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Wave loads and motions of long structures in directional seas Nwogu, Okey U. 1985

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WAVE LOADS AND MOTIONS OF LONG STRUCTURES IN DIRECTIONAL  SEAS  by OKEY U. NWOGU B.A.Sc, University  o f Ottawa, 1983  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in FACULTY OF GRADUATE Department  We a c c e p t to  of C i v i l  this  Engineering  thesis  the r e q u i r e d  STUDIES  as c o n f o r m i n g standard  THE UNIVERSITY OF BRITISH COLUMBIA J u l y 1985 ©  OKEY U. NWOGU, 1985  In  presenting  requirements  this  that  I agree that  available  permission  scholarly  for  partial  purposes or  understood  that gain  by  may his  be or  copying  shall  the  reference  f o r extensive  Department  financial  in  not  of C i v i l  1985  shall  and s t u d y .  of  the  this  granted  by  the  her  be  allowed  Engineering  Head  i t  agree  thesis  representatives.  or p u b l i c a t i o n  make  I further  of  THE UNIVERSITY OF BRITISH COLUMBIA 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5  Date: J u l y  Library  copying  permission.  Department  fulfilment  f o r an a d v a n c e d d e g r e e a t t h e THE UNIVERSITY OF  BRITISH COLUMBIA, freely  thesis  of It  for my is  of t h i s t h e s i s f o r  without  my  written  ABSTRACT  The  effects  of  wave  directionality  on t h e l o a d s a n d  motions of long s t r u c t u r e s i s i n v e s t i g a t e d i n t h i s A  numerical  method  based  on  d e v e l o p e d t o compute t h e e x c i t i n g coefficients  due  to  the  previous The  of  diffraction  of  finite  obtained problem  and  long,  shape.  results obtained  results  functions  arbitrary  forces  using other  are  used  hence  to  the  i s expressed  as  dependent r e d u c t i o n ratio for  techniques.  determine operators  loads  a  the  cosine  power t y p e  r e s u l t s of  and  amplitudes  of  seas. short-crested  seas  results for long-crested  seas.  directionally  averaged,  frequency  presented  f a c t o r and response r a t i o o f a l o n g  t o a d i r e c t i o n a l wave s p e c t r u m  energy spreading  the  transfer  f a c t o r f o r t h e wave l o a d s a n d a r e s p o n s e  the force reduction box s u b j e c t  wave  f o ra structure  f o r t h e body m o t i o n s . N u m e r i c a l r e s u l t s a r e  floating  the  floating  a r e made w i t h  solution  wave l o a d s a n d body m o t i o n s i n  compared t o c o r r e s p o n d i n g  This  oblique  from t h e s o l u t i o n o f t h e o b l i q u e  are  is  hydrodynamic  semi-immersed  motion of t h e s t r u c t u r e i n s h o r t - c r e s t e d The  and  Comparisons  response amplitude  l e n g t h and  theorem  i n t e r a c t i o n of a r e g u l a r  wave t r a i n w i t h a n i n f i n i t e l y cylinder  Green's  thesis.  present  i i  and  a  f u n c t i o n . A p p l i c a t i o n s of  procedure  s t r u c t u r e s as f l o a t i n g b r i d g e s  with  include  breakwaters.  such  long  Table of Contents ABSTRACT  .  i i  L I S T OF TABLES  v  L I S T OF FIGURES  .  .  vi  NOMENCLATURE  viii  ACKNOWLEDGEMENTS 1.  x i i  INTRODUCTION  1  1 . 1 GENERAL  1  1.2 LITERATURE SURVEY 1.2.1  3  DIFFRACTION THEORY  3  1.2.2 EFFECTS OF DIRECTIONAL WAVES  5  1.3 DESCRIPTION OF METHOD 2.  8  DIFFRACTION THEORY  11  2.1  11  INTRODUCTION  2.2 THEORETICAL FORMULATION 2.2.1  13  WAVE DIFFRACTION PROBLEM  13  2.2.2 FORCED MOTION PROBLEM 2.3 GREEN'S FUNCTION  17  SOLUTION  2.4 EXCITING FORCES, COEFFICIENTS  ADDED  19 MASSES  AND  DAMPING  2.5 EQUATIONS OF MOTION  25  2.6 REFLECTION AND TRANSMISSION  3.  COEFFICIENTS  28  2.7 NUMERICAL PROCEDURE  30  2.8 EFFECT OF F I N I T E STRUCTURE LENGTH  35  EFFECTS OF DIRECTIONAL WAVES  39  3.1  4.  21  REPRESENTATION  OF DIRECTIONAL  SEAS  39  3.2 RESPONSE TO DIRECTIONAL WAVES  44  RESULTS AND DISCUSSION  48  iii  4.1 EXCITING FORCES, COEFFICIENTS  5.  ADDED  MASS  AND  DAMPING  48  4.2 MOTIONS OF AN UNRESTRAINED BODY  52  4.3 EFFECTS OF DIRECTIONAL WAVES  53  CONCLUSIONS AND RECOMMENDATIONS  57  5.1 CONCLUSIONS  57  5.2 RECOMMENDATIONS FOR FURTHER STUDY  59  BIBLIOGRAPHY  61  APPENDIX I  .65  i v  L I S T OF TABLES Table 1.  2.  3.  4.  page C o m p a r i s o n o f t h e sway a d d e d mass a n d damping c o e f f i c i e n t s o f a s e m i - c i r c u l a r c y l i n d e r (d/a=») obtained i n the present study with the r e s u l t s of GAR ( G a r r i s o n , 1 984)  68  C o m p a r i s o n o f t h e h e a v e a d d e d mass a n d damping c o e f f i c i e n t s o f a s e m i - c i r c u l a r c y l i n d e r (d/a=°°) o b t a i n e d i n the present study with the r e s u l t s of B&U ( B o l t o n and U r s e l l , 1 973)  69  C o m p a r i s o n o f t h e sway e x c i t i n g f o r c e c o e f f i c i e n t and wave a m p l i t u d e r a t i o o f a s e m i - c i r c u l a r c y l i n d e r (d/a=°°) o b t a i n e d i n t h e p r e s e n t s t u d y w i t h t h e r e s u l t s o f GAR ( G a r r i s o n , 1 984) Comparison of the heave e x c i t i n g f o r c e c o e f f i c i e n t and wave a m p l i t u d e r a t i o o f a s e m i - c i r c u l a r c y l i n d e r (d/a=») o b t a i n e d i n t h e p r e s e n t s t u d y w i t h t h e r e s u l t s o f B&U ( B o l t o n a n d U r s e l l , 1973)  v  70  71  L I S T OF  FIGURES  Figure  page  1.  Definition  sketch  for a rectangular  2.  Definition  sketch  for floating  component  cylinder  cylinder  motions  showing  .  73  3.  Sketch  of c l o s e d  4.  Sketch  showing  5.  A t y p i c a l b o u n d a r y e l e m e n t mesh f o r a r e c t a n g u l a r cylinder (b/a=1 ,d/a=2) Square of r e d u c t i o n f a c t o r r f o r d i f f e r e n t v a l u e s  6.  of  72  surface  73  relationship  between  x, £, and  74  74  0  75  7.  Sketch of a d i r e c t i o n a l  8.  D i r e c t i o n a l spreading function f o r d i f f e r e n t values of the parameter s.. Sway e x c i t i n g f o r c e c o e f f i c i e n t f o r a r e c t a n g u l a r cylinder (b/a=1,d/a=2)  9. 10. 11. 12. 13. 14. 15. 16. 17. 18.  wave s p e c t r u m  Heave e x c i t i n g f o r c e c o e f f i c i e n t cylinder (b/a=1 ,d/a=2)  for a  R o l l e x c i t i n g moment c o e f f i c i e n t cylinder (b/a=1,d/a=2)  for a  Reflection coefficient (b/a=1 ,d/a=2)  75 76 76  rectangular 77 rectangular 77  for a rectangular  cylinder ....78  Sway e x c i t i n g f o r c e c o e f f i c i e n t cylinder (b/a=0.265,d/a=»)  for a  78  Heave e x c i t i n g force, c o e f f i c i e n t cylinder (b/a=0 . 265 ,d/a=»)  for a  R o l l e x c i t i n g moment c o e f f i c i e n t cylinder (b/a=0 .265,d/a==>)  for a  rectangular 79  rectangular 80  rectangular  Heave a d d e d mass c o e f f i c i e n t f o r a cylinder (b/a=0.265,d/a=») vi  rectangular 79  Sway a d d e d mass c o e f f i c i e n t f o r a cylinder (b/a=0.265,d/a=») Sway damping c o e f f i c i e n t f o r a cylinder (b/a=0.265,d/a=»)  rectangular  80 rectangular 81  19. 20. 21. 22. 23. 24. 25. 26.  27.  Heave damping c o e f f i c i e n t f o r a cylinder (b/a=0.265,d/a=»)  rectangular 81  R o l l a d d e d mass c o e f f i c i e n t f o r a cylinder (b/a=0.265,d/a=«) R o l l damping c o e f f i c i e n t f o r a cylinder (b/a=0.265,d/a=»)  rectangular 82  rectangular  Sway r e s p o n s e a m p l i t u d e o p e r a t o r f o r a l o n g f l o a t i n g box (a=7.5m,b=3m,l=75m,d=12m) Heave r e s p o n s e a m p l i t u d e o p e r a t o r f o r a l o n g f l o a t i n g box (a=7 . 5m, b=3m, l = 75m,d= 1 2m) R o l l response amplitude operator f o r a long f l o a t i n g box (a=7.5m,b=3m,l=75m,d=l2m) F o r c e and moment r e d u c t i o n f a c t o r s f o r a l o n g f l o a t i n g box (a=7 . 5m, b=3m, l=75m,d= 1 2m)  82 83 83 84 84  F o r c e r e d u c t i o n f a c t o r s f o r a l o n g f l o a t i n g box (a=7.5m,b=3m,l=75m,d=l2m) i n n o r m a l a n d o b l i q u e mean s e a s  86  R e s p o n s e r a t i o s f o r a l o n g f l o a t i n g box (a=7.5m,b=3m,l = 75m,d=1 2m)  87  vi i  NOMENCLATURE a  = half  a. .  = matrix  A  = displaced  A  beam o f  cylinder  coefficient volume p e r u n i t  = complex a m p l i t u d e  0  A^j  = complex wave  b  = draft  of  of v e l o c i t y  potential  amplitude  cylinder  b. . 3  = matrix  B  = beam o f  c^j  = hydrostatic  Cj  = exciting  C(s),C'(s)  = normalizing coefficients spreading functions  d  = water  f  = circular  1  length  coefficient cylinder s t i f f n e s s matrix  force  coefficient  coefficient for directional  depth frequency  f^ *  = coefficient  Fj  = exciting  F.. ^  = f o r c e i n t h e i t h d i r e c t i o n due t o t h e j t h mode o f m o t i o n o f c y l i n d e r  g  = gravitational  G(fa>,0)  = directional  G(x;£)  = Green's  function  H  = incident  wave  Hj  = system  i  = v/(-D  k  ;  I  0  defined  i n e q n . (2.92)  force  acceleration  spreading  function  \  height  response  function  = p o l a r mass moment per u n i t l e n g t h  k  = incident  K  = Keulegan-Carpenter  of i n e r t i a  wavenumber  viii  number  about  the y  axis  K ,K_  =  D  H  reflection  and  transmission  K ,K,  = modified one  1*  =  length  L  =  incident  m  = mass p e r  m^j  = mass m a t r i x  0  of  Bessel  f u n c t i o n s of  unit  coefficient  = u n i t normal  n.n  = direction  p  =  pressure  q(kl,/3)  =  factor defined  r  = distance  Ty  = r a d i u s of  g y r a t i o n of  r(kl,/3)  =  factor  r'  = distance  Rp  =  Rj^  = response  s  = cosine  S(CJ)  = s p e c t r a l energy  S(co,/3)  = directional  S  D  =  immersed  S  D  =  seabed  S  F  =  free  segments on vector  cosines  S +Sp+S B  R  d i r e c t e d out of  of  fluid  region  n  i n eqn.  (2.103)  between x and J[  between x  force reduction  c y l i n d e r about  the  y  axis  and  factor  ratio  power o f  spreading  wave  body  function  density spectrum  surface  surface  S_  = radiation surface  S  = w a t e r p l a n e a r e a moment of a x i s per u n i t l e n g t h =  and  l e n g t h of c y l i n d e r  n  t  zero  wavelength  number o f  reduction  orders  structure  N  N  coefficients  1  time  ix  inertia  about  the  x  T  = wave p e r i o d  u  = fluid  U  = wind  velocity  vector  speed  U"  = maximum p a r t i c l e  V  = d i s p l a c e d volume o f c y l i n d e r  V  = normal v e l o c i t y  x  = horizontal coordinate  x  = vector  x^  = c e n t r o i d of the w a t e r p l a n e the c e n t r e of g r a v i t y  X  = x c o o r d i n a t e of t h e r a d i a t i o n  m  R  velocity  o f body normal t o c y l i n d e r  axis  of p o i n t ( x , z ) line  measured  from  surface  y  = horizontal coordinate p a r a l l e l  z  = v e r t i c a l c o o r d i n a t e measured upwards f r o m t h e s t i l l water l e v e l  z  axis  = z c o o r d i n a t e of t h e c e n t r e of buoyancy  B  ZQ  = Z c o o r d i n a t e of t h e c e n t r e of g r a v i t y = response  amplitude  (5  = angle axis  P  = principal  77  = water water  77^  = asymptotic  0  7 j  to cylinder  R  , 7 j  T  operator  of i n c i d e n c e measured direction  o f wave  from t h e p o s i t i v e x propagation  s u r f a c e e l e v a t i o n measured level  = reflected  wave  from t h e s t i l l  amplitude  and t r a n s m i t t e d wave  amplitudes  5..  = Kronecker d e l t a  A  = phase  angle  7  = angle  between x-jj_ a n d n; a l s o E u l e r ' s  7'  = angle  between x - £ ' a n d n'  X^j  = damping  3  1  function  coefficient x  constant  u  = nondimensional 2.14)  frequency parameter  u- • D  = a d d e d mass c o e f f i c i e n t  v  = nondimensional 2.14)  6  = a n g l e o f i n c i d e n c e measured direction  p  = d e n s i t y of  4>  = velocity  0^  = complex v e l o c i t y  cj  = wave a n g u l a r  £  = v e c t o r of p o i n t  (£,$) on f l u i d  £'  = v e c t o r of p o i n t  U,-($+2d)]  1  frequency parameter from  (see eqn.  (see eqn.  principal  fluid  potential potentials  frequency  = nondimensional  boundary  amplitude  o f body  or r o t a t i o n  o f body  Hj  = displacement  5^  = complex wave a m p l i t u d e  xi  ratio  motion  wave  ACKNOWLEDGEMENTS  The Dr.  author  Michael  throughout  wishes  t o express  de S t . Q. I s a a c s o n  the preparation  Financial assistantship  support from  the  Natural  to  thesis.  the  form  of  Sciences  and  i s gratefully  xii  gratitude  f o r h i s g u i d a n c e and a d v i c e  of t h i s in  R e s e a r c h C o u n c i l of Canada  h i s immense  a  research  Engineering  acknowledged.  1.  1.1  GENERAL With  the  resources, design in of  of  both  of these  complex  The  ocean  have  structures  unidirectional random  and  different structure  forces, could could  also  accepted  economic  the  acceptable to  the  determination  i n t e r a c t i o n with The  a  structure  the the loads i t s  from  motions  limits.  the  design  incident  wave  offshore  field  t o be  have  and frequencies  different  of  leads to  affect  or rejected  waves  approach  property  a  can  a  i s also  spectra  reduction  case  the  i n wave  force  the  i n construction  as  in feasibility  1  in  by  of long-crested  savings  decisions  significantly  experienced  directional  the  to significant  the  motions  to  may  short-crestedness.  