n a l s e l f - p r o p e l l e d s h i p s , b a r g e s a r e m o r e s u s c e p t i b l e t o d a m a g e b e c a u s e o f t h e i r i s o l a t i o n f r o m t h e t u g b o a t c r e w . B h e n t h e b a r g e b e g i n s t o s l a m , t h e m a s t e r o f t h e t u g may c o n t i n u e h i s t o w , u n a w a r e o f a n y p r o b l e m ( w h e r e a s o n a c o n v e n t i o n a l s h i p s l a m m i n g w o u l d b e n o t i c e d i m m e d i a t e l y , d u e t o h u l l v i b r a t i o n ) . 1 R e c e n t c a s e s o f d a m a g e , o f t h e t y p e d e s c r i b e d a b o v e , r e p o r t e d b y S e a s p a n I n t e r n a t i o n a l ( t h e l a r g e s t o f t h e B r i t i s h C o l u m b i a t o w i n g c o m p a n i e s ) l e d t o t h e i n v e s t i g a t i o n s o f b a r g e s l a m m i n g d e s c r i b e d i n t h i s t h e s i s . , 2 1.2 State of the Art 1.2,1 .Two-Dimensional S t r i p Theory (Heave-Pitch) The theory of s h i p motions i s the b a s i s of c o n v e n t i o n a l a n a l y s i s of s h i p slamming . An understanding of the background of l i n e a r s h i p motion theory i s necessary t o a p p r e c i a t e s i m p l i f y i n g assumptions that w i l l be i n t r o d u c e d to develop a l i n e a r barge slamming model, as w e l l as a " n o n - l i n e a r " s h i p motion model. Sea c o n d i t i o n {generally parameterized by waveheight and wave frequency) i s a s t a t i s t i c a l q u a n t i t y , so t h a t s h i p motions i n g e n e r a l , and slamming motions i n p a r t i c u l a r , a r e d i s c u s s e d i n p r o b a b i l i s t i c terms. 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 by shape, mass, and a p p r o p r i a t e moments of i n e r t i a , heading r e l a t i v e t o wave d i r e c t i o n , and speed, ,, I t has been shown t h a t f o r c o n v e n t i o n a l l a t e r a l l y symmetric s h i p s i t i s reasonable, f o r s m a l l motions, to uncouple the ge n e r a l six-degree-of-freedom motion problem i n t o "roll-sway-yaw*} motions and "h e a v e - p i t c h " motions (surge i s g e n e r a l l y o m i t t e d ) . The coupled heaving and p i t c h i n g f o r m u l a t i o n w i l l be the b a s i s f o r a l l motion: theory c o n s i d e r e d i a t h i s t h e s i s . Coupled heaving and p i t c h i n g motions can be represented as s h i p responses to a f o r c i n g i n p u t ( s u r f a c e waves). The s i m p l e s t case i s t h a t o f s i n g l e - f r e q u e n c y harmonic waves. For l i n e a r t h e o r i e s , response amplitudes f o r harmonic responses w i l l be d i r e c t l y p r o p o r t i o n a l to wave h e i g h t , a l l o w i n g use of 3 s u p e r p o s i t i o n techniques ( S e c t i o n 1 . 2 . 2 ) . The two-dimensional (heave-pitch) l i n e a r model f o r r i g i d c o n v e n t i o n a l s h i p s i n l o n g - c r e s t e d harmonic waves i s w e l l known. I t i s evaluated as a " s t r i p theory", and was f i r s t developed i n p r a c t i c a l computational . form by Korvin-Kroukovsky and J a c o b s 2 , They determined a method t o r a t i o n a l l y account f o r c o u p l i n g between heave and p i t c h motions., Other a u t h o r s , most n o t a b l y Gerritsma and Beukelman 3, and Salvesen, Tuck a n d . F a l t i n s e n * have presented t h e i r own s t r i p t h e o r i e s , of the same form as Korvin-Kroukovsky and Jacobs, but with s l i g h t l y d i f f e r e n t c o e f f i c i e n t s . F u r t h e r study of s h i p motion theory has i n c l u d e d a n a l y s i s of coupled motions f o r f i v e degrees of freedom, by 8ang s and Loukakis and S c l a v o u n o s 6 . S t r i p theory f o r m u l a t i o n s a r i s e from hydrodynamic v e l o c i t y p o t e n t i a l ; t h e o r y . I t i s necessary to s a t i s f y : £ = O (1.1) where: (j) =• (J) (J)^ = t o t a l v e l o c i t y p o t e n t i a l = i n c i d e n t v e l o c i t y p o t e n t i a l ( d u e t o ocean <waves) (f) = d i f f r a c t e d v e l o c i t y p o t e n t i a l ° (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 induced v e l o c i t y p o t e n t i a l with the boundary c o n d i t i o n s : (1) a r i s i n g from the eguations of motion of the body and (2) on f l u i d boundaries, i . e . at the s u r f a c e , on s h i p s i d e s , and at i n f i n i t y . When t h i s i s done, and i t i s assumed t h a t the r e s u l t i n g motions are harmonic and l i n e a r , the r e s u l t can be expressed as coupled equations, with motion amplitudes as unknowns. The c o e f f i c i e n t s of these equations are obtained by i n t e g r a t i o n , i n an a p p r o p r i a t e d i r e c t i o n , of s h i p p r o p e r t i e s . The e f f e c t i s to reduce the hydrodynamic problem to a mechanical v i b r a t i o n problem. A d e t a i l e d treatment o f the v e l o c i t y p o t e n t i a l f o r m u l a t i o n i s presented by Newman7, Hehausen^ and the appendix t o the paper by Salvesen, Tuck and F a l t i n s e n * . The s t r i p theory from the l a t t e r i s used e x c l u s i v e l y i n t h i s t h e s i s , and i s presented i n more d e t a i l , and compared with other s t r i p t h e o r i e s , i n S e c t i o n 2. 1.1. 1.2.2 P r o b a b i l i s t i c Ship Hotion Theory The t h e o r y of motions i n r e g u l a r waves no l o n g e r can be considered by i t s e l f a l o n e . , I t i s only a p a r t of the p i c t u r e , the hydromechanical phase which e s t a b l i s h e s the dependence of a s h i p s motion 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 ) , the s h i p responses t o r e g u l a r waves or "response f a c t o r s " , a re then t r e a t e d by methods of mathematical s t a t i s t i c s i n c o n j u n c t i o n with a measured or assumed spectrum of a r e a l i s t i c i r r e g u l a r sea to g i v e the r e a l i s t i c s h i p motions. Korvin-Kroukovsky and J a c o b s 2 . Although t h i s t h e s i s w i l l not c o n s i d e r i n d e t a i l the methods used t o convert s i n g l e frequency harmonic i n p u t s i n t o p r o b a b i l i s t i c terms (that i s , s h i p response to a s t a t i s t i c a l s e t of waveheight-wavelength combinations), i t i s u s e f u l to o u t l i n e 5 the p r i n c i p l e s i n v o l v e d , s i n c e the development of s t r i p t h e o r i e s and p r o b a b i l i s t i c methods has not occu r r e d independently. I t i s necessary to o b t a i n a spectrum of sea c o n d i t i o n . For the case of heave and p i t c h motions, a u n i d i r e c t i o n a l spectrum of waveheight vs. wave frequency i s r e g u i r e d , adjusted f o r s h i p heading and speed. Response amplitude o p e r a t o r s are computed f o r a s h i p (as a f u n c t i o n of wave frequency and s h i p speed) using s t r i p t h e o r i e s as o u t l i n e d i n S e c t i o n 1.2. 1. When the wave spectrum i s expressed i n terms of wave energy ( p r o p o r t i o n a l to the sguare of the waveheight), and the response amplitude o p e r a t o r s (commonly termed "S. A.O. *s") are known, the s h i p motion energy spectrum can be expressed as f o l l o w s , ( f o r example, f o r heave motions) : s h i p motion enerqy spectrum response amplitude operator (heave motion) heave amplitude/wave amplitude a t one frequency wave energy spectrum I f the wave spectrum i s given by a Ray l e i g h d i s t r i b u t i o n , and m i s the area under the s h i p motion energy spectrum, then the motion amplitudes can fee d e f i n e d as f o l l o w s where: 6 2 average amplitude = L2&Jl^-n0 average of o n e - t h i r d h i g h e s t amplitudes 2.0 -/E An example of the r e l a t i o n s h i p between v a r i o u s s p e c t r a i s shown i n F i g u r e 1 {from Hawscn and T u p p e r 9 ) . P r o b a b i l i s t i c s h i p dynamics i s an important t o o l f o r the s h i p d e s i g n e r , a l l o w i n g him to compare performance over a wide range of sea c o n d i t i o n s . S t . Denis and P i e r s o n 1 0 were the f i r s t to apply l i n e a r s u p e r p o s i t i o n technigues t o s h i p motion t h e o r i e s , while more r e c e n t l y P r i c e and B i s h o p 1 1 have w r i t t e n a comprehensive i n t r o d u c t i o n t o p r o b a b i l i s t i c s h i p dynamics. P r o b a b i l i s t i c methods can a l s o be a p p l i e d to extreme motions (e.g. slamming), and l i c k 1 2 has made the r e q u i r e d a n a l y s i s f o r a s t a t i s t i c a l model of c o n v e n t i o n a l l i n e a r slamming c r i t e r i a , as d e s c r i b e d i n S e c t i o n 1.2.3. B h a t t a c h a r y y a 1 3 i s a u s e f u l r e f e r e n c e f o r the standard c a l c u l a t i o n s r e q u i r e d f o r p r o b a b i l i s t i c motion s t u d i e s . 