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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 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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