of  and  use  but a l s o  d i r e c t i o n s . This  t o a s wave  loads  often  seas  of  m u l t i - d i r e c t i o n a l , i . e . t h e waves n o t o n l y  The  Real  aspect  both  compared  lead  and  a r e , however,  the  calculations  offshore  but i n addition  approach  directionality  structure.  t o withstand  assumes  referred  influence  d u e t o wave  or long-crested.  from  The  involves  of  important  of the s t r u c t u r e .  t o be w i t h i n  amplitudes  sometimes  forces  environment,  often  f o r the safe  structures  not only  traditional  development  s t r u c t u r e s . An  and the response  be d e s i g n e d  the  a need  offshore  the e x c i t i n g  generally  in  has been  various  body  should  growth  there  the design  fixed  the  INTRODUCTION  to  whether  studies.  With  computed  waves. costs. designs the  This It are  recent  2  developments spectra  in  methods  (Borgman  of  part  of  spectra  the  When  semi-immersed  motion), axis).  with  structures,  the  et  forces.  If  incident  wave  is  waves,  design  the  structure  (vertical motion  only  waves  the  and  the  i s often  structure  the  usually  about  the  (beamwise  longitudinal due  to  slender  significantly  Morison's  to estimate  the  forces  s t r u c t u r e . For  body d o e s n o t  used  three  hydrodynamic  the  long  in  sway  forces  separation  problem  linearized  established  the  equation exciting  i s l a r g e enough t o d i f f r a c t  flow  ( K e l l o g g , 1 9 2 9 ) . The  of  infinitely  motion),  also  r e s p o n s e of  field,  basins  use  responds  exciting  but  p r e s e n c e of  al ,1950)  the  and  process.  (angular not  wave  i s soon b e c o m i n g an  i n c i d e n t wave k i n e m a t i c s and  (Morison  theory  directional  : heave  are  associated  neglected  laboratory  the  the  the  b u i l d i n g of  (1975), Leblanc  structure,  There  affect  al  i s i n c i d e n t upon an  roll  of  et  d i r e c t i o n a l wave  wave t r a i n  freedom  and  presence  the  models  offshore a  d e g r e e s of  and  generating  directional  determining  (1969), M i t s u y a s u  Middleton(1982)) capable  of  i s solved  effects using  complete problem  by  assuming a  are  the  often  potential  flow  i s nonlinear  small  amplitude  and wave  train. A this  n u m e r i c a l method b a s e d on thesis  to  hydrodynamic cylinder first  of  solve  for  coefficients a r b i t r a r y shape  extended  to  Green's theorem  the of  an  exciting infinite  in oblique  structures  of  forces  in and  semi-immersed  s e a s . The  finite  i s used  length  results and  then  are to  3  directional  seas  u s i n g the t r a n s f e r f u n c t i o n approach.  wave l o a d s and motions of the s t r u c t u r e i n d i r e c t i o n a l  The seas  a r e compared with those of l o n g - c r e s t e d waves. The a p p l i c a t i o n s of the r e s u l t s of t h i s t h e s i s such  long  structures  as  floating  include  breakwaters,  floating  b r i d g e s and p i p e l i n e s . I t c o u l d a l s o be used i n the study of ship  motions  where Korvin-Kroukovsky's  (1955) s t r i p theory  i s o f t e n used to reduce the t h r e e - d i m e n s i o n a l problem two-dimensional  1.2  LITERATURE  1.2.1  a  one.  SURVEY  DIFFRACTION THEORY  A number of authors ( U r s e l l Kim  to  (1949), MacCamy  (1964),  (1965), Bai (1972), I j i m a et al (1976)) have t r e a t e d  the two-dimensional wave-structure i n t e r a c t i o n  problem.  Much l e s s work has however been r e p o r t e d f o r the case of obliquely  i n c i d e n t waves.  Previous interaction (1970),  studies  of  i n c l u d e those conducted  Bai  (1975),  Leonard  water depth, and by G a r r i s o n (1973), and G a r r i s o n Garrison to  compute  coefficients, coefficients  oblique  et  for  Mei  and  Ursell  depth.  function  procedure  f o r c e s , added mass and damping  reflection a  and  al (1983) f o r f i n i t e  (1984) f o r i n f i n i t e  the e x c i t i n g  Black  (1969), B o l t o n  (1969) used a Green's  and  by  wave-structure  and  shallow d r a f t c y l i n d e r  transmission f l o a t i n g at  4  the  free  surface.  potential  The  a t any p o i n t  continuous  method  i n the f l u i d  distribution  of  s u r f a c e . The G r e e n ' s f u n c t i o n of  unit  strength.  involves  The  expressing  region  sources  equation  solved  numerically  the source  obtain  hence the v e l o c i t y p o t e n t i a l . G a r r i s o n approach Bolton solve  problem  oscillating varying The  based  with  field  then  a  technique  to solve  and t r a n s m i s s i o n  depth.  The  method  amplitude  functional.  The f l u i d  domain  and  of  variables  used  to  method t o cylinder  of  motion  relate  this  problem.  variational principle  technique  t o obtain the  i s used  to define  variables  f o r the e x c i t i n g coefficients  forces  i n water of  a s t h e minimum into  functions  the  of  some  subregions  with  nodal  the v e l o c i t y p o t e n t i a l over  the f u n c t i o n a l yields  used  expressing  i s divided  interpolation  domain. M i n i m i s i n g nodal  and  of the c y l i n d e r .  involves  equation  set  be  extended  s o l u t i o n o f t h e p r o b l e m . B a i (1975) a l s o  governing d i f f e r e n t i a l  the  strengths  a circular  t o t h e wave d i f f r a c t i o n  a n d M e i (1970) u s e d  reflection  a  can  (1984)  the length  on S c h w i n g e r ' s v a r i a t i o n a l  finite  the  with  the  along  r e l a t i o n s were  a variational and  heave  problem  Black  which  (1973) u s e d a m u l t i p o l e  associated  sinusoidally  radiation  far  in  Haskind  source  t o c y l i n d e r s of a r b i t r a r y shape.  and U r s e l l  the  body  b o u n d a r y c o n d i t i o n on t h e body  r e s u l t s i n an i n t e g r a l  this  the  a point  surface  to  i n terms of a  along  represents  the  with  respect  a set of l i n e a r  to  equations  5  which  can  be  variational larger  using  that  i s however efficient  approach case  approach  than  matrix  of A  boundary  diffraction used  by  (1979)  that  Green's  in  et  results  The al  the  (1975)  Ursell  of  field.  The  equations  much  e q u a t i o n method.  banded  Bai  and  et  al  (1975)  can  be  (1983)  in  The  solved used  studying  approach  wave  the  has  in  solve  an the  Finnigan  problems  to  water  procedure  second  the  wave  previously and  and  been  Yammamoto  by  Isaacson  interaction.  complexity  has  for finite  (1973) and  and  Green's  to  wave-structure  which  present  involving  thesis  (1976)  avoids  conditions of  and  this  nonlinear  function  boundary  system  method  for two-dimensional  method  potential  integral  of  integral  Ijima  present  the  cylinders.  problem.  for  and  the  symmetric  used  (1981)  Bai  give  leads to a of  to  multiple  is  to  techniques. Leonard  similar  identity  of  solved  of  satisfy  deriving the  of  finite  are  compared  a  various  depth.  well  The  with  water  depth,  as  as  Garrison  (1984)  for infinite  The those  Bolton water  depth.  1.2.2  EFFECTS OF Previous  structures scattered There determine  studies  in  of  the  directional  i n the have the  DIRECTIONAL WAVES loading seas  are  and few  response and  of  widely  literature. been  response  two of  general structures  approaches in  used  to  short-crested  6  s e a s . The more common a p p r o a c h approach  where  transfer  functions  spectra  linear  theories  which  t o the response  n a t u r e . Time involve Fast  either  Fourier  analysis  is  seas. the  on a  transfer  found  Thompson  vertical  of  of a  when  the  nonlinear  directional  seas  of white noise The t i m e  or  domain  more e x p e n s i v e  than the  (1976) computed  t h e wave  cylinder  diffraction functions.  to  is  filtering  generally  and  large  Linear  wave  approach.  Huntington loads  used  (FFT) techniques.  however  to determine  incident  often  process  the d i g i t a l  f r e q u e n c y domain  are  description  Transform  the  domain  spectra.  interaction domain  frequency  a r e used  relate  Time domain s i m u l a t i o n s wave-structure  i s the  be  in  theory  The  in  was  short-crested  used t o determine  theoretical  results  were  good  agreement  with  experimental  proposed  a hybrid  method  of computing  which  incorporates  measurements. Dean the  (1977)  wave l o a d s  on o f f s h o r e  structures  both the n o n l i n e a r i t y and d i r e c t i o n a l i t y linearized  form  of  Morison's  determine .  the  effect  of  f a c t o r s were p r e s e n t e d  spreading  function.  waves  on  (1982) s t u d i e d  the  loads  f a c t o r s were p r e s e n t e d  on  equation  for  the  the e f f e c t s a  was  directional  reduction  Battjes  o f t h e waves. A  long  of  wall  to  waves.  Force  cosine  power  directional  structure.  for a vertical  used  Reduction  occupying the  7  entire  water d e p t h and  type d i r e c t i o n a l Dallinga directional barge  used  et  al  (1984) i n v e s t i g a t e d  spreading for  the  transfer  functions.  in  equation  was  Hackley  response  water  by  the  reduction compared  in to  significant  the  determine  inline  long-crested  Georgiadis  the  used  to  response seas.  al  of  (1979) used a loading  and  s e a s . The  Fast  determine  the  f o r use i n found  in short-crested There  the  scheme.  (1979)  (1984) u s e d a Monte C a r l o  s e a s . The  determined  accelerations et  to  differential  difference et al  a  was  a  seas  also  a  simulation  to  response.  the a p p r o p r i a t e  short-crested  and  Shinozuka  transverse  directions  in short-crested  T r a n s f o r m t e c h n i q u e was  Morison's equation.  (1982) used  the  simulate  velocities  horizontal  response  solving  the  (1982) b o t h  flexible  and  of a  obtain  Lambrakos  Shinozuka  to  structures  to  of  platform.  wave l o a d s were  using a f i n i t e and  motions  jackup  used  slender  and  obtained  particle  a  frequencies  approach  of  power  effects  and  s e a s . Lambrakos  equation  (1979)  domain  Fourier  wave  of motion  of  (1984) a n d  directional  M o r i son's  cosine  the  loads  was  t h e s e a s u r f a c e . The  structure  time  theory  and G r e a t e d  number o f  describe from  the  the r e s p o n s e of l o n g  cylinders finite  on  transport  diffraction  Bryden  f o r the  spreading function.  Linear  studied  a pipeline  nodal  f o r c e s on  r e s p o n s e of  the  structures in structure  was  8  then evaluated using a d e t e r m i n i s t i c  analysis.  1.3 DESCRIPTION OF METHOD  The  analysis  o f t h e dynamic  response of long  i n d i r e c t i o n a l s e a s c a n be d i v i d e d The  first  diffraction  of  semi-immersed Green's forces  part  involves  a regular  identity  and hydrodynamic  solving  the problem  of the  o b l i q u e wave t r a i n b y a n  infinite  fluid  motion  diffracted  i s used  i sdescribed  wave,  their  the values normal  function  and  of  forced  The  finite  incident  waves f o r e a c h mode o f  second i d e n t i t y i s used  Green's  on  a  boundary  free  to  function  surface  only  has  the  of  and to  Green's consists  radiation  s a t i s f y the  equation which i s t h e two-dimensional  H e l m h o l t z e q u a t i o n . The b o u n d a r y number  to  t h e unknown v e l o c i t y p o t e n t i a l s a n d  body s u r f a c e ,  governing d i f f e r e n t i a l modified  the  i t s n o r m a l d e r i v a t i v e s . The b o u n d a r y  t h e immersed  surface.  of  derivatives  and  i n terms of a v e l o c i t y  o f components due t o  motion of t h e c y l i n d e r . Green's relate  t o compute t h e e x c i t i n g  coefficients.  p o t e n t i a l which c o n s i s t s wave,  i n t o two p a r t s .  c y l i n d e r . An i n t e g r a l e q u a t i o n m e t h o d b a s e d on  second  The  structures  segments.  Application  i sdivided of  the  into a various  boundary  conditions  on t h e v a r i o u s s u r f a c e s y i e l d s a s e t o f  algebraic  equations  which  velocity  potentials.  Bernoulli's pressures  and  equation hence  can  be  i s then  solved  used  the e x c i t i n g forces  to  to  obtain  the  compute  the  and hydrodynamic  9  forces forces  due t o t h e m o t i o n s o f t h e c y l i n d e r . The h y d r o d y n a m i c c a n be e x p r e s s e d  i n t e r m s o f components  i n phase  with  the  body a c c e l e r a t i o n and v e l o c i t y . T h e s e a r e r e f e r r e d t o a s  the  a d d e d mass and damping c o e f f i c i e n t s r e s p e c t i v e l y . The  reflection  determined the  with  surface  the pressure  combined  added with  elevation  and  equations  of motion a r e then  coefficients  of i n e r t i a  motion  solved  response amplitude  two-dimensional obtain  the t o t a l  The results  for  a  multi-directional approach.  The  three  body.  The  r e f e r r e d t o as  oblique  seas using  the  short-crested  frequency  length,  the  t h e body a x i s t o  on t h e s t r u c t u r e .  