1.2.3 Slamming Slamming has been widely s t u d i e d , f o r c o n v e n t i o n a l s e l f - p r o p e l l e d s h i p s , as one case of extreme s h i p response. I n v e s t i g a t i o n s are broadly c l a s s e d i n t o two areas. F i r s t , study o f the "hydrodynamic impact problem 1' i s concerned with experimental and 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 pressures c r e a t e d when a s h i p s e c t i o n (perhaps modelled by a 7 f l a t p l a t e o r wedge) impacts the water s u r f a c e , and the r e s u l t i n g response o f the s h i p g i r d e r t o t r a n s m i t t e d f o r c e s . H o d e l l i n g methods f o r s h i p response (and s t r u c t u r a l damage, where a p p l i c a b l e ) have been developed, and have l e d to the c l a s s i f i c a t i o n o f d i f f e r e n t t y p e s of slamming . behavior. The type a s s o c i a t e d most f r e q u e n t l y with towed barges (and s i m i l a r l y shaped v e s s e l s , such as o i l tankers) i s known as "bottom slamming". While damage i n the p a r t i c u l a r case t h a t l e d t o t h i s t h e s i s was c o n f i n e d t o l o c a l p l a t e c o l l a p s e near the bow, bottom slamming can generate s i g n i f i g a n t bending s t r e s s e s amidships, with magnitudes of the same order as t h e wave bending s t r e s s (Figure 2 ) . L o c a l s t r u c t u r a l response can be modelled as response t o a s i n g l e pressure p u l s e . I n v e s t i g a t i o n of pressures and responses has been undertaken by J o n e s 1 * ,Ochi and H o t t o r l s , and HcLean and L e w i s 1 6 (an e x p e r i m e n t al o b s e r v a t i o n of s t r e s s e s on a f u l l s c a l e s h i p ) . A c u r r e n t review (1977) of s t r u c t u r a l response theory i s contained i n a paper by Nagai and C h u a n g 1 7 . The second area of study, t h a t with which t h i s t h e s i s i s d i r e c t l y concerned, i s slamming occurrence. S t u d i e s of the hydrodynamic impact problem have shown t h a t bottom slamming r e s u l t s i n high v e r t i c a l a c c e l e r a t i o n s , f o r s h o r t time p e r i o d s , and have t h e o r i z e d t h a t the slam i s p a r t l y due to a high r a t e of change of added mass near the f o r e f o o t 1 2 , , U n f o r t u n a t e l y , n e i t h e r t h e a c c e l e r a t i o n nor the added mass v a r i a t i o n can be admitted t o a l i n e a r s t r i p theory model, s i n c e , as we w i l l see i n S e c t i o n 2. 1.1, the l i n e a r theory p r e d i c t s harmonic motions 8 {at the frequency of wave encounter) so that no hig h f r e q u e n c i e s are allowed, and the notion c o e f f i c i e n t s are eval u a t e d only once, i n the s t i l l - w a t e r r e s t p o s i t i o n , so t h a t no change i n added mass i s allowed, I n s t e a d , more s u i t a b l e slamming c r i t e r i a are s u b s t i t u t e d i n l i n e a r models, These a r e , from o b s e r v a t i o n , f a c t o r s present when bottom slamming o c c u r s , and i n order of importance a r e : {1) f o r e f o o t emergence (2) r e l a t i v e s h i p bottom - wave s u r f a c e v e r t i c a l v e l o c i t y and (3) angle between the s h i p hottom and wave s u r f a c e near the f o r e f o o t , , These are i l l u s t r a t e d i n Fig u r e 3, The advent o f very l a r g e o i l t a n k e r s has i n c r e a s e d i n t e r e s t i n extreme s h i p motions, s i n c e the tankers are s u b j e c t to l a r g e bending s t r e s s e s amidships. Recent slamming occurrence models have been presented by T i c k * 2 and O c h i * 9 , P r i c e , Bishop and Tam z° have c o n s i d e r e d slamming motions of an " e l a s t i c " s h i p , v i b r a t i n g i n v a r i o u s harmonic modes. 1,3 O b j e c t i v e s and Scope 1.3.1 C o n v e n t i o n a l L i n e a r Slamming C r i t e r i a The purpose of t h i s p r o j e c t , as o r i g i n a l l y s t a t e d , was to develop a mathematical model f o r p r e d i c t i n g the slamming motion of barges. As the p r o j e c t progressed, the o b j e c t i v e s were narrowed t o the f o l l o w i n g : {1)development of a model t o p r e d i c t the l i n e a r heave and p i t c h response of a r e c t a n g u l a r s e c t i o n barge {an 9 approximation to t y p i c a l barge shapes) i n r e g u l a r harmonic waves, u t i l i z i n g c o n v e n t i o n a l s t r i p theory t e c h n i q u e s . (2) i n v e s t i g a t i o n of the slamming occurrence of such a model, i n terms of c o n v e n t i o n a l l i n e a r slamming c r i t e r i a . E xtension to p r o b a b i l i s t i c motion r e s u l t s was not c o n s i d e r e d i n e i t h e r of the above cases. , 1.3.2 Quasi-Harmonic Hotion A n a l y s i s L i n e a r s h i p motion models, as e x p l a i n e d p r e v i o u s l y , cannot d i r e c t l y i s o l a t e the f a c t o r s a s s o c i a t e d with bottom slamming. In order t o do t h i s , a method was developed which p r e d i c t s response on a stepwise quasi-harmonic b a s i s over a motion c y c l e , e v a l u a t i n g motion c o e f f i c i e n t s on each i n t e r v a l . , . T h i s non-rlinear f o r m u l a t i o n , while s u b j e c t to r e s t r i c t i o n s noted i n Chapter 3, was developed and r e f i n e d f o r the r e c t a n g u l a r s e c t i o n model mentioned above. 10 2. L I N E AB MODEL (HARMONIC B E S P O N S E ) 2. 1 D e s i g n o f a S i m p l i f i e d L i n e a r M o d e l 2.1.1 S a l v e s e n - T u c k - F a l t i n s e n S t r i p T h e o r y T h e t w o - d i m e n s i o n a l ( h e a v e - p i t c h ) s t r i p t h e o r y t h a t w i l l b e u s e d i n t h i s t h e s i s i s f r o m S a l v e s e n , T u c k a n d F a l t i n s e n * ( h e n c e f o r t h r e f e r r e d t o a s S - T - F s t r i p t h e o r y ) . I t h a s b e e n w i d e l y e m p l o y e d f o r c o n v e n t i o n a l s h i p f o r m s , a n d s h o w s r e a s o n a b l e c o r r e l a t i o n w i t h e x p e r i m e n t a l r e s u l t s , a n d w i t h r e s u l t s f r o m o t h e r s t r i p t h e o r y f o r m u l a t i o n s . T h e S - T - F s t r i p t h e o r y w a s c h o s e n f o r t h i s t h e s i s b e c a u s e i t s u s e i s w e l l d o c u m e n t e d , a n d b e c a u s e i t i s o n e o f t h e m o r e r e c e n t s t r i p t h e o r y f o r m u l a t i o n s . T h e S - T - F s t r i p t h e o r y i s a l i n e a r t h e o r y , w h i c h i n t h i s t h e s i s w i l l m e a n t h a t p r e d i c t e d s h i p r e s p o n s e s a r e d i r e c t l y p r o p o r t i o n a l t o w a v e h e i g h t ( r e c a l l t h a t h a r m o n i c m o t i o n was a s s u m e d ) . T h i s i s a r e s u l t o f a s s u m i n g s m a l l ( r e l a t i v e t o s h i p d i m e n s i o n s ) m o t i o n s a b o u t s o m e s t i l l - w a t e r r e s t p o s i t i o n . , M o t i o n c o e f f i c i e n t s a r e e v a l u a t e d o n l y o n c e , f o r e a c h f r e g u e n c y , a t t h a t r e s t p o s i t i o n . F o r c i n g f u n c t i o n s a r e e v a l u a t e d f o r u n i t w a v e a m p l i t u d e , a l l o w i n g t h e s h i p r e s p o n s e s t h u s o b t a i n e d , t h e " u n i t r e p o n s e c o e f f i c i e n t s " f o r h e a v e a n d p i t c h m o t i o n s , t o b e m u l t i p l i e d b y a c t u a l wave a m p l i t u d e , s o t h a t s h i p r e s p o n s e f o r g i v e n w a v e f r e q u e n c y a n d a m p l i t u d e i s t h e r e s u l t . P r i c e a n d B i s h o p 2 1 , i n a n i n v e s t i g a t i o n o f t h e m o t i o n s o f 11 an o i l t a n k e r and a d e s t r o y e r , note the f o l l o w i n g d i f f e r e n c e s between r e s u l t s obtained f o r S-T-F s t r i p theory and the s t r i p theory of Gerritsma and Beukelman 3: "(a) t h e r e i s l i t t l e to choose between the two t h e o r i e s a t high v a l u e s of L/Lw (shiplength/wavelength r a t i o ) s i n c e the e f f e c t s o f f l u i d damping are then s m a l l . (b) t h e i r p r e d i c t i o n s are somewhat d i f f e r e n t a t low v a l u e s of L/L,y where wave response i s the more s e r i o u s problem, s i n c e f l u i d damping i s then dominant. (c) the d i f f e r e n c e s are i n the magnitudes of the resonant peaks {of the s h i p responses) r a t h e r than i n the l o c a t i o n s of those peaks i n terms of L/L^/" Eguation 2.1 i s the l i n e a r form of the coupled motion equations {in the "frequency of encounter" domain) f o r a l a t e r a l l y symmettric s h i p i n harmonic waves. Tab l e I l i s t s the r e s p e c t i v e S-T-F c o e f f i c i e n t s . In p r a c t i c e , f o r a m u l t i - s e c t i o n s h i p , the i n t e g r a l s i g n s are converted to summations. Equation 2. 