of the a n a l y s i s  regular  along  involves wave  sea  spectrum  function. A cosine  extending  train  linear  to  transfer  surface  s p e c t r u m c a n be e x p r e s s e d a s t h e p r o d u c t  spreading  then  the amplitudes  finite  t e r m s o f a d i r e c t i o n a l wave s p e c t r u m . The  one-dimensional  obtain  the  often  of  are integrated  wave l o a d s  second part  are  operator. structure  forces  to  to obtain  the  rigid  used t o  o f t h e body and  for  body m o t i o n p e r u n i t wave a m p l i t u d e  a  is  at  surface.  coefficients of  amplitudes  equation  of  For  wave  are  t o the v e l o c i t y p o t e n t i a l  damping  stiffness  coefficients  a t the free  t h e mass o r moment  linear  equations  Bernoulli's  set to zero mass  hydrostatic  coupled  transmission  the asymptotic  surface.  t h e water  The  the  by e v a l u a t i n g  radiation  relate  and  the  random function  i s described in directional  of the  and  power s p r e a d i n g  a  wave  conventional directional  function  which  1 0  is  independent The  of frequency i s used  exciting  obtained  force  by m u l t i p l y i n g  appropriate  transfer  and  i n t h i s study.  body  the i n c i d e n t function  or  response  wave s p e c t r u m response  o p e r a t o r . The e f f e c t s o f wave d i r e c t i o n a l i t y a  directionally  factor The  t o be a p p l i e d  mean  are seas.  averaged,  also  frequency  compared  are  with the amplitude  i s e x p r e s s e d as  dependent  t o the one-dimensional  square v a l u e s of t h e r e s p o n s e  spectra  force  reduction spectrum.  in short-crested  seas  to corresponding results for long-crested  2.  2.1  long  t r e a t i n g the problem  structures the  with  infinite  an  arbitrary The  interaction  cylinder the  assumed  a  semi-immersed  surface.  The  and  a  regular  shall  oblique  horizontal  the  the  Keulegan-Carpenter amplitude  of  large  field.  the thin  flow  indication is  flow and  to  fluid  irrotational  effects  considered  negligible  confined  An  i s  incident  assumed  body,  of  we  response of first  wave  train  cylinder  of  shape.  diffract  theory.  of the dynamic  i n m u l t i - d i r e c t i o n a l seas,  consider  can  enough  Flow  of  layer  thus  considered  be  using  of the importance  of  Keulegan-Carpenter  fluid  i s defined  motion  to  a typical  to  potential  flow  the  are  on t h e body be flow  separation  number,  as  to  effects  viscosity  boundary  solved  as  separation  effects  problem  number  so  R.  ratio  The  of  dimension  the  of the  that i s K  where  U  m  period  = U T/B  (2.1)  m  i s t h e maximum and  B  i s  range  of frequencies  less  than  Sarpkaya For this  THEORY  I N T R O D U C T I O N  Before  are  D I F F R A C T I O N  two  and  a typical used  and  velocity,  dimension  in this  flow  is  the  of t h e body.  study,  separation  T  K will  should  wave  For the  usually  not occur  be  (see  Isaacson,1981).  rectangular  study,  particle  vortices  section are  cylinders  usually 11  which  formed  at  a r e used i n the  sharp  12  corners. Various authors Jamieson  ( 1 9 7 6 ) ) have however  potential  flow  cylinders For  when  coefficient It  exciting and  still  wave  s m a l l so t h a t  with a f l o a t i n g  superposition  cylinder  oscillation  of  two-dimensional obtained  i s assumed  periodic  three-dimensional  along  problem one.  Even  f o r an i n f i n i t e  structures  ships,  by a f i x e d  added  with  the  (1)  cylinder,  an  infinite  i n an  otherwise  mass  and  damping  motions of the  the complete  cylinder  motion.  two p a r t s :  The wave h e i g h t and o s c i l l a t o r y  a r e assumed  of  the wave-structure  sway and r o l l  i n terms of  near  v i s c o u s damping  into  associated  i n heave,  of motion i s  particularly  an e m p i r i c a l  bodies  vortices.  problem  of  c a n be r e p r e s e n t e d  diffraction  and  forced  problems.  The  axis,  f o r such  to separate  forces  expressed  interaction  motion  and  for floating  oscillating  a linear  results  amplitude  by v i s c o u s damping  convenient  problem  coefficients. cylinder  roll  f o r c e s due t o wave d i f f r a c t i o n  water  between  s h o u l d be i n c l u d e d i n t h e e q u a t i o n s  (2) h y d r o d y n a m i c  cylinder  agreement  experimental  the  frequency  i s usually  interaction  good  Mogridge and  d e s p i t e the formation of the  affected  resonance  et al ( 1 9 7 9 ) ,  found  and  cylinders,  significantly  by  theory fixed  floating  the  (Bearman  of  ignoring  finite  flexible  with  i t s amplitude  the a x i s of the c y l i n d e r , can  be  reduced  though the n u m e r i c a l  cylinder,  they  are  l e n g t h by i n t e g r a t i n g  end e f f e c t s .  Korvin-Kroukovsky's  For non-uniform (1955) s t r i p  of  so t h e to  a  results are extended  along  bodies  to  t h e body such  t h e o r y c a n be  as used  13 w i t h t h e t w o - d i m e n s i o n a l r e s u l t s , F o r head normal  t o the  length  of  magnitude  cylinder  the  axis),  cylinder  the  becomes  as a t y p i c a l c r o s s  the  sectional  wave  direction  crests  along  same  the  order  of  and  the  dimension  A t h r e e - d i m e n s i o n a l model  w h i c h c o n s i d e r s end e f f e c t s w o u l d h a v e t o incident  (wave  wavelength of  p r o c e d u r e i s no l o n g e r a p p l i c a b l e .  seas  be  used  as  the  moves s u b s t a n t i a l l y away f r o m t h e  beam d i r e c t i o n .  2.2  THEORETICAL FORMULATION  2.2.1  WAVE DIFFRACTION PROBLEM A  regular  small  a m p l i t u d e wave t r a i n o f h e i g h t H  a n d a n g u l a r f r e q u e n c y co i s o b l i q u e l y . i n c i d e n t infinitely propagate  long  fixed  horizontal  upon  c y l i n d e r . The  i n water of depth d i n a d i r e c t i o n  waves  making  angle  0  system  i s r i g h t handed w i t h z measured upwards from  still  water  with  the  x axis  level  i s p a r a l l e l to the a x i s  The  origin  still of  ( s e e F i g . 1 ) . The c o o r d i n a t e  The  fluid  is  the  flow  i n c o m p r e s s i b l e and m o t i o n may potential  of the  infinite  the  above or below  i s at the  the  centre  a s s u m e d t o be i n v i s c i d a n d irrotational.  t h e r e f o r e be d e s c r i b e d defined  y  cylinder.  the ( x , y , z ) c o o r d i n a t e system  water l e v e l v e r t i c a l l y  gravity.  an  and t h e x-y p l a n e h o r i z o n t a l . The  axis  of  an  The  fluid  i n terms of a v e l o c i t y  by  u = V#(x,y,z,t)  (2.2)  1 4  where the  u i s the f l u i d  Laplace  and # must  = 0  2  the f l u i d  sufficiently  (2.3)  region.  small  The  so  wave  that  height  linear  boundary  g  constant  The  that  fluid  free  be  equal  to  given  to zero surface  f o r convenience. boundary  a t the f r e e  +  surface.  Hi2  the  velocity  +  square eqn.  the s t i l l  i n s t a n t a n e o u s water can  surface  then  boundary  free  surface  This  fluid  c a n be e x p r e s s e d a s  !f!?  < 2  neglecting  -5) the  t e r m i n e q n . (2.4) and t h e wave  conditions  at  condition  the normal v e l o c i t y of a  in  free  is  (2.4)  (2.4) and (2.5) a r e l i n e a r i z e d by  equations  usual  = R  2  s t e e p n e s s terms  the  the  pressure  the normal v e l o c i t y of  H =& Eqns.  t h e dynamic  s e t equal  kinematic  elevation particle  is  i s the g r a v i t a t i o n a l a c c e l e r a t i o n and R i s the  Bernoulli  requires  theory  equation  | | + grj + ^ ( V # ) where  to  assumed  conditions.  On t h e f r e e s u r f a c e , by t h e B e r n o u l l i  is  wave  a p p l i c a b l e and c o n s e q u e n t l y $ i s s u b j e c t linearized  satisfy  equation  V <I>(x,y,z,t) within  v e l o c i t y vector  (2.5), water  surface  and level  applying  z=0 r a t h e r  elevation  be combined condition  by  to give  z=7j.  the  than at The  two  the l i n e a r i z e d  15  || for  simple The  and  = 0  hence S  (2.6)  harmonic motion.  immersed  surface,  at z = 0  body  surface  is  the normal v e l o c i t y D  must  equal  assumed  impermeable  of the f l u i d  on t h e body  zero  a  ||=0 where  n is  directed and  on S  a  direction  normal  i n t o t h e body. The  impermeable  addition  satisfy  surface  horizontal  a t z=-d  (2.8)  t o t h e above b o u n d a r y c o n d t i o n s , condition  $  has  to  at the f a r f i e l d  t o ensure  t o assume t h e v e l o c i t y  potential  solution.  It to  body  giving  a radiation  a unique  the  s e a b e d i s assumed  || = 0 In  to  (2.7)  B  is  convenient  be o f t h e form * = #  where  0  + *a  4> and 4>j, a r e 0  incident  (2.9) the  velocity  potentials  and d i f f r a c t e d waves r e s p e c t i v e l y .  wave p o t e n t i a l  i s given  •„(,.y.«,t)  r  The  by l i n e a r wave t h e o r y  - Ret^gH  c  for  angular  k  is  the  frequency  wave  number  incident  as  ° ^ d f "  x e x p { i (kxcos0+kysin/3-o>t)} ] where  the  which i s r e l a t e d  u by t h e d i s p e r s i o n  relation  (2.10) t o the  16  k tanh(kd) The  radiation  diffracted is  given  a  (2.11)  condition  + ikcos/3  away from  = 0  that  the  at a f i n i t e  distance,  modes due  analysis  to  radiation  condition  The  find  fluid  *(x,y,z,t)  (2.12) boundary  from t h e o r i g i n  An  a t which  R  is considered periodic  t h u s be d e f i n e d  by  x'  = x/a,  d'  = d/a,  For  now  and  beam o f the c y l i n d e r , z'  = z/a,  u = ^g -, 3  y'  as  nondimensional  „  variables  a.  = y/a,  k'  =  ka (2.14)  v = kasin/3  the primes  i t i s understood  nondimensional.  h e n c e f o r t h be  I.  = R e [ ^ i r g ^ > ( x , z ) e x p { i (kysin/3 - cot)}] (2.13)  convenience,  variables  the  i n time  i s a l s o convenient to n o n d i m e n s i o n a l i z e the the h a l f  body  approximate  i s given i n appendix  of t h e c y l i n d e r . A  is  where  of t h e t h e  sufficiently.  i s applied  motion  <t> c a n  X  infinite  t h e optimum d i s t a n c e X  as a l o n g t h e a x i s  potential,  the  to the presence  a r e assumed t o have d e c a y e d  using  the  cylinder  a t x = ±»  numerical approximation,  the evanescent  It  ensures  by  truncated  well  which  waves a r e t r a v e l l i n g  j-— In  = £p  have been d r o p p e d that  Dimensional  the  from  variables variables  b a r r e d where n e c e s s a r y f o r c l a r i t y .  the are will  17 The  boundary  potential  can  V 0«  now -  2  value be  problem  2  the  diffracted  s t a t e d i n nondimensional = 0  v 4>k  for  i n the  form  fluid  as  (2.15a)  -g-pj— = u<t>*  a t z=0  (2. 1 5b)  •g-^- = 0  a t z=-d  (2.15c)  •g^- = ikcos/3tf>  at x = ± X  a  d(f>n  (2.15d)  R  d<t>  0  1TT  -cTfr  =  o  n  (2.l5e)  S B  where  *° • ° c S s h U d f C  The  three-dimensional  been r e d u c e d equation  2.2.2  to the  Laplace  (2.16)  equation  two-dimensional  (2.3)  modified  has  now  Helmholtz  FORCED MOTION PROBLEM an  h e a v e , sway and motion  infinitely roll  is periodic  the c y l i n d e r .  The  mode i s g i v e n  by  E (y,t) = £^  cylinder roll  *  as  long c y l i n d e r  shown i n F i g . 2.  i n time  displacement  or r o t a t i o n  k  s  with  the  complex a m p l i t u d e  k=1,2,3 c o r r e s p o n d i n g  modes  oscillating  in  Each  of  as w e l l as a l o n g  ReU exp{iUy-cot)}]  k  and  exp(ikxcos^)  (2.15a).  Consider  where  ) ]  {J  mode  t h e a x i s of in  the  I \' } 2  of o s c i l l a t i o n to the  r e s p e c t i v e l y . Throughout  sway,  kth  (2.17) of  the  heave  the f o l l o w i n g  18  development, with  the upper t e r m s i n the c u r l y  e a c h o t h e r , and s e p a r a t e l y t h e  with  each other.  by a, w h i l e  i-  3  (•, and  corresponds  The v e l o c i t y is  given  V  lower  have been  2  t o the r o l l  o f t h e body  apply  terms  apply  nondimensionalized angle  in radians.  surface i n the d i r e c t i o n  n  by  3 _, = Z |fn k=1  n  £  brackets  =  R  e  [  1  k  3 -i"a£ n exp{i(vy-ut)}] k=1 k  (2.18)  k  where  n  1  = x  )  n  n~ = n„ n and  n  3  , n  vector  }  = (z-e)n  - xn  x  z  )  are the d i r e c t i o n  n on t h e immersed  (2.19)  c o s i n e s of t h e u n i t  body  the p o i n t about which the r o l l normal  velocity  surface  must e q u a l  of  the the  s u r f a c e and  normal  (0,e) d e n o t e s  m o t i o n i s p r e s c r i b e d . The  fluid normal  on  the  velocity  immersed  body  of  body  the  yielding  ^ 3n This  = V  on S.  n  boundary c o n d i t i o n i s s a t i s f i e d  position  o f t h e body  position  o f t h e body.  From potential  equations  (2.20)  B  r a t h e r than  (2.18) and  at  (2.20),  at the the  equilibrium  instantaneous  the f o r c e d motion  f o r t h e k t h mode o f m o t i o n c a n be e x p r e s s e d = Re[-icja ^.^. 2  exp(iUy-cot)} ]  as  (2.21)  19  The  linearized  boundary c o n d i t i o n  can  t h u s be e x p r e s s e d a s 30v  The  potentials  value  is  in  and  for  hence  order  method  f o r the  <j>^ (k= 1 ,2, 3 , 4 ) .  the f l u i d  </>(Jj.)  surface  (2.22)  the  forced  governed  motion  by  eqns.  (2.22).  integral  may be a p p l i e d o v e r  in  body  FUNCTION SOLUTION  used as t h e b a s i s  region  is  c  boundary  potentials  problem  0j (k=1,2,3)  ( 2 . l 5 a - d ) and e q n .  