1 i s r e p r e s e n t a t i v e of a two-degree-of-freedom coupled mechanical v i b r a t i o n system, with c o n s t a n t c o e f f i c i e n t s at a s p e c i f i e d frequency, and can be s o l v e d f o r the harmonic motion amplitudes by standard methods {Appendix A): ( 2 . 1) A / B t C = 2x2 matrices of c o e f f i c i e n t s {real) frequency of encounter amplitudes of heave and p i t c h motions {complex) amplitudes of heave and p i t c h f o r c i n g f u n c t i o n s (complex) 12 2.1.2 S i m p l i f y i n g Assumptions ( S t r i p Theory) Barge shapes are i n g e n e r a l much simpl e r t h a t those of c o n v e n t i o n a l s h i p s . Often the h u l l i s of n e a r l y constant c r o s s s e c t i o n shape over most of the l e n g t h , with s h o r t bow and s t e r n s e c t i o n s . S e c t i o n a l c o e f f i c i e n t s are c l o s e to u n i t y (except f o r c r o s s s e c t i o n s near the bow). I t would thus appear reasonable t o model a barge with a s e r i e s of r e c t a n g u l a r ( u n i t s e c t i o n a l c o e f f i c i e n t ) c r o s s s e c t i o n s (Figure 4 ) . The advantage to t h i s s i m p l i f i c a t i o n a r i s e s from the f a c t t h a t the c a l c u l a t i o n s of 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 , and i n e r t i a l c o e f f i c i e n t s , as r e g u i r e d f o r the S-T-F s t r i p t h e o r y , are q u i t e complex f o r a g e n e r a l c u r v i l i n e a r s h i p form. The r e d u c t i o n i n c a l c u l a t i o n time (on a d i g i t a l computer) t h a t i s obtained by s u b s t i t u t i n g r e c t a n g u l a r s e c t i o n s i s s i g n i f i g a n t , and u s e f u l f o r a " f i r s t approximation" technique. The r e d u c t i o n i n computing time becomes more s i g n i f i g a n t f o r the quasi-harmonic model of Chapter 3. The S-T-F c o e f f i c i e n t s f o r a r e c t a n g u l a r - s e c t i o n s h i p are presented i n Table I I i n s i m p l i f i e d form. F i g u r e 5 d e s c r i b e s the method by which a r e c t a n g u l a r s e c t i o n i s f i t t e d to a t y p i c a l c u r v i l i n e a r s e c t i o n , a t two w a t e r l i n e s . T h i s technique g i v e s reasonably c o r r e c t c r o s s - s e c t i o n a l areas f o r the r e c t a n g u l a r s e c t i o n approximation. 13 2.1.3 S e c t i o n a l Added Bass C o e f f i c i e n t s S e c t i o n a l added mass (for heave motion) can be p h y s i c a l l y d e s c r i b e d as that mass of water e n t r a i n e d (per u n i t length) by an i n f i n i t e l y long s h i p of constant c r o s s s e c t i o n when i t moves v e r t i c a l l y on a f r e e s u r f a c e (Figure 6)• The added mass depends on the p o s i t i o n (at some i n s t a n t ) of the o s c i l l a t i n g shape, i t s t r a n s v e r s e v e l o c i t y and a c c e l e r a t i o n at t h a t i n s t a n t and on c e r t a i n shape parameters. For l i n e a r models, and quasi-harmonic n o n - l i n e a r models as d i s c u s s e d i n Chapter 3, where motion c o e f f i c i e n t s are e v a l u a t e d at an instantaneous p o s i t i o n , the e f f e c t s of v e r t i c a l v e l o c i t y and a c c e l e r a t i o n a r e i g n o r e d , and the s e c t i o n a l added mass i s d e f i n e d as constant at a p a r t i c u l a r value of freguency of encounter. For g e n e r a l c u r v i l i n e a r s h i p s e c t i o n s , s e c t i o n a l added mass i s determined by a v a r i e t y of v e l o c i t y p o t e n t i a l methods (from s o l u t i o n s of Equation 1.1). L e w i s 2 2 was the f i r s t to present a s u i t a b l e a n a l y t i c technique. More exact methods, i n c l u d i n g the conformal mapping o f s h i p s e c t i o n s , have been used by Macagno 2 3 and V u g t s 2 * . Conformal mapping techniques have the disadvantage of r e q u i r i n g e x t e n s i v e computation. The values of s e c t i o n a l added mass, f o r r e c t a n g u l a r c r o s s - s e c t i o n s , t h a t w i l l be used i n t h i s t h e s i s are from Vugts, reproduced i n Newman7 (Figure 7a)., They are d e f i n e d f o r a range of values f o r frequency and breadth/depth r a t i o . The numerical values f o r modelling were obtained by a polynomial curve f i t to the graph. (Appendix C). The s e n s i t i v i t y of motion responses 14 t o 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 dis c u s s e d i n Chapter 4.. For the purpose of most l i n e a r s h i p motion t h e o r i e s , the s e c t i o n a l added mass f o r heave motion i s used f o r p i t c h i n g as w e l l , when m u l t i p l i e d by an a p p r o p r i a t e moment arm, 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 S e c t i o n a l damping (f o r heave motion) i s the damping f o r c e per u n i t length) on an i n f i n i t e l y long s h i p moving v e r t i c a l l y on a f r e e s u r f a c e . In mathematical terras, the problem of computing s e c t i o n a l damping i s s i m i l a r to that of computing s e c t i o n a l added mass. .,• S e c t i o n a l damping occurs i n the p o t e n t i a l flow model as a r e s u l t of waves generated by s h i p motions, s i n c e v i s c o u s damping i s not p e r m i t t e d i n the s o l u t i o n , , I t should be noted, however, t h a t damping a l s o occurs due to the r a t e of change of added mass at a s e c t i o n as the s h i p moves forward, so t h a t a s e c t i o n a l damping c o e f f i c i e n t equal t o zero does not n e c e s s a r i l y cause u n r e s t r a i n e d motion amplitudes. T h i s i s not t r u e where 0=0. (see, f o r example, P r i n c i p l e s of Naval A r c h i t e c t u r e * 8 , pp. 6 36)., The damping 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 are from the same s o u r c e 7 as the added mass c o e f f i c i e n t s , and are e v a l u a t e d n u m e r i c a l l y by a f i t to the graph (Figure 7b). I t i s c l e a r from the f i g u r e t h a t damping c o e f f i c i e n t s are s m a l l at high encounter f r e q u e n c i e s , so that s h i p responses are not very dependent on damping, as s t a t e d e a r l i e r . Motion s e n s i t i v i t y i s d i s c u s s e d i n more d e t a i l i n Chapter 4. 15 S e c t i o n a l damping f o r p i t c h i s obtained by m u l t i p l y i n g heave damping c o e f f i c i e n t s by an a p p r o p r i a t e moment arm. 2.1.5 Dynamic S i m p l i f i c a t i o n s The above s i m p l i f i c a t i o n s f o r a r e c t a n g u l a r c r o s s - s e c t i o n model may be extended by seme approximations f o r the i n e r t i a ! p r o p e r t i e s of the barge., The r e c t a n g u l a r - s e c t i o n barge w i l l be t r e a t e d as s e c t i o n s of constant d e n s i t y , with mass c e n t e r s on the r e s t p o s i t i o n w a t e r l i n e . Thus, the mass can be w r i t t e n as and the moment of i n e r t i a about the t r a n s v e r s e a x i s through the x=0 p o s i t i o n as: (Figure 8) Small changes i n the v e r t i c a l p o s i t i o n of the c e n t e r of g r a v i t y w i l l have a n e g l i g i b l e e f f e c t on motion r e s u l t s . V a r i a t i o n s i n l o n g i t u d i n a l p o s i t i o n s of the CG w i l l not be c o n s i d e r e d . 2.2 Equation of n o t i o n 2.2.1 Coordinate System and Wave Parameters F i g u r e 9 shows s h i p c o o r d i n a t e s 0'x*z* f i x e d t o the s h i p c e n t e r of g r a v i t y a t 0', and f l u i d c o o r d i n a t e s Oxz, where 0 l i e s on the undisturbed l e v e l s u r f a c e of the f l u i d , and t r a n s l a t e s i n 16 the p o s i t i v e x - d i r e c t i o n with speed U. Have speed c i s a l s o d e f i n e d i n the p o s i t i v e x - d i r e c t i o n . Only head seas w i l l be c o n s i d e r e d here. The wave p r o f i l e i s shown i n F i g u r e 10. The f o l l o w i n g wave and freguency parameters w i l l be used: , , 2-fT wave number k .= —.— wave frequency W freguency of encounter CUe = CO * kO shiplength/wavelength r a t i o = L/L„ Note that freguency of encounter can be expressed i n terms of shiplength/wavelength r a t i o as (for head seas) : 2.4 2.2.2 Complex Amplitude Formulation For the harmonic motions t h a t are assumed, complex n o t a t i o n i s a u s e f u l mathematical d e v i c e , Magnitudes and phase angles are e a s i l y preserved i n complex n o t a t i o n , so t h a t v e c t o r i a l a d d i t i o n of p i t c h and heave i s p o s s i b l e . Motion amplitudes (displacements and v e l o c i t i e s ) are understood to be the r e a l p a r t only of complex e x p r e s s i o n s . The wave displacement (in the v e r t i c a l d i r e c t i o n ) i s 17 Ship motions are harmonic, at the same frequency as the wave, so t h a t heave and p i t c h amplitudes are ( r e s p e c t i v e l y ) at x=z=0: where K, , K 2 a r e the " u n i t response amplitudes" obtained from the s o l u t i o n of Equation 2,1 f o r % = 1. -2.2*3 Measures of Response In S e c t i o n 1.2*3 three l i n e a r slamming c r i t e r i a were presented. Only two w i l l be used here, to s i m p l i f y the a n a l y s i s ( f o l l o w i n g T i c k 1 2 ) . These are: (1) R e l a t i v e displacement (with r e s p e c t to the wave surface) of the f o r e f o o t , which i s d e f i n e d f o r our r e c t a n g u l a r s e c t i o n model by (X/«,T^) . For a slam to o c c u r , the f o r e f o o t must be emerged. Note that i t i s not neccessary f o r the f o r e f o o t t o be w i t h i n the p h y s i c a l boundaries of the r e c t a n g u l a r s e c t i o n model. (2) R e l a t i v e v e r t i c a l v e l o c i t y of the f o r e f o o t . a slam occurs when (1) above i s s a t i s f i e d and the r e l a t i v e v e r t i c a l v e l o c i t y has the c o r r e c t phase r e l a t i o n s h i p (ship bottom approaching wave s u r f a c e ) with a magnitude exceeding a t h r e s h o l d v e l o c i t y V d, g i v e n by B h a t t a c h a r y a 1 3 as: Equation 2.9 a r i s e s from hydrodynamic impact t h e o r y , where B 1 i s a t y p i c a l l e n g t h s c a l e f o r movement of the f r e e s u r f a c e . , 18 The above c o n d i t i o n s can be s p e c i f i e d i n terms of the u n i t response amplitudes f o r the barge as f o l l o w s ( f o r some ): V e r t i c a l displacement of s h i p a t f o r e f o o t ( r e s t p o s i t o n w aterline) V e r t i c a l v e l o c i t y of s h i p at f o r e f o o t Haveheight at bow V e r t i c a l v e l o c i t y of wave s u r f a c e a t the bow R e l a t i v e v e r t i c a l displacement at f o r e f o o t 2.13 \*,Cxf>Trt)=z-y = Kfafa) +Kxfa)xf-e!kx*)e'ra#. -? 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 f o r e f o o t 2.if-V, (pcf)t) * 2 „ = -CiAJ e («, Ccoe) + Kz fa)Xf ~JLik**) I t i s a l s o u s e f u l t o d e f i n e the " u n i t d r a f t c r i t i c a l waveheight" which i s t h a t wave amplitude, f o r d r a f t a t the f o r e f o o t e q u a l to u n i t y , t h a t causes the f o r e f o o t t o j u s t emerge a t one p o i n t on the harmonic motion c y c l e (derived i n Appendix B) . , 19 Equations 2.3 .through 2.14 are presented g r a p h i c a l l y i n F i g u r e 11. The r e l a t i o n s h i p between equation 2.13 and 2.14-l i k e l y t o s a t i s f y the l i n e a r slamming c r i t e r i a i s i l l u s t r a t e d i n F i g u r e 12. 20 3. NGN-LINEAB HODEL (QOASI-HARMONIC RESPONSE) 3.1 Background What i s d e s i r e d i s not a ri g o u r o u s s o l u t i o n f o r a n o n - l i n e a r problem, but r a t h e r a s o l u t i o n of a s u b s t i t u t e l i n e a r problem which would approximate a t r u e s o l u t i o n Korvin-Kroukovsky and J a c o b s 2 The n o n - l i n e a r s h i p motion theory presented here uses the l i n e a r S-T-F s t r i p theory c o e f f i c i e n t s from Chapter 2 ( f o r the r e c t a n g u l a r s e c t i o n model), ev a l u a t e d at a number of time i n t e r v a l s over a motion c y c l e . I t i s termed non-^linear because the motion amplitudes obtained are not d i r e c t l y p r o p o r t i o n a l to the waveheight. Host authors have c o n c e n t r a t e d attempts at n o n - l i n e a r motion a n a l y s i s , f o r coupled heaving and p i t c h i n g , on the det e r m i n a t i o n of s e c t i o n a l c o e f f i c i e n t s f o r added mass and damping (for example G o l o v a t o 2 5 ) . P r i n c i p l e s of Naval A r c h i t e c t u r e (pp. 614) r e f e r s to an analogue computer n o n - l i n e a r technique, but d e s p i t e the suggestion of Korvin-Kroakovsky and Jacobs (above), a s u b s t i t u t e l i n e a r problem seems to have been ig n o r e d . The importance of improved methods f o r e v a l u a t i o n of motion c o e f f i c i e n t s i s apparent when the wave p r o f i l e on a s h i p i s observed. The " r e s t p o s i t i o n " assumption f o r the l i n e a r model c o e f f i c i e n t s i s not a p a r t i c u l a r l y v a l i d one when v a r i a t i o n s i n 21 wave height, r e l a t i v e to the design w a t e r l i n e , are s i g n i f i c a n t . Even i f a b s o l u t e s h i p amplitudes ( p a r t i c u l a r l y pitch) are s m a l l , the r e l a t i v e v e r t i c a l displacements can be l a r g e . ( I t should be noted t h a t t h i s i s not the same s o r t of n o n - l i n e a r a n a l y s i s r e f e r r e d t o f o r r o l l i n g motions of s h i p s , where the l a r g e a b s o l u t e angles of r o l l r e g u i r e c o n s i d e r a t i o n ) . F i g u r e 14a shows v a r i a t i o n s i n wave p r o f i l e from the r e s t p o s i t i o n . A f u r t h e r 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 of motion c o e f f i c i e n t s i s a s s o c i a t e d with the d i s c u s s i o n of bottom slamming, i n S e c t i o n 1.2.3. Slamming i s c h a r a c t e r i z e d by high v e r t i c a l a c c e l e r a t i o n s near the bow, caused by a high r a t e of change of added mass. tihile the c o n v e n t i o n a l l i n e a r theory a l l o w s n e i t h e r of these f e a t u r e s , both are p o s s i b l e when c o e f f i c i e n t s are ev a l u a t e d with r e s p e c t t o a c t u a l wave p r o f i l e . T h i s i s d i s c u s s e d f u r t h e r i n Chapter 4. The disadvantages of a n o n - l i n e a r approach such as t h i s a r e : (1) s u p e r p o s i t i o n techniques are not d i r e c t l y a p p l i c a b l e , s i n c e motions are not purely harmonic. Thus, p r e s e n t a t i o n of motion response data i n a p r o b a b i l i s t i c form i s much more complicated than with c o n v e n t i o n a l l i n e a r t h e o r i e s . (2) computational time i s i n c r e a s e d , s i n c e motion c o e f f i c i e n t s are computed at many time i n t e r v a l s . 22 3.2 Method 3.2.1 C a l c u l a t i o n of Motion C o e f f i c i e n t s The method t h a t w i l l be used here to c a l c u l a t e the u n i t response c o e f f i c i e n t s , f o r a s h i p at some p o s i t i o n and frequency o f encounter, i s as f o l l o w s ; (1) f i x the s h i p i n some r e l a t i v e p o s i t i o n t o the wave, moving with speed 0 (2) f o r each s e c t i o n , c a l c u l a t e the e f f e c t i v e l e n g t h and mean d r a f t , using s u i t a b l e approximations, t o f i n d the s e c t i o n a l c o n t r i b u t i o n to the motion c o e f f i c i e n t s and f o r c i n g terms (3) s o l v e f o r the u n i t response c o e f f i c i e n t s , at t h a t p o s i t i o n . , Step (2) r e q u i r e s e v a l u a t i o n of the s e c t i o n a l p r o p e r t i e s f o r each s e c t i o n . F i g u r e 13 i l l u s t r a t e s the f o u r p o s s i b l e cases t h a t w i l l be c o n s i d e r e d , f o r the r e l a t i v e p o s i t i o n of each s e c t i o n . The corresponding values of e f f e c t i v e s e c t i o n l e n g t h and mean d r a f t a r e , r e f e r r i n g t o the n o t a t i o n o f Fig u r e 13, based on simple l i n e a r approximation... These a r e , r e s p e c t i v e l y : CAStT I Z 3 Z,(Xr) 2 4- o o The l i n e a r approximation f o r mean d r a f t i s reasonable when 23 the wavelength i s of an order l a r g e r than the s m a l l e s t s e c t i o n dimension (AX„) , so t h a t two or more wave c r e s t s are not w i t h i n the boundaries of one s e c t i o n . . An allowance has been c o n s i d e r e d here f o r the f r e e b o a r d a s s o c i a t e d with each s e c t i o n , so t h a t i f the c a l c u l a t e d mean d r a f t exceeds the o r i g i n a l s t i l l water d r a f t p l u s the f r e e b o a r d , the mean d r a f t i s s e t t o t h a t maximum value (see F i g u r e 14b) . F i g u r e 15 shows the f l o w c h a r t f o r c a l c u l a t i o n of motion response c o e f f i c i e n t s f o r a s h i p at a f i x e d p o s i t i o n . F i g u r e 16 d e t a i l s the modified b i s e c t i o n r o u t i n e r e g u i r e d t o c a l c u l a t e t h a t p o i n t (x*) on a s e c t i o n (case 2 or case 3) t h a t i n t e r c e p t s the wave s u r f a c e . 