A  the  k=1,2,3  boundary  2.3 GREEN'S  on  i n v o l v i n g Green's  numerical  The s e c o n d  a closed  surface  t o the boundary  i t s normal  evaluation  S containing  the  theorem  the  fluid  o f t h e p o t e n t i a l <f>(x)  values  of the  3t/>(JL)/9n.  derivative  of  form o f G r e e n ' s  t o r e l a t e the values  region  identity  potential  This  can  be  expressed as  *<i>  = ^  where G ( x ; £ ) the  point  (£/$)  over  surface free  as  fU(i)f§(x;i)  i s an a p p r o p r i a t e (x,z)  being  i$(JL)G(x;I)]dS  "  Green's  considered  which the i n t e g r a t i o n i s  S  surface  comprises  t h e immersed  function,  (2.23)  x  denotes  and £ d e n o t e s t h e p o i n t performed.  body  S„, the r a d i a t i o n surface F  surface  The  closed  S , t h e mean fi  S_., a n d t h e s e a b e d S_. R D  shown i n F i g . 3. When t h e i n t e r i o r  within,  eqn.  (2.23)  point  x approaches  reduces  to  the  the boundary following  from  integral  20  equation  • (£> - i SSi^V^liV The  Green's  equation the  function  (2.15a)  point  " U i *''i {  which s a t i s f i e s  i n an unbounded  )G{  '  )]dS  (2 24)  the modified Helmholtz  fluid  and  is  singular  x=i. i s g i v e n by G ( x ; £ ) = -K (vr)  (2.25)  0  where K is  0  i s the modified  the distance r  The  =  function  between  |1 -  x| =  K (x)  Bessel  function  the p o i n t s [ U - x )  +  2  of order  zero and r  x and £  ( S - Z )  ]  2  1  /  (2.26)  2  - I n x a s x —s» 0. The  0  which s a t i s f i e s  Green's  the two-dimensional Laplace  function  equation  G ( x ; l ) = In r is  at  thus obtained Since  the  computationally  (2.27)  a s 0 —> 0 ° . seabed  is  assumed  more e f f i c i e n t  and an a l t e r n a t i v e  Green's  horizontal,  t o exclude  function  i t  is  t h e seabed from S  which takes  into account  symmetry a b o u t t h e s e a b e d c a n be d e f i n e d G(x;£) = -[K (;/r) + K ( * » r ' ) ] 0  where  r'  is  the  (£/~($ 2d)) which +  r«  If  distance  eqn.  to  = |£' - x| =  be i n c l u d e d  (2.25) u s e d  between  i s the r e f l e c t i o n  the depth v a r i a t i o n s  have  (2.28)  0  [ U - x )  2  the points  of £ about t h e seabed:  + ($ + 2 d + z ) ]  are significant,  i n S and t h e Green's  instead.  x and £' =  2  the  (2.29)  l / 2  seabed  function  would  g i v e n by  21 The i n t e g r a l numerically fluid  to  equation give  the  and h e n c e p r o v i d e  (2.24)  can  potential  the solution  now  be  evaluated  </> a t any p o i n t t o the  i n the  boundary  value  problem.  2.4  EXCITING FORCES, ADDED MASSES AND Once  the  velocity  DAMPING  potential  h y d r o d y n a m i c p r e s s u r e c a n be computed Bernoulli  the  the  linearized  = - p | | = iw/o$  forces  and  integrating surface  from  obtained,  equation  p The  is  COEFFICIENTS  (2.30)  moments  per u n i t  length  a r e d e t e r m i n e d by  the hydrodynamic p r e s s u r e over  t h e immersed body  S„. a  The e x c i t i n g incident height  force  and s c a t t e r e d  i s given  F  per unit  length  waves and i s p r o p o r t i o n a l  by  j HMf * j =  n  Fj(j=1,2)  respectively of  while F  equations  Fj(y,t)  =  denotes 3  (2.9) and  the  the  t o t h e wave  {i - 3 I -  d s  ,2  B  where  w h i c h i s due t o  sway  and  (2  heave  force  denotes the r o l l  moment.  Substitution  (2.13) i n t o e q n .  (2.31) y i e l d s  pg§{f[2}Re[;(0 + >„)n .exp{i(vy-cjt)}dS]|1:3' |(2.32) 2  o  S  t  :]  B  The d i m e n s i o n l e s s e x c i t i n g  force  amplitude  i s given  by  31)  22  F.(y,t) Cj  = — ^•pgHa  The e x c i t i n g  force  Ull' }  (- >  2  = J ( * o + * « > n j dS B  2  33  s  could  a l t e r n a t i v e l y be d e f i n e d  by  F • (y,t)  r~7"{i}  lCjl  =  {1l 3 ' 2 }  cos(,y- t A.) W  +  (2.34)  2PgHa where t h e p h a s e a n g l e A j i s d e f i n e d Aj There  =tan"  are  also  motions of  the  amplitude due  Im(C..)  1  hydrodynamic cylinder  forces  which  are  associated  * j  i  n  ji=3' j 2  d  s  1  2  S  ;  one  component  in  can a l s o  phase w i t h  F  where  i j  p^j  ( 2  -  3 6 )  forced  motion  i  B  2  force  3  « .n exp{i(vy-wt)}dS]  )i=3' | This  2  '  B  pcj ^|^a}Re[/  i  the  of the f o r c e  3- ' '  Substitution of t h e e q u a t i o n for the p o t e n t i a l s (2.21) i n t o e q n . (2.36) y i e l d s F y"  to  o f m o t i o n c a n be e x p r e s s e d a s  =  S  with the  proportional  m o t i o n . The i t h component  t o t h e j t h component  i j  (2.35)  Re(Cj)  of c y l i n d e r  F  by  be e x p r e s s e d  i n phase w i t h  3=1,2,3  (2.37)  i n t e r m s o f two  the a c c e l e r a t i o n  components;  and the o t h e r  the v e l o c i t y  *--OijSj and  X^j  -  X  i j j  are  3" ' '  E  the  1  added  2  mass  3  (  and  2  '  3  8  )  damping  23  coefficients eqn.  (2.38)  F  respectively.  Substitution  o f e q n . (2.17)  into  gives  i j  {*}  =  R e [ { c j 2 M  ij*j  +  i  a  ,  i j^j  X  )  jj='j  e  x  P  {1 (  f'Y-^t)} ]  2  i=1,2,3  3  Comparing  eqn.  (2.37)  with  eqn  (2.39)  n o n d i m e n s i o n a l added mass a n d damping M  i j  (2.39) gives  c o e f f i c i e n t s as  = R e [ / 4>.n. dS] S 3  m  pa  the  (2.40)  B  = Im[J 4>.n. dS]  — pwa m  s  where t h e c o n s t a n t  (2.41)  B  m i s given as  2 f o r ( i , j ) = (1,1) a n d (2,2) m = {3  f o r ( i , j ) = ( 1 , 3 ) a n d (3,1)  (2.42)  4 f o r ( i , j ) = (3,3) The H a s k i n d an  (1953) r e l a t i o n s  alternate  Applying  way  of  s e e Newman,1962)  calculating  the  Green's theorem t o t h e d i f f r a c t i o n  j - 5 r r " *«-9n  ( 0  Substituting  1 +  Applying eliminates  )  d  S  the  boundary  force  provide  exciting  forces.  potential  3 = 1/2,3  0  90K *j7Jn-  into  potential  gives  (2.43)  e q n . (2.32)  gives  ) d s  condition  the d i f f r a c t i o n  exciting  =  t h e above e x p r e s s i o n 9#/ (0O-3W  the  (also  {  given  by  eqn.  2  '  4  4  )  (2.l5e)  from t h e e x p r e s s i o n f o r  24  d<j> •  / S  There  is  coefficient  at  "  0  30o  j ? T  a  direct  (2.45)  relation  and t h e amplitude cylinder  to  between  about  as the r a t i o  the amplitude  the  damping  o f t h e waves g e n e r a t e d  symmetrical  |$^| c a n be d e f i n e d  |x|=°°  r)dS  B  also  oscillating ratio  U O - S T P  by  an  x=0. An a m p l i t u d e  o f t h e wave  of o s c i l l a t i o n  amplitude  of the c y l i n d e r ,  that i s  where for  to  |7j^| i s the amplitude  of t h e r a d i a t e d  t h e i t h mode o f o s c i l l a t i o n By  equating  the  energy  infinity,  the oblique  to  the  the  does  negative not vanish  does not s a t i s f y motion energy x=-X  R  in oscillating  radiated  ratio  potential  across  the  a control  | x | =°°  cylinder  surface at  ( s e e Newman,1977)  by e v a l u a t i n g radiation since  can a l s o  surface  the i n c i d e n t  the radiation condition. i s proportional  the e x c i t i n g force  be r e l a t e d  the i n t e g r a l i n eqn. (x=-X ). R  wave  Since  t o t h e square  f l u x , e q n . (2.45) c a n be i n t e g r a t e d to give  at  of the c y l i n d e r .  c a s e . The e x c i t i n g f o r c e  amplitude at  integral  flux  i t c a n be shown t h a t  for  (2.45)  t h e work done  waves  over  c o e f f i c i e n t as  The  potential  the  forced  root  of the  the depth a t  25  Equations direct the  (2.47) and  relation  damping  (2.48) c a n  between t h e  be  combined  exciting  to  provide  force c o e f f i c i e n t s  a and  coefficients  X. . I il  =  c  1  Equations  results  EQUATIONS OF The  (m.  2  J  1  now  .+fi..)  -  and  c^j  1  J  matrix  coefficients or  present.  k  a  be  ico\..  of  obtained  l  are  be  the  t h e mass and  noted  frequency  assumed  the  due  to  the  the  equations  form  i = 1,2,3  (2.50)  stiffness  Additional forces be  that  and  the  hydrostatic  due  i n c l u d e d i n eqn. for  the  case  i s important  would  to  (2.50)  of  roll  particularly  have t o  that  in deriving the  the  be  cylinder  a d d e d mass was  of motion v a r y i n g s i n u s o i d a l l y  cylinder  thought  on  included  applications.  coefficients amplitude  of  A  resonance  was  are  1  respectively.  It should  solving  c. . ] S . = F . ( y , t ) J J  +  •J  near  It  check  cylinder  by  of m o t i o n  v i s c o u s damping may  practical  useful  the  n o n l i n e a r v i s c o u s damping  in  (2.49)  1 / 2  .  motion, the  cos/3]  d  provide  equations  m^j  if  2  response  where  moorings  )tanhkd sinh2kd  +  MOTION  waves c a n  of m o t i o n . The  [-co  1  obtained.  dynamic  exciting  3 2 j_1  <  (2.47)-(2.49)  numerical  2.5  f — m pcua  of as  as  well  the  ratio  as of  in the  time. incident  and  damping  flexible  with  along  length  of  sin/3 c a n  be  The  the  term  wave l e n g t h  to  its  the  26  the  wave l e n g t h  cylinder  has  cylinder  and  along an  the  axis  of  the  cylinder.  i n f i n i t e wavelength along  hence c o r r e s p o n d s  to a  flexible  the  A  rigid  axis  of  cylinder  the with  0=0°. The  components of m m.  1  D -mz,  'G  where m  i s the  coordinate moment o f be  of  I where axis.  0  r^  the  m  0  are  given  as  (2.51)  0  ~  unit of the  length  of  the  g r a v i t y and y axis  per  I  body, z^ i s the  0  unit  i s the  polar  length.  I  z  mass may  0  as = m(r  calculating  stiffness  -mz.  about  2  y  + z*) G  the  •  radius  hydrostatic  equilibrium  0  centre  i s the  The  mass m a t r i x  mass p e r  inertia  expressed  the  of  gyration  stiffness  forces  small  m a t r i x components a r e  c. . = D 1  0  0  0  C 2 2  0  C  2  3  matrix  required  position for  of  to  is  restore  amplitude given  the  (2.52)  body a b o u t  the  determined the  body  y by  to i t s  displacements.  The  as  0 C  (2.53)  23 C  3  3  where ' c  2 2  =  pgB  c  2 3  =  pgBx  (2.54a)  f  (2.54b)  27  = pgA[(S,,/A) + z  Q  where B i s t h e beam of t h e c y l i n d e r , the  waterplane  symmetrical of  line  about  x=0, z  f i  is  the waterplane  a x i s per u n i t  2  stability  positive.  metacentre centre  bodies  i s t h e z c o o r d i n a t e of t h e  centre  of  inertia  l e n g t h , and  about  the  x  (S^/A)  (2.55)  3  +  requires  (2.54c), z  f i  i t  z^ f o r t h e f l o a t i n g of  motion  t h e complex a m p l i t u d e s  given  wave  frequency  terms  of  is  evident  the  body t o be  oscillation,  and d i r e c t i o n  of  the c o e f f i c i e n t that  c  3  3  the  stable.  (2.50) c a n now be s o l v e d t o  t e c h n i q u e . The a m p l i t u d e in  that  has t o be l o c a t e d h i g h e r t h a n t h e  obtain  described  volume p e r u n i t  = B /12  in roll  equations  inversion  zero  of  for  a r e a moment  From e q n .  of g r a v i t y  The  to  centroid  length, that i s  = / x dx B Static  x^ i s t h e  equal  buoyancy, A i s the d i s p l a c e d is  be  and  (2.54c)  - z ]  f i  £j  for  u s i n g a complex  o f body m o t i o n  response  amplitude  is  any  matrix often  operator  d e f i n e d as  Z.(u,/3) = The body  (2.56)  response, a m p l i t u d e motion  travelling  due  to  operator represents the amplitude a unit  i n d i r e c t i o n /3.  amplitude  of  wave o f f r e q u e n c y co,  28 2.6 REFLECTION AND TRANSMISSION COEFFICIENTS Another for  such  two q u a n t i t i e s  structures  reflection  and  as  of p h y s i c a l  floating  transmission  a r e o b t a i n e d by e v a l u a t i n g  coefficients.  this  a s y m p t o t i c wave a m p l i t u d e f r o m :  the  cylinder  related  for  is  wave by a f i x e d  equation,  the  body.  wave  amplitude  i n t o e q n . (2.57)  is  p o t e n t i a l by < - > 2  the equation f o r the forced yields  motion  57  potentials  t h e a s y m p t o t i c wave a m p l i t u d e  e a c h mode o f m o t i o n exp{iUy-ot)}]  = Ret ^-a ^ t> (x. 0) 2  i<  i  wave a m p l i t u d e r a t i o  r  previously  defined  (2.58)  by  eqn.  (2.46)  now g i v e n as  l*il evaluated  =  TaTtl  at x=±X . R  =  ^i^i  ( x  and t h e v e l o c i t y 0(x,z) =  A  o  I  '°>  At the r a d i a t i o n  modes a r e assumed t o have d e c a y e d I)  (1) t h e o s c i l l a t i o n s o f  "g- f f ^ ' Y ' O ' t )  TJ. The  at  i n i t s t h r e e modes, a n d (2) t h e r e f l e c t i o n and  t o the v e l o c i t y  Substituting (2.21)  The c o e f f i c i e n t s  R  of t h e i n c i d e n t  =  the  ( x = ± X ) . There are c o n t r i b u t i o n s t o  Bernoulli's  *  are  t h e component wave a m p l i t u d e s  radiation  From  especially  breakwaters  the  transmission  surfaces  interest  { 2  surface,  -  5 9 )  the evanescent  sufficiently  (see appendix  p o t e n t i a l s a r e of t h e form  cosh[k(z+d)] cosh(kd)  e  x  p  (  ±  i  k  x  c  o  s  g  )  a  t  x  = ±X  (2.60)  R K  29  where A Given  i s t h e complex  0  a m p l i t u d e of the p o t e n t i a l  = s i n h ( 2 k d ) + 2kd  2  the  coefficient  orthogonality  A  i s given  °i  =  A  4  can  0  condition  be  obtained  of the  sinhtSkdi^Zkd  applying  cosine  exp(±ikxcos/5)  obtained be the  to  in a similar  obtained  waves  (x=-X ).  