3.2.2 Quasi-Harmonic O s c i l l a t i o n s In the preceding s e c t i o n i t was shown how u n i t response c o e f f i c i e n t s c o u l d be computed at i n s t a n t a n e o u s s h i p p o s i t i o n s ( r e l a t i v e to the wave). I t i s thus p o s s i b l e t o compute a s e r i e s of u n i t response c o e f f i c i e n t s f o r v a r i o u s r e l a t i v e p o s i t i o n s , chosen i n some sy s t e m a t i c manner. T h i s would not, however, d i r e c t l y p r e d i c t the response of the s h i p , s i n c e r e l a t i v e p o s i t i o n changes over a motion c y c l e . I t i s not e n t i r e l y c l e a r how to use the i n stantaneous values of u n i t response c o e f f i c i e n t s t o o b t a i n the time-dependent response. For the purpose of t h i s s e c t i o n , we w i l l c o n s i d e r equation 2.1 as an o r d i n a r y d i f f e r e n t i a l equation with t i m e - v a r y i n g c o e f f i c i e n t s . The technique f o r i t s s o l u t i o n that w i l l be used 24 here can be d i s c u s s e d i n terms of the e q u i v a l e n t mechanical o s c i l l a t o r (two coupled degrees of freedom)., when we make the r e q u i r e d s i m p l i f i c a t i o n s i n terms of the hydrodynamic model of the s h i p , , (1) A s h i p o s c i l l a t i n g i n harmonic waves w i l l r e t u r n to a s t e a d y - s t a t e o s c i l l a t i o n of given amplitude and phase ( r e l a t i v e t o wave motion) r e g a r d l e s s of i t s i n i t i a l p o s i t i o n (2) As the r e l a t i v e p o s i t i o n of the s h i p changes i n s m a l l increments, the c o r r e s p o n d i n g u n i t response c o e f f i c i e n t s f o r those p o s i t i o n s vary i n s m a l l increments (phase and amplitude) . The f i r s t assumption i s i n t u i t i v e l y r e a s o n a b l e , and expresses the s t e a d y - s t a t e s o l u t i o n of the coupled equations f o r s m a l l v a r i a t i o n s i n u n i t response c o e f f i c i e n t s (the second assumption), When taken t o g e t h e r , the above assumptions imply s m a l l p e r i o d i c v a r i a t i o n s i n the c o e f f i c i e n t s o f the motion equ a t i o n . The a c t u a l s i z e of these v a r i a t i o n s i s d i s c u s s e d i n Chapter 4. During a c y c l e of s h i p motion/ over a s m a l l time increment, the p o s i t i o n of the s h i p changes by a c o r r e s p o n d i n g l y s m a l l amount. , Ie w i l l apply t h i s f a c t t o i n t r o d u c e what are termed here "quasi-harmonic" o s c i l l a t i o n s , which «e c o n s i d e r as the o s c i l l a t i o n s due to the u n i t response c o e f f i c i e n t s , a t one p o s i t i o n . While t h i s i s not r i g o r o u s l y t r u e , i t appears t o g i v e u s e f u l i n f o r m a t i o n about s h i p motions. F i g u r e 17 shows the g r a p h i c a l c o n s t r u c t i o n of a 25 quasi-harmonic response curve, i n t h i s case f o r r e l a t i v e v e r t i c a l displacement at the bow. Shen there i s on l y s m a l l v a r i a t i o n i n u n i t response c o e f f i c i e n t s , the r e s u l t i n g motions are very n e a r l y harmonic. However, the motions are not q u i t e c o r r e c t s i n c e , as seen on the f i g u r e , the s h i f t from t, to tz i m p l i e s t h a t the s h i p has been o s c i l l a t i n g with the u n i t response c o e f f i c i e n t s of t, f o r a complete c y c l e , which i s not the case. I n s t e a d , i t has beem o s c i l l a t i n g with those parameters f o r a period approximately equal t o A t (where A t i s much s m a l l e r than the motion p e r i o d ) . Shen c o n s i d e r e d c a r e f u l l y , i t can be seen t h a t each new p o s i t o n depends on the cumulative p r e v i o u s motion, which makes the r i g o r o u s l y c o r r e c t s o l u t i o n very complicated. I t i s i n t e r e s t i n g t o c o n s i d e r a s i m p l e r case as an i l l u s t r a t i o n , shown i n F i g u r e 18a as a f o r c e d s i n g l e - d e g r e e - o f - f r e e d c m s p r i n g and. mass system, with a s m a l l v a r i a t i o n i n s p r i n g s t i f f n e s s over one p a r t of the c y c l e , The s t i f f n e s s v a r i a t i o n corresponds t o the s m a l l changes a s s o c i a t e d with the hydrodynamic c o e f f i c i e n t s of the e q u a t i o n s of motions 2.1. The r e s u l t i n g reponses a r e shown i n F i g u r e T8b, while F i g u r e 19 shows the c o n s t r u c t e d quasi-harmonic motion c y c l e . Although the procedure i s not r i q o r o u s l y c o r r e c t , some u s e f u l i n f o r m a t i o n i s obtained, I n t e r e s t i n g l y , a p a r a l l e l t o the quasi-harmonic response technique d e s c r i b e d above can be found i n a paper on s a t e l l i t e dynamics, a f i e l d removed from the marine environment. P.C. Hughes 2 6, i n a c o n s i d e r a t i o n of quasi-modal a n a l y s i s of 26 vibration of a deploying s a t e l l i t e boom, neglects the boom deployment rate to consider "instantaneous" modal c h a r a c t e r i s t i c s . While i t i s inco r r e c t to consider a forced harmonic motion cycle as a "mode" of vib r a t i o n , the consideration of "instantaneous" motions i s s i m i l i a r . Hughes states that "... i f the extension rate ( i . e . , time dependence) i s s u f f i c i e n t l y gradual, a modal viewpoint w i l l give useful r e s u l t s " . 27 4. BESOMS 4.1 Purpose The development of the r e c t a n g u l a r - s e c t i o n barge model of the previous chapters has not c o n s i d e r e d i n d e t a i l the u s e f u l n e s s of the model i n p r e d i c t i n g ( g u a n t i t a t i v e l y ) motions o f a c t u a l f u l l - s c a l e barges, T h i s would r e q u i r e l a r g e amounts of experimental data which i s u n f o r t u n a t e l y not a v a i l a b l e . ., In t h i s s e c t i o n , the only comparisons with f u l l s c a l e motion r e s u l t s a r e q u a l i t a t i v e , f o r an example from Beference 21, where a loaded o i l tanker ( i . e . a v e s s e l with a shape s i m i l a r to t h a t of a barge) i s c o n s i d e r e d . The primary focus o f the r e s u l t s presented here i s t w o f o l d . F i r s t , the s e n s i t i v i t y of motion r e s u l t s t o v a r i a t i o n s of s e c t i o n a l parameters i s i n v e s t i g a t e d , so t h a t an i n d i c a t i o n of which ( i f any) are " c r i t i c a l " i s o b t a i n e d . Secondly, numerical r e s u l t s f o r slamming c r i t e r i a ( for both harmonic and quasi-harmonic motions) are presented f o r an example, so t h a t the two t e c h n i q u e s may be compared.... 