Substitution  potential  given  R  TJ The  r  reflected is  The  the  )  the  /«.cosh[ k (z+d) ]dz J  x=+X  (2.62)  R  reflection  presence  of  and a  at  the asymptotic the  of  the  by e q n . (2.13)  body a r e  c o e f f i c i e n t can  wave a m p l i t u d e o f radiation  surface  form o f t h e d i f f r a c t e d  into  e q n . (2.57)  i s d e f i n e d as t h e to the incident  wave  yields  ,0)_ e x p { i U y - u t ) } ]  coefficient  wave a m p l i t u d e  negative  transmission  fixed  manner. The r e f l e c t i o n  = Re[.§0,(-X  reflection  l  (2.63)  by e v a l u a t i n g  scattered  g  r a t i o c a n t h u s be e v a l u a t e d a s  e a c h mode o f m o t i o n . due  #  function  ISjl = ^ I A O J I  coefficients  2  as  at The wave a m p l i t u d e  by  hyperbolic  f o r the j t hp o t e n t i a l  (  K  -d  for  z=0.  that Jcosh [k(z+d)]dz -d  and  at  (2.64) ratio  of the  wave a m p l i t u d e a n d  t h u s g i v e n by K  IV I  R  = -i-^-  H/ 2  = |0„(-X O) | R f  (2.65)  30  The  transmission  amplitude positive  of  the  =  T  0  wave  K After  R  as  reflection  boundary  the  freely  floating  =  3 |0,(-X ,O) + Z $.Z.(u,/3)| j=1 3 D  square  of  transmission  K  T  =  |0 (X ,O)  and  body  R  the  transmission  are  determined  (2.68a)  O  R  3 + 0,(X ,O) + Z S j Z j ( w , 0 ) |  (2.68b)  R  PROCEDURE  to evaluate  i s divided  into  considered constant  value at the midpoint replaced  and  reflection  D  order  remembering  of body m o t i o n by s o l v i n g  K  NUMERICAL  (2.66)  (2.67)  a  O  by  by  motion,  respectively  In  principles,  = 1  of for  90/3n  the  are related  coefficients  given  (2.65) and  i n a wave i s p r o p o r t i o n a l t o t h e  2  the  (2.66)  s i d e of eqns.  o b t a i n i n g the amplitudes  equations  at  0)|  c o n s e r v a t i o n of e n e r g y  + K  wave  (2.62).  amplitude,  coefficients  R f  waves  is similarly  R  on t h e r i g h t  the energy  the  scattered  ( x = X ) and  R  e v a l u a t e d u s i n g eqn. Applying  2.7  surface  and  |# (X ,0) + * , ( X  The e x p r e s s i o n s  that  i s due t o t h e a s y m p t o t i c  incident  radiation K  are  coefficient  the i n t e g r a l  N segments w i t h over  equation  (2.24),  the v a l u e of  e a c h segment  o f t h e segment.  by t h e summation  equation  Eqn.  and e q u a l (2.24)  the  <t> o r t o the can  be  31  i 1  0 (x.) = k  ar  N  90i  Ig^i^^dS  2 {*k(£jU  - ^  / G(x.;x.)dS}  j  j k=1,2,3,4  where  t h e summation  (2.69)  i n e q n . (2.69) i s p e r f o r m e d i n a c o u n t e r  c l o c k w i s e manner a r o u n d t h e b o u n d a r y .  Eqn.  (2.69)  can  be  r e w r i t t e n as N  • 5 where  b<j>[  m  {  (  a  1  i j  +  i j  6  )  0  j  ijJn  +  h  k)  }  3  S^j i s the K r o n e c k e r d e l t a 6^  (1 = {  =  k=1,2,3,4  0  function  (2.70)  g i v e n by  i - j (2.71)  The c o e f f i c i e n t s a ^ j a n d b ^ j a r e d e f i n e d a s i j  a  ij InlKot^r.j)  -  4  b. • =  + KoUrl^JdS  •) + . K o U r ! -)]dS  S^oivr.  (2.72) (2.73)  r . . and r ! . a r e g i v e n a s 3  3  1  1  y  r-j  = [(Xj-x.)  rlj  = [(Xj-x^  + (  2  -z.) ] 2  2  j  l  /  + (z +2d+z.) ]  2  2  angles  (2.75)  o f e a c h segment. The  9G/9n may be e x p r e s s e d a s  !§<£i?£j> where  l / 2  j  x^ and X j a r e e v a l u a t e d a t t h e m i d p o i n t gradient  (2.74)  2  - H  c  o  s  7  +  If'  COST'  7 a n d 7' a r e a s shown i n F i g . 4 a n d c o r r e s p o n d between  respectively,  (2.76) t o the  n . a n d r=x.-x., and between n'. and r'=x'.-x. -3 3 - i ' -D -3 - 1 that i s  32  COST  n . • ( x .-x. ) ^ — ~ 3 ~*  =  COST'  =  =3—7  1  -  (2.77) (2.78)  1  where n. = n i + n k ~1 ~ ~ n'. = n i - n k —3 x— z— X  X j i s the point  and is  (2.79)  2  (xj(zj+2d)).  The u n i t  normal v e c t o r n  g i v e n by n = || i -  || k  (2.80a)  The above e x p r e s s i o n c a n be a p p r o x i m a t e d a s Az • _ 2S i The d e r i v a t i v e  Ax , 715" -  (2.80b)  o f t h e G r e e n ' s f u n c t i o n i s g i v e n by  IfKodr) = - J » K , (vr)  (2.81 )  where K , i s t h e m o d i f i e d B e s s e l i*j,  the  integrals  in  a p p r o x i m a t e d by e v a l u a t i n g normal  derivative  coefficient  at  K,Ur..)  1  K,  v  the  (2.72)  Green's  the midpoint  of  and  When  (2.73)  are  function  and i t s  e a c h s e g m e n t . The  (x .-x. )n„ + ( z . - z . ) n l A S . 3 i x ] i ' z' 3  Ur! .) Trr!.  i]  Substituting given  eqns.  one.  a ^ j i s thus given as  a. 3 . = - v- Trr.j -  f u n c t i o n of order  [ (x .-x. )n + ( z . + 2d+z.)n ]AS . D i x 3 1 2 3 the  approximation  i n eqn. (2.80b),  a ^ j becomes  i * j  (2.82a)  f o r the d i r e c t i o n  cosines  33  = -v  a.. *D  where  v  (T  (x .-x. ) A z . D D  Az. = D  z ... D 1  z. D  A  =  x  x.  =  [(A  X j  +  b  i : j  i=j,  singular.  -  1  Z  0  +  2  b.^  the  7  given  as  j  )  2  ]  i s given  0=0°,  l  /  as  before  »  as  i n egns.  (2.72)  -{lnUr/2)  ASi>AS. = - y l l l n - ^ - i  about  coefficients  +  components  0  f o r 0=0°  and  formula f o r  as  the diagonal  vr^O  + 7  ~  1 - K {2i»(z.+d)}]  (2.86)  0  function  are given  are  (2.85)  to the t y p i c a l  seabed  (2.84)  coefficients  1  reduces  Green's the  become  K ('i>r')  the asymptotic  7}  1  the problem the  (2.73)  K,[2v(z.+d)]Ax.  b  Using  and  (2.83)  and Stegun,l964)  = £  H  i * j  0  and using  a..  7T  (2.82b)  2  + K (»rJj)]ASj  i s Euler's constant,  1 1  symmetry  X  integrals  0  where  ( A  = "^[Kod^r.j)  K (*>r)  one.  )  (see Abramowitz  Q  For  j  i#j  1  Evaluating the nonsingular  9K (j>r' )/3n K {vr)  1  ( z - + 2 d + z . ) A x .] D D  1  +  j  (z:-z.)Ax.] D D  I • )  coefficient  When  [ ( x .-x . ) A z . D I D  7rr!^j  ASj The  .)  ,rr. . ID  1  K  -  K,(vr.  given  taken by  two  i n eqn.  into  dimensional (2.27)  account,  with the  34  1 i j [ (XyX.)Az.  b j  - (2j+2d+z )AXj] i  = l ( l n r . j + In r l ^ A S j  t  i * j  (2.87)  i * j  (2.88)  For i = j Ax. a  ii  b.. With  the  provides  AS. AS. = — i t l n - T i - 1 + In 2(z.+d)]  N equations The  a^j  and b ^ j now  relating  various  the remaining  the values  e q n . (2.15)  into  N equations  e q n . (2.70)  2  N 1  Z ( . . + 6. •+^r -b. . ) * •  +  a  a  N  3  Z ( a j=N2+1  1  of < $ > and 9</>/9n o v e r  needed  2«  to solve  f o r 4> and  boundary condition's given  yields  N2 ,,x S ( a - . + S.-U-  +  N  4  K  i  i  3  (a.  j=N4+1  known, e q n . (2.70)  (k) (k) +6..+2gab .)^ ' + I ( a . . + 6 ,mcos0b.. >^ j=N3+1 W  J  , u  i  3  N  I  (2.90)  b o u n d a r y c o n d i t i o n s a r o u n d S +S +S_  d<j>/dn. S u b s t i t u t i o n o f t h e v a r i o u s in  .+.6. J  3  i  3  3  .+ikcos0b. ,)0-(k)= ;  J  i s d e f i n e d as  i  1  N  J  3  (k) b..t\  2  - Z  j=N1+T  J  for (k) where f .  '89)  ( 2  coefficients  S +S+S_. provide  2*U*+d)  =  K>  J  J  i=1,....,N;  k=1,2,3,4  (2.91)  35  k=1,2,3 f  (k)  (2.92) k=4  The  expressions  are given  on  the  right-hand  s i d e of  the  above  equation  as a/4) TTn^  =  ik  c o s h [ k ( z .+d) ] osh(kd) exp( ikcos/3x .)  cos/3  sinh[k(z  .+d)]  Ax. exp(ikcos/3x . ) — . 3 AS  coshUd)  k  Az.  C  (2.93)  and AZj/ASj  k=1  -AXj/ASj  k=2  Az (z . - e ) — : 3 AS Fig.  5  shows  constants  a  N1,  typical  N2,  N3  Ax . x.—1 3 AS . D  +  k=3  discretized  and  N4  (2.94)  boundary  shown. Eqn.  with  the  (2.91) y i e l d s  N  (k) equations solved  f o r N unknown <f>\  using  a  matrix  unknown v e l o c i t y forces,  (k=1,2,3,4) v a l u e s  inversion  p o t e n t i a l s on  added mass and  damping  and  transmission coefficients  the  expressions  2.8  E F F E C T OF Let  structure is  given  i n the  boundary.  coefficients,  can  now  preceding  be  be  to obtain  the  The and  exciting reflection  determined  using  sections.  F I N I T E STRUCTURE LENGTH  us now of  the  technique  w h i c h can  consider  finite  assumed t o be  the  l e n g t h , 1.  much g r e a t e r  f o r c e s and The than  response  length of the  the  incident  of a  rigid  structure wavelength.  36  The  force per u n i t  2  by e q n .  (2.32) a s  ^]ll' \ 2  1  j = 1,2,3  exciting  corresponds  forces  structure along  i s given  = pq%{l }Re[C.(to,ti)exp{i( ,y-tot)}]  Fj(y,t) where  length  is  (or  to  the  sway,  heave  and  moment).  The  total  force  on  the  by i n t e g r a t i n g t w o - d i m e n s i o n a l  force  obtained  i t s length,  (2.95) roll  i g n o r i n g end e f f e c t s 1/2  F.(t)  =  /  F.(y,t)dy -1/2  3  Substitution  of eqn.  (2.96a)  3  (2.95) i n t o e q n .  (2.96a) y i e l d s  , , 2sin(- -isin^) F.(t) = pg§l\l C?e x p ( - i u t ) \ ]Z' J ^ -1 klsin/J 13--*  ?\ \ J  k  p  a  2  a  for  p*0°.  product  The above e x p r e s s i o n of  structure,  the  force  per  can  be  unit length,  and a f a c t o r r ( k l , / 3 ) d e f i n e d 2sin(^isin/3) ^  the  load  structure  unit  f o r a given  obliqueness  angle  of r  The  influences  zeros  at  given  of  against  l e n g t h c a n be s e e n . The behavior  large by  as  a  reduction  l e n g t h due t o t h e f i n i t e  6 shows a p l o t separate  the l e n g t h of the  t o be  approach,  of t h e waves f o r a g i v e n 2  the  0=0°  r ( k l , 0 ) can be c o n s i d e r e d per  as  (2.97)  1 factor  of  (1*0°  klsin/3  r(kl,/3) =  The  thought  (2.96b)  due  to  the  1 5 ° , 30° and  60°.  k l and /3 on t h e l o a d p e r u n i t  f a c t o r r(kl,/3)  values  l e n g t h of the  structure length. F i g .  k l f o r /3=0°,  of  or  of  of k l w i t h  has an  an  oscillatory  infinite  number o f  37  ^sinj3  For  an  length  = n it  infinite  n=1,2,...  span  structure,  (2.98)  the t o t a l load  t e n d s t o z e r o . The maximum t o t a l  load  per unit  occurs  on  a  span of l e n g t h 1 = L/2sin/3 where L=27r/k  Fj(t) The  (2.99)  i s t h e w a v e l e n g t h . T h i s maximum f o r c e i s  = P g j ^ C j U ^ T f g  motions of a r i g i d  s e a s c a n be d e s c r i b e d addition seas,  to  the  cylinder  can  also  to  and  motions about  rotational  section  added  with  motion along damping  sway,  structure  heave are  coefficients forces motions the  (2.100)  i n oblique  and  yaw  damping  variation  of a r i g i d  obtained  roll by  respectively.  of  of  multiplying  c a n be o b t a i n e d  coefficients from  a  flexible  a  finite the The  strip  and  to the  coefficients  f o r the p i t c h  the sectional  heave and sway m o t i o n s u s i n g  a  correspond  with the length of the s t r u c t u r e .  and h y d r o d y n a m i c  derived in  The added mass  cylinder  motions  axis  of the amplitude of  ( 0 = 0 ° ) . The h y d r o d y n a m i c and  pitch  the y  coefficient  to the o s c i l l a t i o n s  i n beam  and  motion along  the l e n g t h of t h e c y l i n d e r .  seas  length  t h e z and x axes  sinusoidal  coefficients  c a s e o f beam the  mass  a  2  modes p r e s e n t  surge,  the t r a n s l a t i o n a l  2.4 c o r r e s p o n d s  cylinder  of f i n i t e  t h e sway, heave a n d r o l l  cylinder  {]ll' }  exp(-icot)  i n t e r m s o f s i x d e g r e e s o f f r e e d o m . In  corresponding  The  ^  2  for  length sectional exciting and  yaw  coefficients for theory  approach  38  described exciting  in  Bhattacharyya  exciting  moment  (1978).  coefficients  The  yaw  and  pitch  are given as  1/2 F where  j + 3  (t)  j=4,5  =  J  corresponds  respectively.  Substitution  two-dimensional  forces  Fj ( t ) + 3  where q ( k l , 0 )  (2.101)  to  the  of  (2.95) i n t o  yaw  and  expression e q n . (2.101)  pg5al Cj-q(kl,/3)exp(-icjt) 2  pitch  modes  for  the  yields  j = 1,2  (2.102)  i s d e f i n e d as  2i (klsin/3)  =  j=1,2  . yF.(y,t)dy  -1/2  2  [^sinj3cos(^sin/3)  sin(^sin/3)]  0*0'  q =  (2.103) 0=0'  3. EFFECTS OF DIRECTIONAL  3.1 REPRESENTATION Before  proceeding  structures  and  to  in directional  mathematical The  OF DIRECTIONAL  seas,  representation  preceding  response  chapter  of  a  WAVES  SEAS  determine  the  we  first  shall  of d i r e c t i o n a l dealt with  structure  response  the  exciting  subject  pattern  i s h i g h l y complex and i r r e g u l a r .  surface  i s often modelled  long-crested point  waves o f a l l p o s s i b l e  from a l l d i r e c t i o n s .  assumed  to  Gaussian  process.  The  implies  symmetry  about  realistic A relative  however  by a l i n e a r  be  a  f o r small  zero  The  mean,  the s t i l l  amplitude  long-crested  wave  surface  of water  a  T h i s complex  a  elevation ergodic  Gaussian level  train  travelling  relation  A random sum of l i n e a r  random process  which  at  be r e p r e s e n t e d  where A i s t h e complex wave a m p l i t u d e w i t h  dispersion  is  i s only  waves.  t o t h e p o s i t i v e x a x i s may  t h e wave number  of  approaching a  angle  0  by  r ? ( x , y , t ) = Re [A exp{ i (kxcos0+kysin/3-cot)} ]  is  wave  superposition  stationary,  assumption  regular  exhibit  frequencies  sea  a  forces  to  waves. Ocean waves  sea  present  seas.  unidirectional which  of  (3.1)  a random p h a s e , k  r e l a t e d t o t h e f r e q u e n c y co by t h e l i n e a r (eqn. 2.11).  