4.2 Model S p e c i f i c a t i o n s Three r e c t a n g u l a r - s e c t i o n models w i l l be c o n s i d e r e d . . The f i r s t i s an approximation t o the loaded tanker, from Beference 21, and the other two are based on the "Seaspan 250" barge, i n loaded and l i g h t c o n d i t i o n , r e s p e c t i v e l y . Small v a r i a t i o n s of 28 s e c t i o n a l p r o p e r t i e s are c o n s i d e r e d f o r each model, as r e q u i r e d . For the tanker model, s i n c e i t i s s p e c i f i e d only by i t s o v e r a l l displacement and l e n g t h , i t was not p o s s i b l e to r a t i o n a l l y f i t r e c t a n g u l a r s e c t i o n s . Instead, a simple three s e c t i o n model was developed, based on t y p i c a l tanker dimensions, and the motion r e s u l t s compared with those from Reference 21. Note that these r e s u l t s were ob t a i n e d from a 5 0 - s e c t i o n model, with a c c u r a t e c a l c u l a t i o n ( f o r a c u r v i l i n e a r s h i p s e c t i o n ) of s e c t i o n a l added mass and s e c t i o n a l damping c o e f f i c i e n t s , using S-T-F s t r i p theory. F i g u r e 20 compares the model s p e c i f i c a t i o n s with those from Reference 21. The Seaspan 250 was chosen f o r modelling because i t i s t y p i c a l of c o a s t a l barges, and was s u b j e c t e d t o severe slamming damage ( p l a t e c o l l a p s e , r e f e r r e d to as " s e t t i n g up" of the bottom p l a t i n g ) . The r e c t a n g u l a r s e c t i o n models f o r the barge were s p e c i f i e d using the f i t t i n g method of Chapter 2, with the number of s e c t i o n s chosen so t h a t more d e t a i l i s achieved near the bow and s t e r n s e c t i o n s . For the l i n e a r s t r i p theory model, i t i s only necessary t o s p e c i f y one r e c t a n g u l a r s e c t i o n where the a c t u a l s e c t i o n shape remains constant over some l e n g t h , whereas the n o n - l i n e a r model r e q u i r e s s m a l l s e c t i o n dimensions ( A X h ) over the e n t i r e l e n g t h . Table I I I and IV show the dimensions f o r the r e c t a n g u l a r s e c t i o n models of the Seapan 250 i n t y p i c a l l i g h t and loaded c o n d i t i o n s , r e s p e c t i v e l y . The loaded model p r i m a r i l y serves as an example of motion response f o r a form s i m i l a r t o the loaded tanker d i s c u s s e d p r e v i o u s l y , while the l i g h t model i s of more 29 i n t e r e s t f o r slamming r e s u l t s . F i g u r e 21 shows the how p r o f i l e of the 25 0, compared with those of the r e c t a n g u l a r s e c t i o n model, F i g u r e 22 i s from the l i n e s plan of the 250, showing c r o s s - s e c t i o n shapes a t s t a t i o n s s p e c i f i e d i n F i g u r e 23. The r e s p e c t i v e i n e r t i a l data f o r the r e c t a n g u l a r s e c t i o n models f o r l i g h t and loaded c o n d i t i o n i s given i n Table I I I and IV. 4,3 L i n e a r S t r i p Theory R e s u l t s 4.3,1 Motion S e n s i t i v i t y t o V a r i a t i o n of S e c t i o n a l Parameters The measures of response, as d i s c u s s e d i n S e c t i o n 2.2.3, th a t w i l l be used here are: (1) a b s o l u t e v e r t i c a l displacement {at the bow, f o r d r a f t s e q u a l t o Tf and zero) (2) r e l a t i v e v e r t i c a l displacement at f o r e f o o t (3) 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 f o r e f o o t {4) u n i t d r a f t c r i t i c a l wave amplitude at f o r e f o o t . , The r e s u l t s from l i n e a r s t r i p theory f o r the r e c t a n g u l a r s e c t i o n model of the loaded tanker are shown i n F i g u r e s 24 through 26 with the corresponding responses from Reference 2 1. U n i t heave r e s u l t s , while o f the same q u a l i t a t i v e form as those from the more p r e c i s e s t r i p t h e o r y , are s h i f t e d somewhat {near L / L w =1), and f a i l to reproduce the response "peak" at 1>/Lw = 1.5. A s h i f t i n the same d i r e c t i o n i s observed f o r a b s o l u t e displacement a t x=L/2, while f o r x=I/4 the r e l a t i v e magnitudes become very d i f f e r e n t , and s h i f t e d o p p o s i t e l y . {Note that f o r x=L/4, the comparison i s not e n t i r e l y v a l i d , s i n c e the tanker was t r a v e l l i n g at a 45 degree heading angle. The more exact 30 v e r s i o n of s t r i p theory i n c o r p o r a t e s t h i s f e a t u r e , while the r e c t a n g u l a r - s e c t i o n model of Chapter 2 does not* The value of U from Reference 21 was 3 m/s)• The l a r g e d i f f e r e n c e s f o r the x=L/2 case are probably due t o v a r i a t i o n s i n v a l u e s of the u n i t response c o e f f i c i e n t f o r p i t c h i n g , magnified by the l a r g e d i s t a n c e . U n f o r t u n a t e l y , these p i t c h i n g c o e f f i c i e n t s were not i s o l a t e d i n Reference 21,, In a d d i t i o n t o e r r o r s i n t r o d u c e d by the rough 3 - s e c t i o n approximation t o the tanker model, the f o l l o w i n g should a l s o be c o n s i d e r e d : (1) v a r i a t i o n s i n the L/Lu; parameter are d i r e c t l y p r o p o r t i o n a l to the assumed w a t e r l i n e l e n g t h 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 l i m i t s , f o r the s e c t i o n a l added mass and s e c t i o n a l damping c o e f f i c i e n t s of F i g u r e 7, must be c a l c u l a t e d {Appendix C ) . For the r e c t a n g u l a r - s e c t i o n tanker model (L = 384 m) these a r e : . U = 3 »/s U -3 mfc U-Z./Z M/S T=iom 4.4 M 9.o T*lOm 2.9 +8 S.4-I t i s p o s s i b l e to s h i f t t h e responses f o r the 3 - s e c t i o n model c l o s e r t o those obtained f o r the more exact s t r i p theory model, through c a r e f u l manipulation of s e c t i o n a l parameters, but u n l e s s a more r a t i o n a l comparison of the two models i s known ( i . e . more d e t a i l about the a c t u a l tanker dimensions), the 31 exercise would be of limited value. It i s unfortunate that further comparative re s u l t s between ship motion data from other sources (experimental or otherwise) and the rectangular section model were not avai l a b l e , but the i n d i c a t i o n (above) that r e s u l t s 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 to further consider the rectangular section model. I t would, of course, be necessary to correlate r e s u l t s very c a r e f u l l y before quantitative prediction f o r f u l l - s c a l e ships or barges i s considered. The remainder of t h i s section considers motion results for the rectangular section models of the Seaspan 250. Since a l l measures of response (with the exception of z r) are l i n e a r with respect to wave amplitude, unit response amplitudes only s i l l be considered. Three values of U have been selected, on the basis of t y p i c a l barge slamming speeds mentioned i n Reference 1. ("...the incidence of noticable slamming decreases with draft and speed, and i s seldom a problem with . barges which tow at speeds between 9 or 11 knots at drafts over 8 feet or so"). The correspondence between L / L ^ , and frequency of encounter (uJe) for those speeds i s shown in Figure 27, noting that a l l cases considered are "head seas", with the ship advancing opposite to the d i r e c t i o n of save motion. Respective frequency l i m i t s are also shewn, f o r the .light and loaded barge described i n Section 4.2. Where JL/LW values plotted for the motions exceed 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 set to constant values (a =0.5, fa =0.0) respectively.,. 32 F i g u r e 28 shows the a b s o l u t e v e r t i c a l displacement at the bow, f o r the loaded barge, as a f u n c t i o n of L/L^, and U. Maximum amplitudes occur i n a l l cases near i n t e g r a l L/i.^, v a l u e s . . The s h i f t towards the r i g h t of response "peaks" ( f o r example, near L / I w - 3 ) i s the r e v e r s e of what we might expect f o r higher speeds to reach e q u i v a l e n t encounter frequency values. Reference 21 s t a t e s : "...while i t i s i n t u i t i v e l y expected t h a t these c a l c u l a t e d resonance v a l u e s of are almost independent of 0 t h i s does not appear to be assured by the l i n e a r t h e o r y " . L / L w values near zero imply i n f i n i t e l y long waves. F i g u r e s 29, 30, and 31 i l l u s t r a t e e f f e c t s of parameters of the r e c t a n g u l a r s e c t i o n model on values of a b s o l u t e v e r t i c a l d isplacement. Reducing the number of s e c t i o n s from 8 to 5 s h i f t s the curve to the l e f t with s m a l l v a r i a t i o n s i n amplitude (Figure 29), The e f f e c t s of added mass and, damping v a r i a t i o n shown i n F i g u r e 30 appear to be r e l a t i v e l y s m a l l ( p a r t i c u l a r l y at low v a l u e s c f L/Z.