sea surface  c a n be c o n s i d e r e d  waves o f d i f f e r e n t  frequencies  t o be a d i s c r e t e and d i r e c t i o n s  77 = R e [ Z Z A ^ e x p { i ( k ^ c o s / 3 j + k ^ s i n / 3 ^ - c o ^ ) } ] i  39  (3.2)  40  where  k^ d e n o t e s  travelling  in  amplitude. infinity  be  direction  while  frequencies  number of h a r m o n i c s  the d i f f e r e n c e  between a d j a c e n t  t e n d s t o z e r o , t h e summation  replaced  component  /3j, co^ i t s f r e q u e n c y a n d A ^ j  I f we l e t t h e t o t a l  and d i r e c t i o n s can  t h e wave number o f t h e i - t h wave  by an i n t e g r a l  in  tend  to  frequencies eqn.  (3.2)  o v e r a c o n t i n u o u s r a n g e of  and d i r e c t i o n s  r?(x,y,t) = R e [ J/exp{ i (kxcos/3+kysin/3-a>t) }dA(co, 0) ] where dA r e p r e s e n t s t h e d i f f e r e n t i a l two-dimensional (/3,/3+d|3).  (co,/3)  The  elevation  its  mean  space  square  wave a m p l i t u d e  bounded  value  of  in  ico,co+dco)  by the  (3.3)  water  the and  surface  i s g i v e n by  T7  7  = i//dA(w,/3)dA*(u,/5) = U  S(co,/3)dcod/J  (3.4)  -TTO  where  dA*(co,/3)  is  the  S(co,/3) i s a d i r e c t i o n a l energy d e n s i t y  complex  wave  conjugate  spectrum.  o f dA(to,/3) and  Since  i n t h e waves i s p r o p o r t i o n a l  the  average  t o t h e s q u a r e of  t h e wave a m p l i t u d e , t h e p r o d u c t S ( C J , j3)dcod/3 c a n be c o n s i d e r e d to  be t h e c o n t r i b u t i o n  to the t o t a l  mean e n e r g y  t o waves w i t h f r e q u e n c i e s between co and co+dco, directions  between  directional  wave s p e c t r u m  The  and  one-dimensional  integrating directions  /3  the  /3 dj3. +  A  d e n s i t y due  travelling  sketch  of  a  in  typical  i s shown i n F i g . 7. spectrum,  directional  wave  S(co) c a n be o b t a i n e d by spectrum  over  a l l  41  7T  S(w) The  = / S(cj,/3)d0  one-dimensional  wave s p e c t r u m  measurements o f t h e in  space;  buoy.  free  In  order  complicated evaluating  to  techniques. directional  analysis  of  the  analysis  of  elevation, floating  elevation  at a s i n g l e  point  water  slope  Smith  Sand of  curvature  measurements  Holthujsen  (1981)).  is  convenient  spectrum  often  i n terms  o f an e n e r g y  the o n e - d i m e n s i o n a l  S(u,0)  =  to  methods  for  the from  of  Regier  and  the  (1980)).  (1964), M i t s u y a s u  stereophotographs  more  observation point  water  surface  the motions  Longuet-Higgins  f r o m an a r r a y o f guages  of  the  to  elevation a t an  elevation  means  resort  common  surface  ( 1 9 7 1 ) , D a v i s and  about  include  of  by  the  to  most  measurements  (e.g.  and  has  velocities  and  of a h e a v i n g  information  one The  the motions  analysis  Panicker  It  from  e t a l (1978),  the  buoy  Cartwright  4.  determined  wave s p e c t r a  orbital  (e.g. F o r r i s t a l l  3.  be  obtain  o f t h e waves,  horizontal  2.  surface  can  f o r i n s t a n c e by r e c o r d i n g  directionality  1.  (3.5)  et et  the  al al  of a  (1961),  (1975)).  water  surface  ( e . g . Borgman  (1969),  (1977)).  (e.g. Cot6  express  et  al  (1960),  the d i r e c t i o n a l  spreading function  wave  applied  to  spectrum  S(u)G(u,0)  where G(co,|3) i s a d i r e c t i o n a l  (3.6) spreading  function.  42  It  follows  from  eqn.  (3.5)  that  G(w,0)  must  satisfy  7T  / G(u,0)d0 = 1  (3.7)  -it  Various  one-dimensional  frequency  describe  ocean  most commonly  the  waves. The  Bretschneider,  These  spectra  Isaacson also  been  described and  several  various  authors.  outlined  below  1. C o s i n e - s q u a r e d St.  D e n i s and  which  2.  spectrum  A  C(e)  that  eqn.  used and  detail  used  ones  include  JONSWAP  spectra. and  a r e not g i v e n h e r e . T h e r e  have  for  the  in  G(CJ,0)  commonly  proposed  by  used  ones a r e  spreading  function  formulation (1953) p r o p o s e d a of f r e q u e n c y  f o r I 0 I < TT/2  ( 0  i s centred  (3.81  otherwise  about  0=0°.  formulation et  al  (1961)  proposed  the  following  spreading function  = C(s) c o s  where 8 i s m e a s u r e d propagation.  have been  Sarpkaya  of  ( | cos 2 0 =I *  Longuet-Higgins directional  hence  few  Pierson  Cosine-power  in  formulations  i s independent  G(0) The  Pierson-Moskowitz  are  (1981)  spectra  C(s)  (3.7)  2 s  (0)  from t h e  (3.9) principal  direction  i s a normalizing coefficient  i s s a t i s f i e d and  i s g i v e n by  that  of  wave  ensures  43  C(s) r  is  the  spreading that  s  =  ] 2/TT r ( s + ^ ) r  gamma  On  the  +  (3.10)  )  Fig. 8  fordifferent  describes with  s  function.  function  direction  (  the degree  basis  of  of s p r e a d about  their  measurements  et al (1975) f o u n d  depend on t h e d i m e n s i o n l e s s  the p r i n c i p a l  2  5  /  s to  (3.11)  m v ,  5 b  f o r T-<I  frequency  = d i m e n s i o n l e s s modal  U = wind speed  the parameter  f o r I>I  5  b  where T = d i m e n s i o n l e s s  f o r wind d r i v e n  frequency  ( 0.116(T)~ * s = { _ _ ( 0.116(1) ( T J m  m  v a l u e s o f s . I t c a n be seen  s — r e p r e s e n t i n g l o n g - c r e s t e d waves.  o c e a n waves, M i t s u y a s u  T  shows t h e d i r e c t i o n a l  m  m  = Uf/g  frequency  a t 19.5m above  = Uf^/g  sea l e v e l  Hasselmann  et al (1980) on t h e b a s i s o f  the  from  Joint  (JONSWAP) f o u n d t h e  the  parameter proposed  a different (1969)  formula used  on  f / f  m  rather  than  obtained  I  and  f o r s. an  alternative  cosine  power  g i v e n as  G(6)  The  Sea Wave P r o j e c t  s t o depend m a i n l y  Borgman function  North  data  •ir  (s)  cos  2 s  (0)  f o r |0|<7r/2  ={  (3.12) otherwise  normalizing coefficient C(s)  =  -1 H  r(s+  l>  r(s+£)  C'(s)  i s given as (3.13)  44  3. SWOP f o r m u l a t i o n Cote  et al (1960) p r o p o s e d  which  i s dependent  data  obtained  a directional  on b o t h  from  frequency  the  Stereo  spreading  and d i r e c t i o n  Wave  function based  Observation  on  Project  (SWOP). 1 + acos20 + bcos40]  for|0|<7r/2 (3.14)  0  otherwise  where a = 0.50 + 0.82exp(-^u*) b = 0.32exp(-2^ ) a  co = n o n d i m e n s i o n a l  frequency  = Uco/g  3.2 RESPONSE TO DIRECTIONAL WAVES The length and  exciting due  to  direction  force  on  a regular  a  rigid  o b l i q u e wave t r a i n  finite  o f f r e q u e n c y co  = Hj(a>,0)7?(t)  (3.15)  where Hj(co,j3) i s a c o m p l e x - v a l u e d  system  response  function  by e q n . (2.96b) a s  Hj(co,/3)  Since  the  linear,  =  frequency  to  spectrum  interaction  t h e v a l u e o f any f o r c e  be  due  to  but p r o p a g a t i n g  spectrum  {ill' }  wave  S (co,0) by  process at  components  from a l l p o s s i b l e  S„ (co) i s t h u s  related  *  2  pql{l^Cj(u,fi)r(kl,p)  wave-structure  we e x p e c t  frequency  force  of  0 c a n be e x p r e s s e d a s  Fj(t)  given  structure  a at  (3 16)  i s assumed given  wave  that  same  directions.  The  to the incident  wave  45  S where the  2  jH^ (co, j3) |  subscript  noted  that  i s the t r a n s f e r  2  j will  water  Gaussian  process,  function.  For convenience,  h e n c e f o r t h be d r o p p e d and i t s h o u l d  a l l following  S i n c e the  (3.17)  (u,0)d0  (co) = / |H.(co,/3) | S  p  expressions are v a l i d  surface  elevation  the  forces  is  will  be  f o r j = 1,2,3.  assumed  possess  to a  be  a  Gaussian  probability distribution. Using in  the  eqn. (3.6),  form o f t h e d i r e c t i o n a l wave s p e c t r u m  given  e q n . (3.17) r e d u c e s t o (3.18)  The  factor  i n the brackets  directionally the  =  0- 0  root  the  usually The  can  of  force  the  of  d i s measured  from  and i s t h u s r e l a t e d t o 0 by  force  spectrum (rms) from  of  can  be  obtained  by  over  t h e f r e q u e n c y co. The  the  force  represents  w h i c h extreme v a l u e  a  predictions  made.  effects  of  wave d i r e c t i o n a l i t y  be e x p r e s s e d a s a f o r c e  ratio  function,  dependent,  (3.19)  force  mean s q u a r e v a l u e  characteristic  O  a frequency  O  mean s q u a r e v a l u e  integrating  are  transfer  p r i n c i p a l wave d i r e c t i o n 0 6  The  averaged  represents  the  frequency  transfer  function  in  function  for long-crested,  reduction  on t h e wave  factor defined  as  loads the  dependent, d i r e c t i o n a l l y averaged  short-crested normally  seas  to  the  i n c i d e n t waves  transfer  46  7T  J  R  = —  2  the  rms  G(co, 0)d0  2  (3.20) 2  r a t i o R^^ c a n  value  corresponding  |  |H(co,0)|  F  A body r e s p o n s e  0)  |H(CJ,  of  the  results  a l s o d e f i n e d as  response  the  ratio  of  i n s h o r t - c r e s t e d seas  to  for long-crested seas, that i s  / /|Z(co,0) | G ( w , e ) S ( c o ) d 0 d c o 2  r?  R^  = °~*  (3.21) 7| Z(w, 0) | S 0  (u)dw  2  where Z ( t o , 0 ) previously The  is  i n eqn.  first  at  any  0  angle  function  Substitution  R  is  =  2  in  = C'(s)  the  case  distribution approach  |H(o>,0)|  transfer  cos 0  (3.22)  2  2  (3.12)  is  for  directional used the  in  / 2  / cos 0  cos  2  spreading  this  transfer  f u n c t i o n (3.12) i n t o eqn. *  The  force  as  cosine-power  eqn.  an  sinusoidal  horizontal cos0.  to  on  study.  function  (3.20) y i e l d s  2s (0)d0  (3.23)  s  -TT/2  F  For  defined  force  s t r u c t u r e w i t h the  of the e x p r e s s i o n s  spreading 2  a  i s t h e wave  proportional  independent  given  (3.22) and  of  t h u s be e x p r e s s e d  frequency  operator  t h e l e n g t h n e g l e c t e d . The  |H(w,0)| The  amplitude  (2.56).  segment  along  f u n c t i o n can  response  example c o n s i d e r e d  infinitesimal variation  the  n  the  of  will  oblique be  cut  mean i n c i d e n c e , t h e off  s t r u c t u r e from  one  to  ensure  directional  that  s i d e o n l y . I f the  the  waves  principal  47  direction  of  wave  integrated  to give  propagation  i s zero,  e q n . (3.23) c a n be  C (s)  (3.24)  C(s+1 ) For  any  length,  given the  (3.20) w i l l force the  structure dependence  arbitrary  shape  on 0 i s no l o n g e r  reduction  factor.  transfer function /  to  finite  and e q n . give  the  S u b s t i t u t i o n of t h e e x p r e s s i o n f o r  (3.16) i n t o e q n . (3.20)  2  J  and  explicit  have t o be i n t e g r a t e d n u m e r i c a l l y  *  yields  2s  |C .(w,0) 1 r ( k l , / 3 ) c o s ( 0 ) d 0  -TT/2  2  the cosine-power  2  z s  3  C (s)  for  of  (3.25)  ICj^O)!2 type  spreading  function.  4.  RESULTS AND DISCUSSION  4 . 1 EXCITING FORCES, ADDED MASS AND DAMPING COEFFICIENTS A  computer  the p r e c e d i n g forces,  compare  the  other The  cylinder water of  coefficients accuracy  solution first  the  element  case  0.4.  results  of  Bai's  and e f f i c i e n c y  surface,  The angle  f o r c e and by  a rectangular  section  reflection  1,  (1975) u s i n g a are  finite  plotted  as  results  solid  a  under t h e M i c h i g a n  for  the  exciting  and  angle  a  force coefficients  matrix  System  of  dimension  an 88  mesh w i t h a CPU t i m e procedure  48  i s thus  central  (MTS) t o  f o r a given  of i n c i d e n c e . B a i (1975) u s e d  free  s u r f a c e and 16 node  3.0s on t h e Amdahl V 8 - I I Terminal  and  a r e shown a s p o i n t s .  s u r f a c e h a d 40 node p o i n t s on t h e  computer. The p r e s e n t  in  coefficients  a r e r e p r e s e n t e d by t h e  approximately  element  of  o f i n c i d e n c e 0 f o r ka=0.1, 0.2 and  processor  370  method  20 node p o i n t s on t h e r a d i a t i o n  finite  i n order to  9-12 show a c o m p a r i s o n  Bai  p o i n t s on t h e body s u r f a c e y i e l d i n g  node  cases  and  of t h e p r e s e n t  coefficients  the present  discretized  N=76. I t took  exciting  reflection  (b/a) r a t i o  (d/a=2). F i g s .  obtained  while  is  t o half-beam  (1975) r e s u l t s  dashed curves The  the  the  and  for several test  considered  depth  technique.  function  determine  described in  techniques.  with a d r a f t  of f i n i t e  to  coefficients,  t h e computed e x c i t i n g  with  on t h e p r o c e d u r e  s e c t i o n s was u s e d  hydrodynamic  transmission  with  program based  solve  wavenumber  element,  325  o f 12s on an IBM relatively  quite  49 efficient. The and  computed  the  Bai's was  sway a n d heave e x c i t i n g  reflection  coefficient  (1975) r e s u l t s . consistently  The r o l l  greater  w i t h a maximum d i f f e r e n c e larger  set  present  results.  the  From F i g s . coefficients vanishing 0=0°.  The d i f f e r e n c e  constant 0=90°,  while  coefficients cos/3  with  exciting  up t o c e r t a i n  for  slightly  angle  ka=0.1.  The  presented  coefficient by B a i (1975) a  much  that  with  the e x c i t i n g  increasing force  force  or  moment  occurs was  before decreasing exciting to  reflection  be  to  force  of  at  fairly zero  at  ( o r moment)  proportional  coefficient  angle  force  a n g l e of i n c i d e n c e  coefficient  0 seemed  increasing  to diminish  to  decreases  incidence  before  t o one a t 0 = 9 0 ° . exciting  rectangular  f o r c e and h y d r o d y n a m i c  cylinder  with  infinite  d e p t h were a l s o  results  of  Garrison  used a Green's radiation  moment  with  computations.  sway a n d r o l l  at, any a n g l e  with  increasing The  the  closely  7.5%. The u s e o f  9-11 i t c a n be s e e n  heave  quite  i s expected  mesh i n B a i ' s  decrease  coefficients  d i d n o t s i g n i f i c a n t l y change t h e  a t 0 = 9 0 ° . The maximum  The  that  of about  o f node p o i n t s  use of a f i n e r  agree  exciting  than  force  computed  o f 0.265a  and  conditions  of the c y l i n d e r  function  i n the present  compared  and  surface procedure  the free thus only.  of  a  i n water of with  13-21. G a r r i s o n  which s a t i s f i e s  discretization used  draft  (1984) i n F i g s .  function  boundary  a  coefficients  the  (1984)  s u r f a c e and  requires The  i s relatively  the  Green's simple  50  while  the G r e e n ' s  c o m p l e x and water the  i s only v a l i d  depth  present  d=7r/k+b, procedure  discretized 40  the  body  function  of  incidence  results.  A  where k i s t h e wavenumber  i s used  in  radiation  30°  The  and  infinite  the  s u r f a c e and  coefficients parameter 6 0 ° . The  quite  water depth.  node p o i n t s on  frequency  agree  The  40  of  well  a d d e d mass and  Garrison's results  results  however  relations. location  The  of  empirically. variations the  present The  slightly  distance of  potential  with  the c r e s t  force  decreasing  should  The  of  force (1984)  sway  15%  t o 25%.  with  added  while  Garrison's  the  was  with  improve  Haskind to  the  estimated  higher  the  the from  sensitive  which  elements  at  coefficients with  increasing  moment o c c u r s  intuition  moment t o o c c u r and  were  exciting  maximum r o l l  agrees  results  the  a  order  accuracy  of  method.  tendencies, The  of up  better  use  angles  substantially  much  radiation  as  exciting  as much a s  deviated  with d i f f e r e n c e s  The  of  d e v i a t e d by  agreed  the  plotted  damping c o e f f i c i e n t s g e n e r a l l y  60°  coefficients  points  Garrision's  mass c o e f f i c i e n t damping  are for  with G a r r i s o n ' s r e s u l t s .  roll  free surface, node  computed  with  The  16  ka  show good a g r e e m e n t at  (1984) i s q u i t e depth.  surface.  0=0°,  coefficients  f o r water  to simulate  the  the  by G a r r i s o n  infinite  s u r f a c e had  node p o i n t s on  on  f u n c t i o n used  when t h e the  a t about  show  the  angle  of i n c i d e n c e .  ka = ?r/4. T h i s  since  one  trough  of a wave i s a t  s i d e s of  the  would e x p e c t  cylinder.  expected  result  the maximum the  origin  51  The  added  mass c o e f f i c i e n t s  t h e damping c o e f f i c i e n t incidence  for  most  of  damping c o e f f i c i e n t s crests energy  are  decreased the  tended with  normal t o the  i s propagated  increase, while  increasing  frequency  should vanish at  to  range  from  0=90° s i n c e  the  of  studied.  a x i s of t h e c y l i n d e r  away  angle  the  and  cylinder  The wave  hence  no  the  ±x  in  directions. The  exciting  coefficients circular and  and  and  results  were  paper.  The  angles  of  modelled  with  Garrison  are  incidence  with on  segments on  40  ratios  of  infinite  the  results  (1984)  estimated  results  coefficients,  amplitude  i n water  compared  (1973),  points  wave  cylinder  are  force  from  hydrodynamic  of a  depth  semi-immersed were  of B o l t o n and 1-4.  Tables  the  figures presented  35°  node p o i n t s on  the  radiation  the  s u r f a c e of  Ursell  in  shown f o r ka=0.25, 0.75 0=0°,  computed  and  and  5 5 ° . The  the  free  surface  and  the c y l i n d e r .  Garrison's  1.25  with  boundary  surface, 16  in his  was  40  node  straight  line  Agreement  between  the  different  methods i s g e n e r a l l y good w i t h d i f f e r e n c e s  less  than  It is  15%.  amplitude  ratios  indicates  that  decreases, cylinder  the  interesting  increase as  the  waves  with  to  angle  wavelength generated  become more a m p l i f i e d .  note  by  of  that  the  of  wave  incidence. This  along  the  the  motions  cylinder of  the  52 4.2  MOTIONS OF The  AN  UNRESTRAINED BODY  equations  of  a m p l i t u d e s of motion l=75m) rigid  in and  water  the d i s p l a c e d  be  at the s t i l l g i v e n as Figs.  depth  a  i s assumed t o be for  i s pV where V  volume. The  c e n t r e of g r a v i t y  i s assumed t o  water  and  level  the r o l l  modes  respectively.  function  r a d i u s of  at the  ka—>0 and  for  beam  free  decreasing,  surface.  d e c r e a s e s as  seas  increases  ka  The  gyration  30° and  heave and  with  f o r t h e sway,  amplitudes  vertical  up  to  amplitudes  ka. There  low  have  the  motions  of a  sway a m p l i t u d e  ka  are  6 0 ° . At  motions  i n c r e a s e s . The  w h i l e the r e s p o n s e  decrease with i n c r e a s i n g  The  of ka f o r 0 = 0 ° ,  (ka:$0.1), t h e sway and  as  factor  b=3m,  damping c o e f f i c i e n t s  a m p l i t u d e s as t h e h o r i z o n t a l  response  box  the  mass of t h e box  roll  as  particle  is  heave  maximum amplitude  maximum  before  f o r 0=30° and  are l o c a l  zeros  60°  of  the  f o r o b l i q u e waves c o r r e s p o n d i n g t o t h e z e r o s of r(kl,j3).  The  excessively  high.  present  practical  in  computations. observed and  d=12m. The  give  (a=7.5m,  22-24 show t h e a m p l i t u d e s o f m o t i o n  frequencies same  of  box  to  19.5m.  and  plotted  solved  of a l o n g f l o a t i n g  (/3=0°) a r e u s e d . The  is  heave  were  hence t h e added mass and  beam s e a s  is  motion  roll  modes was  that  In  roll  This  amplitude  i s because situations  solving  the  t h e heave r e s p o n s e  resonance  is  v i s c o u s damping w h i c h was  neglected  in  e q u a t i o n s of motion, i s uncoupled  responses while coupling weak e x c e p t c l o s e  at  the  between  t o the r o l l  the i t was  from t h e  t h e sway and  resonance  is  sway roll  frequency  53 where  there  is a  EFFECTS  4.3  OF  There of  are  wave  sinusoidal  The  that  factors  compared  (2)  and  angle  of  of  the  The  square  plotted  as  a  6.  For  r(kl,/3) length the in  a  the  with  ka.  i f we The  of  the  ignore  parameter  discussed  previously  of  r(kl,0)  incidence  with  ka in the  results  in  the  given  in  the  reduction kl  for  length  of  finite of  i f there of  and  is  per  section  4.1.  of The  the  It  in  factor per  also  unit of  results  length  as  sectional  ka  forces  forces  with  the  incidence  /3 h a s  been  combination  of  sectional  force  v a r i a t i o n with  the  force  reduction  total  is  60°  variation  unit the  the  factor  and  loads  no  sectional angle  30°  wave  along  r(kl,/3)  length,  incidence.  loads  the  15°,  the  of  cross-section.  factor  /3=0°,  the  two-dimensional  a  v a r i a t i o n of  v a r i a t i o n of  frequency  factor  the  along  in  (1)  seas:  reduction  angle  wave  structures  results  even  with  a  reduction  force  reduction  waves  force  decrease  increases  the  forces  for  the  two-dimensional  the of  to  long  v a r i a t i o n of  the  amplitude.  long-crested  wave  structure  in  oblique  sectional  of  of  given  by  incidence  function  results for  the  structure  r(kl,/3).  Fig.  to  the  sway  contribute  experienced  integration  length  the  two  v a r i a t i o n of  with  in  WAVES  seas  structure,  forces  drop  DIRECTIONAL  loads  short-crested  the  sudden  the angle  factor  R_. F  The  frequency  been  computed  4.2.  The  for  computed  the R„  dependent long values  force  floating for  the  reduction box  factor  described  cosine  power  in type  Rp,  has  section energy  54  spreading  function  25(a)-(c)  i s plotted  f o r t h e sway,  heave  respectively.  The  assess  influence  the  computations.  numerical  and  A  of  integration  in  was  eqn.  ka  used  (3.25)  =  o  moment)  i n order  degree /3  in Figs.  (or  f o r s=1,3,6  direction  rule  of  forces  the  principal  Simpson's  function  roll  r e s u l t s a r e shown  short-crestedness. the  as a  of 0°  with  wave  was  to carry an  to  used i n out  the  interval  of  10°. At  low  frequencies,  approaches  a  that  heave  for  the  limiting  low v a l u e s  significant factor thus  makes  The  that  for  frequencies.  This  R|  At  higher  (or is  =  slightly  0.866  force as  s  the  than  This  along  due the  be  the forces  Battjes  of  is  a  t o the length  spreading seen  from  approach  the  derived  an  (1982)  form  /3 Al  as  0  t h e sway,  factors  R  p  at  reduction  result  was  kl^=°  heave  a l l converge  the asymptotic  ( o r moment)  ka—>0.  2 s  there  high  as  (ka>1),  reduction less  seas.  fact  of d i r e c t i o n  mostly  I t can a l s o  factor  the  for directional  asymptotic  27rC(s)cos  force  variation  increases,  i s given  confirms  (or k l ) increases,  to account  as  reduction  i s independent  structures.  frequencies  moment)  roll  ka  for long-crested  expression  force  sinusoidal  f o r long  force  of one. T h i s  of t h e heave  i t important  25(a)-(c)  results  exciting  reduction  particularly Figs.  value  of ka. As  r(kl,/3).  the heave  roll  force  to a value  which  value.  factors expected  (4.1)  and  The  approach since  sway a value  and of  the sectional  55 sway and cos/3  roll  at  factors  low  The  of  0.4  at  sway and  computed  shown  for  factor  has  force  low  in  Figs. mean  of  0.79  ka=0  /3 = 30°  slight that  for  reduction the  of  waves a p p r o a c h t h e  factor  the  for  The  for  Bretschneider  force  was  to  mean  waves  ratio  for  case  of  1.0  the  = T^f 1 6 f  where  H  s  i s the  f r e q u e n c y . The Figs.  °  , V  5  from  the that  the  As  ka  only. force  fact  reduction  mean  power  4  0  the  f  box  energy  wave h e i g h t  sway,  a  heave  been  subject  to  a  spreading.  i s given  as  (4.2)  °  p l o t t e d as  waves  has  _ exp[-|(4 )" ]  significant  for  The  o  4  (f/f )  r e s u l t s are  27(a)-(c)  f o r /3 = 0 ° .  motions  i n c i d e n t u n i d i r e c t i o n a l wave s p e c t r u m  S ( w )  at  0  floating  spectrum with c o s i n e  0.866 f o r  cos/3 .  body  the  reduction  to  normal  of  for  factor  side  and  ratio  included  force  heave  and  reduction  o f f to ensure  one  also o  The  arises  cut  were  reduction  compared  from  asymptotic  the  up  o  compared  d i f f e r e n c e between the  response  computed  The  structure  t o an  ka  (/3 = 3 0 )  are  sway  heave f o r c e  function  oblique  i n c r e a s e s , up  the  heave  o  the  spreading  increases,  incidence  o  The  0.985  26(a)-(b).  f o r /3 = 30°  n o r m a l mean i n c i d e n c e . was  reduction  factors  mean i n c i d e n c e  frequencies,  a value  force  increasing  reduction  oblique  normal  c o m p a r i s o n . At  The  to  ka=2.  c a s e of  r e s u l t s are  (ka<0.1).  modes d e c r e a s e w i t h  heave  f o r one  factors  (or moment) i s p r o p o r t i o n a l  frequencies  for a l l three  to a value  the  e x c i t i n g force  and  f  0  i s the  function and  roll  of  peak s  in  responses  56 respectively  assuming  wave h e i g h t H =2m the  and  computations.  f r e q u e n c i e s between of  10°  were  are  0.22m, 0.32m and  normal a peak In  u s e d . The  43%,  and  roll  responses  compared ratios  numerical  integration,  0.26Hz and  an  interval  in long-crested heave  and  v a l u e of t h e sway, heave  in short-crested s increases,  v a l u e of one  of the s t r u c t u r e  the l o n g - c r e s t e d  angle  five  seas roll  2 7 ( a ) - ( c ) show r e d u c t i o n s of  s e a s . As  a limiting  o  f o r t h e sway,  Figs.  significant  f = 0 . 2 H z were used i n  amplitudes  respectively  a m p l i t u d e s of motion approach  rms  i n t h e rms  to long-crested approach  the  0.60rad  respectively. 41.5%  frequency  0.14Hz and  responses 42.5%  mean i n c i d e n c e . A  results  seas with the  indicating  s=1  response that  in short-crested  as s—>-<=°.  and  the seas  5. CONCLUSIONS AND RECOMMENDATIONS  5.1 CONCLUSIONS The  effects  m o t i o n s of  long  of wave d i r e c t i o n a l i t y s t r u c t u r e s has  A numerical developed  to  coefficients oblique  been  method b a s e d  compute  the  train  with  Green's  f o r c e s and  the  interaction  an  infinitely  arbitrary  quite  a p p l i e d to cases  and  can  be  loads  theorem  semi-immersed c y l i n d e r of general  the  and  studied.  exciting  associated with  wave  on  on  has  hydrodynamic  of  a  long,  shape.  regular floating  The of  been  method  is  v a r i a b l e water  depth. Numerical  results  been compared w i t h finite  finite  compared  to  and  a  cylinder The  those  and  present  wide r a n g e of  frequencies  required  have  using  a  also  been  water d e p t h method  w e l l as  Garrison  for  method i s q u i t e e f f i c i e n t  favorably with  high  have  section cylinder in  multipole  procedure  w h i c h compare  very  a  method  (1975)  infinite  c y l i n d e r as  function  Bai  results  for  (1973) u s i n g  present  a  by  for  a  (1984)  rectangular  a semi-immersed c i r c u l a r c y l i n d e r .  present  conditions.  