«i), suggesting t h a t the approximations made i n Chapter 2 (and Appendix C) are reasonable. The e f f e c t of v a r i a t i o n of d r a f t of the afterbody s e c t i o n i s shown i n F i g u r e 31, f o r the loaded barge. The a f t e r b o d y terms i n the l i n e a r s t r i p theory model are i n c l u d e d , a c c o r d i n g to Reference 4, only because " ... The computed r e s u l t s seem to agree b e t t e r with experiments when these terras are i n c l u d e d " . There i s some reason to doubt t h e i r r i g o r o u s a d m i s s i b i l i t y when the s t r i p theory i s d e r i v e d from the v e l o c i t y p o t e n t i a l f o r m u l a t i o n . F i g u r e 32 shows the u n i t c r i t i c a l wave amplitude (J£ ) f o r 33 the l i g h t barge, compared t o the a b s o l u t e v e r t i c a l displacement. The values f o r near W/L^ - 0 are of some concern, s i n c e we might expect \j/0 —i** co f o r i n f i n i t e l y long waves (that i s , f o r e f o o t emergence should not o c c u r ) . The s h i f t i n peak amplitudes r e f l e c t s the f a c t t h a t z i s an ab s o l u t e q u a n t i t y ( r e l a t i v e to r e f e r e n c e s axes Oxz) while ~zK i s r e l a t i v e to the wave s u r f a c e . 4.3.2 L i n e a r Slamming C r i t e r i a R e s u l t s i n t h i s s e c t c n are f o r the " l i g h t l y loaded" model Seaspan 2 50, at U=2.06 m/s. The l i n e a r slamming c r i t e r i a ( r e l a t i v e 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 ) are i n v e s t i g a t e d f o r wave amplitude and wavelength, v a r i a t i o n s . The t h r e s h o l d v e r t i c a l slamming v e l o c i t y V 0 ( S e c t i o n 2.2*3) , f o r 0 •= 2.06, has the f o l l o w i n g v a l u e s , f o r t y p i c a l beam dimensions fi' = 5 m V0= 3,91 Ws B' - IO m V0* S.90 m/s B' =Z4.+ m V0 = 3.SS „/s For motion c y c l e s , time w i l l be expressed i n a non-dimensionalized form where: T drr A l l phase angles are expressed r e l a t i v e to the ocean wave (cosine wave at t = 0 ) . . 34 F i g u r e s 33 and 34 show the harmonic motions ( f o r one c y c l e ) , at the minimum values of Vc from F i g u r e 32, f o r the l i g h t barge. , The r e g i o n where slamming occu r s , and the maximum r e l a t i v e v e r t i c a l slamming v e l o c i t y (magnitude), a t f o r e f o o t impact, are shown. The wave amplitudes f o r slamming to occur can then be c a l c u l a t e d (since r e s u l t s are l i n e a r with amplitude) as : L/Lui^l.8 2.51 -» 3.78 ~ 6*18 ~ L / L u ) * 2 . $ 2.OS ,» 3 to ~> S.OZ Since the wave amplitude/wavelength r a t i o f o r the s t e e p e s t waves t h a t occur i n nature i s approximately 0.071 the corresponding maximum wave amplitudes ( f o r L =. 92.7 m) a r e : L/L.0—I.Q maximum amplitude - 3.7 m L/c-tu - 2.9 maximum amplitude - 2.3 m The f i g u r e s i n d i c a t e t h a t , because the r e l a t i v e phase d i f f e r e n c e between 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 i s always 90 degrees, the f o r e f o o t d r a f t (T^) i s an important f a c t o r . The e f f e c t of i n c r e a s i n g T^ i s t o s h i f t the z P curve down, thereby d e c r e a s i n g r e l a t i v e v e r t i c a l slamming 35 v e l o c i t y at forefoot impact (Vc ) (see Appendix C) . 4.4 Quasi-Harmonic Motion Results 4.4.1 Motion S e n s i t i v i t y I t should f i r s t he v e r i f i e d that the assumptions of Section 3.2.2 for the non-linear theory are v a l i d for the rectangular section model. Relative velocity w i l l not be considered, since d i f f e r e n t i a t i o n of an approximate r e s u l t ( i . e . the quasi-harmonic r e l a t i v e v e r t i c a l displacement) w i l l only magnify small piecewise slope changes. The quasi-harmcnic curves are presented here i n the same form as the computed r e s u l t s , and no smoothing routines have been employed. Results i n t h i s section correspond to the minimum values of Vc from Figure 32. Figure 35 shows that f o r unit wave amplitude, the quasi-harmonic curve i s a good approximation to the pure harmonic form (calculated using r e s t - p o s i t i o n c o e f f i c i e n t s ) . The large "jump" at pt. A i s discussed in the next section. Figure 36 i l l u s t r a t e s the p e r i o d i c i t y of the guasi-harmcnic waveform, while Figure 37 and 38 show the reasonably constant variation i n magnitudes and phase, respectively, of the unit response c o e f f i c i e n t s for heave and pitch over one motion cycle. Figure 39 i s an i n d i c a t i o n that the number of sections i s not a c r i t i c a l parameter f o r the quasi-harmonic technique. Although not i l l u s t r a t e d here, i t can also be v e r i f i e d that 36 the s t a r t i n g point, or i n i t i a l l y assumed values of unit response c o e f f i c i e n t s , have no e f f e c t on the motion cycle (transients are removed a f t e r the f i r s t time increment)» , 4.4.2 Slamming C r i t e r i a The a b i l i t y of the guasi-harmonic technique to d i r e c t l y predict slamming, for the rectangular section model, i s dependent on two features associated with the se c t i o n a l added mass data from Reference 7, Chapter 2, and the way in which i t was f i t t e d for the computer model. F i r s t , the l i m i t on breadth/draft r a t i o (2 < B/T < 8) a r t i f i c i a l l y r e s t r i c t s shallow draft sections that "occur" during a motion c y c l e . Secondly, the variations in slope of the curve (from Figure 7a) i n the numerical f i t may lead to errors i n added mass determination, as the d r a f t , and therefore <jJpf~ , varies. Figure 40 demonstrates the nonlinear (w.r.t. ^ ) aspect of the motion responses, noting that similar waveforms are observed. The decreasing time i n t e r v a l (Figure 41) produces an unsteady response when the forefoot impacts the water surface. The amplitudes of those responses are too large to be a r e a l i s t i c representation. The rapid increase i n added mass that i s partly responsible for slamming (as described i n Section 1.2.3) i s i l l u s t r a t e d in Figure 42, where the added mass for a rectangular section near the bow {N = 18 from Table III) increases rapidly near the point of forefoot submergence. 37 5. CONCLUDING REMARKS 5.1 Summary This thesis investigates two pa r t i c u l a r aspects of s t r i p theory. Hhile s i m p l i f i c a t i o n s have been applied to make analysis easier, i t i s not intended to imply that the general application of s t r i p theory to ship motions i s a simple problem. F i r s t l y , the rectangular section barge model i s designed, making use of complex notation for the ship and wave displacements, and applied to two examples. Comparison with a loaded tanker, as tested with the s t r i p theory from Beference 21, shows q u a l i t a t i v e r e s u l t s that appear s a t i s f a c t o r y . Although no comparison i s possible, a method for laying out the equivalent rectangular section model i s presented f o r the Seaspan 250 barge, Secondly, the quasi-harmonic motion formulation i s presented, and r e s u l t s compared with the pure harmonic case f o r a rectangular section model, , As well as demonstrating good agreement with harmonic r e s u l t s at moderate amplitudes, the guasi-harmonic model appears to indicate, q u a l i t a t i v e l y at lea s t , a d i r e c t method for predicting slamming motions, at the point of forefoot submergence. 38 5.2 Recommendations f o r F u r t h e r Work Although t h i s t h e s i s has l i m i t e d i t s e l f t o a very s p e c i f i c case, t h a t of r e c t a n g u l a r s e c t i o n bodies i n harmonic waves, the work presented here lends i t s e l f to expansion. Two p a r t i c u l a r d i r e c t i o n s f o r f u r t h e r work are e v i d e n t . Both would r e q u i r e comparison with r e s u l t s from other more e x t e n s i v e s h i p motion experiments or s t r i p t h e o r i e s . F i r s t , the r e c t a n g u l a r s e c t i o n approximation needs to be t e s t e d more f u l l y , e i t h e r with a c t u a l towing tank or f u l l - s c a l e r e s u l t s or with r e s u l t s from a s t r i p theory f o r a c u r v i l i n e a r s h i p form. Secondly, the quasi-harmonic f o r m u l a t i o n of Chapter 3 c o u l d be a p p l i e d using s t r i p theory c o e f f i c i e n t s f o r a more g e n e r a l s h i p shape, p r e f e r a b l y one f o r which r e s u l t s have p r e v i o u s l y been obtained f o r slamming behaviour. Despite the computation time r e g u i r e d , the technique may prove i t s e l f u s e f u l and should be developed as an approximate n o n - l i n e a r s o l u t i o n . Extension to other degrees cf freedom i s a l s o a p o s s i b i l i t y . 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", Internat-ional Shipbuilding Progress, Vol. 14, 1967. 4. Salvesen, N., Tuck, E.O., and Faltinsen, 0., "Ship Motions and Sea Loads", Trans. SNAME, Vol. 81, 1971, pp. 250-287. 5. Wang, S. "Dynamical Theory of Potential Flows With a Free Surface: A Classical Approach to-Strip 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 Strip Theory of Ship Motions, Including the Calculation of Mean Added Forces and Moments", Journal of ShiD Research, Vol. 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", Vol. 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. Price, W.G. and Bishop, R.E.D., "Probabil ist ic Theory of Ship Dynamics"; Chapman and Hal l , London, 1974. 12. Tick, L . J . , "Certain Probabilities 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 Vehicles", Wiley and Sons, Toronto, 1978. 14. Jones, N., "Slamming Damage", Journal of Ship Research, Vol. 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',1 Marine Technology, January, 1973. 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. (editor), "Principles of Naval Architecture", 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. Price, W.G., Bishop, R.E.D. and Tarn, P.K.Y., "On the Dynamics of Slamming", Transactions of the Royal Institute of Naval Architects, Vol.)2©, 1978. '•• O 21. Price, W.G. and Bishop, R.E.D., "A Unified Dynamic Analysis for Ship Response to Waves", Trans. RINA, Vol. nS , 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 Osci l lat ions", David Taylor Model Basin Report No. 1074, 1956. 26. Hughes, P.C., "Deployment Dynamics of the Communications Technology Satel l i te - A Progress Report", Transactions of the Canadian Aeronautics and Space Institute, March 1974, pp. 10-18. Appendix A Solution, of equation 2 . 1 : \_-wlA - iufeB * c] Solve'for unit response coefficients K, (cu^) . K2(l^e) where: or - Z/XcOe) T2 CWe~)~ l e t [ D ] -ccue B - c] (oue) - D > ] X,W) which can be expressed as: K = &.-T, ;~' GPS where: Or + Cu B e 3= - <»l AIM. - CUJ^B,^ - ^ A 2 L - i L U 9 B 2 / + Cz, Appendix B Unit draft c r i t i ca l wave amplitude y c : Definition: For harmonic motions, unit draft c r i t i ca l wave amplitude is that wave amplitude that will cause the forefoot relative vertical displacement to equal zero at one point on a motion cycle, where the forefoot position is defined at x=x^ ., _z=-l, i .e . unit draft. at some #-# t * t * , z,(xf, ttt*)*0 so that % ~ / The minimum value .o-CJs£_that wil l satisfy equation Bl (above) wil l be d e s i g n a t e d - ^ Recalling that only the real part is considered, we have Thus. Appendix C Numerical curve f i t for sectional added mass and damping coeff icients: Sectional added mass and sectional damping coefficients are calculated (see FORTRAN routines, Appendix D) by use of the following polynomial, f i t ted to the curves of Figure 7. y = p,60 ,+ ?*(*,)* f>3(*J+ p* fx!+Pf(**)3+p< p7 *2 +• p9 M +"'• Pd 6</)Z*i +: ?io (Xt)**, + Pa X, Xz + p/£ . where: V % - ~ = — (respectively, for added mass and damoing) The respective coefficients P ' . , ^ a r e given in Appendix D. For both sectional added mass and sectional damping coeff ic ients, the range of frequencies considered wil l be limited to those shown in Figure C-l and Figure C-2. In addition, the values of breadth/draft ratio are restricted 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 ) A ft,, B = B/2 C = 3' /2 £\2I — B /2 = B2, = -C„ = '// r where: fanc/x - jfe 6g + m / * w * - * ^ / 6 A J x * Ua* /x%Jx r g4<V vfc'faf. %lx*bn* ^sfxBJx. C/2 YY sectional added mass coeff icient sectional damping coeff icient CC^}^N - respective sectional coefficients for aftermost section 0 - 0 ii A" ) ft position of aftermost section is f\UiB„ when U=0 (respectively) Table I - Coefficients for Equation 2.1 (continued) Forcing terms: Tz = % fe'**e""T {*[?9B -cu6*„a.-U>,)] 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 valid for head seas only. Table II - Coefficients for a Single Section of the Rectangular Barge Model (derived from Table I) Main-body coefficients (Integral terms) Afterbody coefficients 6 2 2 = C„ = Q / -r -U 2 2 -DA -DA 99 ^ « ^  C/2 -O f C4 L/ / A\Z/*, tJ -rAsr* O O o o where: D, D 3 2 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 - UJUJe <Zn - icUh^ = ; — 4- c Uu)cc* LOADED B.n LIGHT T. LOADED LIGHT 1 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 -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 n -53.6 -48.8 •43.9 -9.0 •34.1 -29.3 •23.0 -17.0 •11.0 -5.0 6.0 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 5 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 12.0 18.0 21.0 24.4 29.3 34.1 39.0 43.9 48.8 53.6 Table III - 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 inear slamming c r i ter ia 56 Figure 5 - Method for f i t t ing rectangular section to curvil inear section Figure 6 - Vert ical Oscil lations of a Rectangular Section on a Free Surface 58 59 © eg n An+1 The center of gravity of section n is assumed to be located at x=xn, z =0. jTDh — ~>BnJ£,, m a s s °f section n (T~ ) , m n f y 2 a ? j_ mor y-Y/'n (x»>o)~ 12 [ ' » * x " J ab( thus: moment of inertia of section-about transverse axis through < v ° > — f * r > n and: YY Figure 8 - Moment-! of inertia approximations for the rectangular section barge model Figure J.Q; - Harmonic wave notation z,V t z'(t) Figure 11a - Absolute vertical displacement and absolute vertical-veloci-ty Figure 11c - Relative vertical displacement and relative vertical velocity 'A'denotes the region where slamming may occur, i . e . where v^ 0, z r 0 at the forefoot . z rj i s the amplitude of z r ( t ) about z r = - T f Jv rj i s the amplitude of v r ( t ) about vr=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 '.' i n i t i a l forefoot submergence, given by: Figure 12 - Phase re lat ionships for l i n e a r slamming c r i t e r i a 63 z r(x^)'<0 \ . J CASE" 1 - z r ( X l ) z (x )< 0 r v j , ' t x 1 r t 2 ( \ i ) <0 CASE Jt\ N. S X i t: x X r 2r (x . r )>0 z r(xT.)>.0 s CASE <3-X 1 x" p z r (x R 4<0 z r ( x v ) > 0 CASE-.4 1 V V > 0 X 1 X r Figure 13 - Cases of sectional emergence considered 64 freeboard wave s t i l l water Figure 14a - Wave profi le~and~sti l l , water prof i le V l 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 -**-— X* r X l * X* T r % ( x r ) / 2 T i + T r : x r 2.1 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-,+xr)/2 x = ( x ^ j / 2 x = (x 1 +x r )/2 x l •• 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 QSin wt ^ x=0 Figure 18a - Forced single-degree-of-freedom system, to i l l u s t r a t e quasi-harmonic v ibrat ion Figure 18b - Forced o s c i l l a t i o n at d i f fe rent values of spring s t i f fnes s Figure 19 - Quasi-harmonic motion cycle for single-degree-of-freedom system <JD B=35 m B=45-m B=45 m B=35 tn 1 30 nr •162 rai- 162 m 30 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 PLATE 1 SEASPAN 250 - OVERALL DIMENSIONS PLATE 2 SEASPAN 250 - CROSS-SECTIONS Figure 22 PLATE 3 SEASPAN 250 - BOW PROFILE Figure 23 Figure 24 - Unit .heave response, tanker, U=9 m/s 72 rectangular section model 0 1 2 3 4 5 6 7 L / L w Figure 25 - Absolute vertical displacement at x=L/4, tanker, U=2.12 m/s 73 Figure 26 - Absolute vertical displacement at x = L/2, tanker, U=3 m/s 1.0 Y Figure 28 - Absolute vertical displacement at forefoot, loaded barge, 11=2.06,4.11, 6.17 m/s L/L w Figure 29 - Absolute vertical displacement at forefoot, light barge, 5 and 8 sections. 11=2.06 m/s Figure 32 - Unit draft c r i t i ca l wave amplitude and absolute vertical displacement, l ight barge, U=2.06 m/s Fiqure 33 - Relative vertical displacement and relative vertical velocity, l ight barge, U=2.06 m/s, L/U=l:8 ,3.0 Figure 34 - Relative vertical displacement and relative vertical velocity, light barge, U=2.06 m/s, L/L =2.9 OO 00 Figure 39 - Quasi-harmonic relative vertical displacement, for 13 and 20 sections, l ight barge, U=2.06 m/s, L/L =2.,9 I O . O r 8.0 • -8.0 • - i o : o *• Figure 40 - Ouasi-harmonic relative vertical displacement for three wave amplitudes, U=2.06 m/s, L/Lw=2.9 oo 00 CO 00 to 

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