by  rectangular  obtained  circular  Green's  from t h e  obtained  d e p t h . The  Ursell  semi-immersed using  those  e l e m e n t method f o r a  w a t e r of  Bolton  obtained  The  present  frequencies  due  to give accurate  a l l the  covering  t o the  usual  i s not  gives results  r a n g e of as  The  present  results over  node  a  design  efficient  l a r g e number of  results.  57  previous  the  procedure  and  for  points  procedure  is  58  however n o t v a l i d the  body  typical  axis  finite  to obtain  length.  structure linear  The  transfer  the  wave  function is  averaged  and  ratio  a response  wave  For  power  the  given  cosine-squared  of  The heave  The  structures of  factor  of  rigid  using the of wave  dependent,  f o r t h e wave  motions.  function.  a  effects  frequency  numerically  The  loads  reduction  f o r the cosine-power  Response  ratios  i n c i d e n t wave  the  sway  and  were  spectrum  roll  from 0.87 a t ka=0 t o 0.41 distribution  reduction  factor  t o 0.40 a t ka=2. The r a t i o  the s t r u c t u r e  a  body  structure,  for  ka=0  on  along  spreading.  factors varied  incidence.  as  Bretschneider  reduction a  integrated  motions  approach.  f o r the  a  and  reduction  type d i r e c t i o n a l spreading for  along  o f magnitude as a  s e a s have been o b t a i n e d  f a c t o r s have been e v a l u a t e d  computed  loads  loads  expressed  directionally  with cosine  wavelength  r e s u l t s have been  in short-crested  directionality  also  the  dimension.  two-dimensional  body a x i s  seas s i n c e  becomes o f t h e same o r d e r  cross-sectional  The the  f o r head  with  varied  force  at  ka=2  normal  mean  from  of the amplitudes  f o r the s p e c i f i e d s h o r t - c r e s t e d  1.0  at  of motion sea  state  w i t h a c o s i n e - s q u a r e d d i s t r i b u t i o n were 57%, 57.5% and 58.5% of  the response  roll and  modes  for  long  seas, f o r t h e sway, heave and  respectively. A further  amplitudes  {($0*0°).  in long-crested  of motions  These r e d u c t i o n s relative  reduction  i s obtained are quite  structure  of the forces  for oblique  significant  lengths  and  mean waves  particularly  need  to  be  59  considered As  i n the  the  design  process.  parameter  short-crestedness short-crested  s  which  describes  increases,  the  loads  approach  the  results  seas  the  and  degree  motions  for  of in  long-crested  seas.  5.2  RECOMMENDATIONS  There are be  made  to  numerical  FOR  several areas  improve  the  scheme u s e d  diffraction elements.  FURTHER  i n which  present  i n the  problem c o u l d This  STUDY  however  further studies  method. The  s o l u t i o n of  be  improved  requires  an  accuracy  the  by  could  using  of  the  oblique  wave  higher  order  increased  computing  effort. The  present  directionality  on  study the  considered  loads  and  though h y d r o d y n a m i c  coefficients  structures  sinusoidal  with  procedure  could  response in  of  Additional be  train.  considered. nonlinearity  mode to  using  been shapes.  a  t o m o o r i n g s and  the  body  even  A  for  numerical  the  dynamic  floating  exciting by  wave  presented  determine  the  given  rigid  of  bridge  forces  present  viscous  and  method.  damping  could  i n the a n a l y s i s .  present For  have  s t r u c t u r e such as  coefficients  f o r c e s due  included The  seas  effects  m o t i o n s of a  developed  a flexible  short-crested  hydrodynamic  be  the  steep  method waves,  Developing and  assumes  a  nonlinear  a theory  that  directionality  of  small  amplitude  effects  have  incorporates the  waves  wave to  both  be the  i s however  60 quite  difficult.  oblique  waves c o u l d  hybrid  method  used t o i n c l u d e Finally, out  The p r e s e n t  such  be e x t e n d e d as t h a t  the e f f e c t s  results.  seas  to  to  d i f f r a c t i o n theory f o r  nonlinear  proposed  o f wave  experimental  t o measure t h e l o a d s  short-crested  linear  by Dean  a  (1977) c a n be  investigations could  verify  and  directionality.  a n d r e s p o n s e of l o n g help  waves  be  carried  structures  the p r e s e n t  in  theoretical  BIBLIOGRAPHY 1.  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Math. 2, p p . 218-231.  APPENDIX  ANALYSIS  TO D E T E R M I N E OPTIMUM R A D I A T I O N  Consider of  an  t h e o b l i q u e waves  infinitely  long c y l i n d e r  with  each  mode o f m o t i o n  the  axis  of the c y l i n d e r .  forced  motions  a>  by  two-dimensional of  xz  A  the  m  i t st h r e e  ° °cSsh(kd) C  tan(kd) m  m  as  )]  modes along  associated  with the  exp{ i (kysin/3-tot)} ]  (11)  relation  to  the  angular  (eqn. 2.11).  0 ( x , z ) c a n be e x p r e s s e d  in  The terms  as  exp(ikxcos/3) +  cos[k  (z+d)]  cosTkm d )  =  as w e l l  i s related  a n d k* a r e w a v e n u m b e r s m -k  i n time  oscillation  as  which  potential  Vm m= 1 k  of  dispersion  a  where  i n any one  potential  an e i g e n f u n c t i o n e x p a n s i o n  *( '> "  the  = Re[#(x,z)  k i s t h e wavenumber  frequency  by  periodic The  DISTANCE  generated  c a n be e x p r e s s e d  *(x,y,z,t) where  I  exp(-k*x)  x>0  (12)  d e f i n e d by (13)  g  and k* m A  0  [k  2  m  i s the complex  field the  =  and  the  evanescent  + (ksin/3) ] 2  amplitude  of  coefficients  (14)  l / 2  the A  potential  at  the  far  are included to account f o r m modes o f w a v e m o t i o n n e a r t h e c y l i n d e r .  65  m  66  Since  the lowest  eigenvalue  amongst  a l l the evanescent  defined  as d(x)  k* g i v e s  modes,  a  the slowest  decay  factor  decay  can  = exp(-k*x)  be  (15)  where J In  order  the  value  < k*d <  to achieve a t x=0,  d i s t a n c e .X  R  X  given  (16)  Tr  a decay  the i n f i n i t e  =  maximum  k*d give  distance  eqn.  good  0=0°.  results  (17) g i v e s  noted water  that of  infinite be  used  optimum  distance  following  distance  i s used  X  R  a  +  2  ( k * ) ]  o f four, t i m e s The above i n water  times  truncated  depth to  at  a  i s obtained was  depth.  of  Bai  cannot  expression f o r deep  water, (1975)  should  about the  infinite  boundary.  for  radiation  water  the  conditions  H 2  i s the half-beam  +  to  be u s e d i n  source  information the  when  found  In deep  large.  A pulsating  useful  study  =  the depth  expansion  for truncation  in this  (17) 2  i s too  0=0°.  obtain  empirical  /  of f i n i t e  which  for  1  approximation  eigenfunction  [(ksin0) where  i s  o r 0.01  — 2  a distance  an  rather  The  boundary  ^  = 7r/2 and  o f exp(-27r)  by  [(ksin/3) A  rate  (18) (7r/ma) ] 2  1 / 2  of t h e c y l i n d e r and m  i s given  as  67  ka<0.5 0.5£ka<1.5 ka>1.5  (19)  68  M.J pa  ka 0.25  0.75  1 .25  0°  present results  \../pcoa  2  2  GAR  present results  GAR  5  1 .97  2.10  0.57  0.60  35  2.04  2.16  0.46  0.53  55  2.08  2.21  0.30  0.38  5  1 .00  0.93  1.31  1 .39  35  1.14  1.19  1 .40  1 .51  55  1 .84  1 .74  1 .44  1 .56  5  0.45  0.43  0.93  0.99  35  0.61  0.59  1.01  1.14  55  1 .09  0.93  1 .34  1 .40  Table!. C o m p a r i s o n o f t h e sway added mass and damping c o e f f i c i e n t s of a s e m i - c i r c u l a r cylinder (d/a=°°) o b t a i n e d i n t h e p r e s e n t s t u d y w i t h t h e r e s u l t s o f GAR (Garrison,1984)  69  M  ka 0.25  0.75  1 .25  2 2  0°  present results  5  /pa  X /pcoa  2  2  2 2  B&U  present results  B&U  1 .38  1 .38  1 .99  1 .96  35  1.61  1 .60  2.51  2.38  55  2.64  2.32  3.23  3.06  5  0.97  0.94  0.94  0.88  35  1 .04  1 .06  0.93  0.92  55  1 .43  1 .32  1.10  1 .02  5  1 .01  0.98  0.49  0.44  35  0.92  0.90  0.39  0.40  55  0.98  0.90  0.46  0.42  T a b l e 2. Comparison of t h e heave added mass and damping c o e f f i c i e n t s of a semi-circular c y l i n d e r (d/a=°°) o b t a i n e d i n the present study with the results of B&U (Bolton and Ursell,1973)  70  1^1 present results  ka 0.25  0.75  1 .25  GAR  present results  GAR  5  0.75  0.77  0.18  0.19  35  0.63  0.65  0.18  0.19  55  0.44  0.46  0.18  0.19  5  1.17  1.18  0.85  0.89  35  1 .07  1.11  0.97  1 .02  55  0.94  0.94  1.17  1 .26  5  0.99  0.99  1.19  1 .26  35  0.95  0.95  1 .37  1 .56  55  0.91  0.90  1 .89  1 .96  Table 3. C o m p a r i s o n of t h e sway e x c i t i n g f o r c e c o e f f i c i e n t and wave a m p l i t u d e r a t i o o f a s e m i - c i r c u l a r cylinder (d/a=°°) obtained i n the present study with the r e s u l t s o f GAR (Garrison,1984)  71  U|  |c l  2  2  ka 0.25  0.75  1 .25  0° 5  present results  present results  B&U  B&U  1 .40  1 .40  0.34  0.35  35  1.41  1 .40  0.42  0.43  55  1 .29  1 .32  0.58  0.58  5  0.95  0.94  0.71  0.70  35  0.87  0.87  0.78  0.80  55  0.77  0.76  1 .02  1 .00  5  0.68  0.67  0.85  0.84  35  0.54  0.57  0.85  0.87  55  0.49  0.49  1.10  1 .07  -  T a b l e 4. C o m p a r i s o n of t h e heave e x c i t i n g f o r c e c o e f f i c i e n t and wave a m p l i t u d e r a t i o of a s e m i - c i r c u l a r cylinder (d/a=°°) obtained i n the present study with the results of B&U ( B o l t o n and U r s e l l , 1 9 7 3 )  Figure  1.  Definition  sketch  for a rectangular  cylinder  73  incident wave  reflected wave  f£2(heave)  transmitted wave  _* (roll)  '  3  /  ^(sway)/  ///////////// Figure 2. Definition component m o t i o n s  sketch  for floating cylinder  S  showing  F  SR  s Figure  3.  Sketch of c l o s e d  D  surface  74  Figure  4.  Sketch  showing  relationship  between x, £ , a n d £'  j=N1  j=N3 i  =  N  2  j»1  l<l»l'l'l'l-l'l'l'l'l'l'l'l'l'l'l'l'l'l-  J-N  j=N4l  Figure  5 .  cylinder  A t y p i c a l boundary (b/a=1,d/a=2)  e l e m e n t mesh f o r a  rectangular  Figure  7.  S k e t c h of a d i r e c t i o n a l  wave  spectrum  76  0(degrees) Figure values  8. Directional of the parameter  spreading  function  for  s  different  1  2-1 ka=0.1 BAI(1975)  Angle of incidence, /S (degrees) F i g u r e 9. Sway e x c i t i n g cylinder (b/a=1,d/a=2)  force  coefficient  for a  rectangular  0  15  —r30 30  45  - T —  75 75  60 60  90  Angle of incidence, /? (degrees) Figure 10. Heave exciting force r e c t a n g u l a r c y l i n d e r (b/a=1,d/a=2)  coefficient  for  0.5-  0.4  —  ko=0.1 BAI(1975)  —  ka=0.2 BAI(1975)  — ka=0.4 BAI(1975) El PRESENT RESULTS 0.3o 0.2-  0.1-  0.0-+  15  30  45  60  /5  Angle of incidence, (1 (degrees) Figure 11. R o l l exciting moment coefficient r e c t a n g u l a r c y l i n d e r (b/a=1,d/a=2)  90  for  78 1.2  ka=0.1 BAI(1975) k a = 0 . 2 BAI(1975) k a = 0 . 4 BAI(1975) PRESENT RESULTS  0.8  a:  o.6-  0.4  —r15  Figure cylinder  -~T~ 30  —r-  60  45  Angle of incidence, /S (degrees)  12. R e f l e c t i o n (b/a=1,d/a=2)  coefficient  i 75  for  90  a  rectangular  0.6  A  0.5-  P R E S E N T RESULTS G A R R I S O N (1984)  /? = 0 °  0.4  c_f 0.3  0.2  0.0-  Figure 13. Sway exciting force r e c t a n g u l a r c y l i n d e r (b/a=0.265,d/a=»)  coefficient  for  a  79 1.5 A PRESENT RESULTS GARRISON (1984)  o A " " \ ^ ^  0.5-  1  1  1  r  r — -i  r• i  0.5  i  i  i  i  -i  r-  1  j •  1.5  ka  Figure 14. Heave exciting force coefficient r e c t a n g u l a r c y l i n d e r (b/a=0.265,d/a=°°)  for  0.35-  A PRESENT RESULTS  0.30  GARRISON (1984) 0.25  a--l  /  ^  = 0  °  -TT-lo^^-  0.20o  0.15-  0.10-  0.05  0.00 1.5  0.5  Figure 15. R o l l exciting moment coefficient r e c t a n g u l a r c y l i n d e r (b/a=0.265,d/a=° ) o  for  a  80 0.5 A PRESENT RESULTS GARRISON (1984)  0.4  0.3 H O  >9. 0.2 H  o.H  0.0  0.5  —I—  1  ka  —r~ 1.5  Figure 16. Sway added mass c o e f f i c i e n t c y l i n d e r (b/a=0.265,d/a=»)  f o r a rectangular  0.30  0.25 H  A PRESENT RESULTS GARRISON (1984)  0.20 H o ^  0.15-  0.10-  0.05  Figure 17. Sway damping c y l i n d e r (b/a=0.265,d/a=»)  coefficient  for a  rectangular  81  A PRESENT RESULTS -  GARRISON (1984) \fl=60°  3-  \ A  D  —g.  ^-^^^  ^ ^ ^ ^ ^  1-  1  ,  ,  1  1  ,  ,  0.5  ,  1-  i  .  .  .  .  .  .  i  i  i  1.5  1 ka  F i g u r e 18. Heave added mass c o e f f i c i e n t c y l i n d e r (b/a=0.265,d/a=»)  for a  rectangular  A PRESENT RESULTS  2.5  GARRISON (1984)  2o 3 v9. 1.5  AVV  0.5-  o-  0.5  1  1.5  ka  F i g u r e 19. Heave damping c y l i n d e r (b/a=0.265,d/a=»)  coefficient  for a  rectangular  82  0.4-1  A PRESENT RESULTS GARRISON (1984)  Figure 21. R o l l damping cylinder (b/a=0.265,d/a=»)  ka coefficient  for  a  rectangular  83  for  a  long  84  Figure 24. R o l l response amplitude f l o a t i n g box (a=7.5m,b=3m,l=75m,d=12m)  operator  f o r a long  0.5-  Figure 25. F o r c e a n d moment r e d u c t i o n f l o a t i n g box (a=7.5m,b=3m,l=75m,d=12m)  factors  f o r a long  85  ka F i g u r e 25.(cont.) l o n g f l o a t i n g box  F o r c e a n d moment r e d u c t i o n (a=7.5m,b=3m,l=75m,d=12m)  factors  for a  86  (a) SWAY  Figure 26. F o r c e r e d u c t i o n f a c t o r s f o r a l o n g f l o a t i n g box (a=7.5m,b=3m,l=75m,d=l2m) i n n o r m a l a n d o b l i q u e mean s e a s  87  Figure 27. R e s p o n s e ratios (a=7.5m,b=3m,l=75m,d=l2m)  for a  long  floating  88  Figure 27.(cont.) Response (a=7.5m,b=3m,l=75m,d=l2m)  ratios  f o r a long  f l o a t i n g box  

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