Hydrodynamic Design
of High Speed Catamaran Vessels
by
Shiran Purvin
B.Sc Mechanical Engineering
Technion, Israel Institute of Technology
SUBMITTED TO THE DEPARTMENT OF OCEAN ENGINEERING IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE IN NAVAL ARCHITECTURE AND MARINE ENGINEERING
AT THE
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
FEBRUARY 2003
JUL 1 520
C 2003 Massachusetts Institute of Technology.
All rights reserved
LIBRARIES
A uthor:.................................................
Departiy nt of Glean Engineering
/
January 17, 2003
C ertified by:........................................
Paul D. Sclavounos
Professor of Naval Architecture
Thesis Supervisor
Accepted b y :..............................................................................
. ....
Arthu ?'aggeroer
Professor of Ocean Engineering
Chairman, Committee for graduate students
BARKER
7'
-
V
,ev
.r
Hydrodynamic Design of High Speed Catamaran Vessels
by
Shiran Purvin
Submitted to the department of Ocean Engineering on January 17, 2003
in Partial Fulfillment of the Requirements for the degree of
Master in Science in Naval Architecture and Marine Engineering
Abstract
This thesis examines the hydrodynamic design of a slender, semi-displacement,
generic catamaran with a variable separation ratio (the ratio of the transverse distance between
the demi-hulls to the length of the vessel) and the influence of this separation ratio upon the
vessel behavior at sea. The hydrodynamic aspects studied were the hull calm water resistance,
the seakeeping characteristics of the vessel with and without passive lifting appendages, and
the structural loads developed as a result of an incident wave spectrum. Some measurements
were compared with a similar monohull (a monohull with exactly the same displacement and
length). The catamaran used for the research had a generic hull form and displaced 7500 ton.
The hydrodynamic characteristics of this vessel were analyzed by a general purpose, potential
flow, time domain, Rankine Panel Method, FORTRAN code, called SWAN-2 (Ship Wave
Analysis). SWAN-2 is a state-of-the-art Computational Fluid Dynamics (CFD) code,
developed in MIT in recent years, and was utilized practically as a numerical towing tank.
Thesis Supervisor: Paul D. Sclavounos
Title: Professor of Naval Architecture
Hydrodynamic Design of High Speed Catamaran Vessels
Acknowledgments
The following thesis was written during the year 2002 (January - December) in the
'Laboratory for Ship and Platform Flow', at the Ocean Engineering Department, MIT.
I would like to express my appreciation to all who helped me while preparing this paper and
supported me throughout my graduate studies.
I thank my thesis advisor, Professor Paul Sclavounos, for his help and guidance. His insights
and advice throughout this last year inspired me towards the completion of this paper.
I thank Dr. Sungeun Kim who helped me with SWAN and other hydrodynamic and technical
related problems. His help is greatly appreciated.
I thank my lab partners, Yile Li, Talha Ulusoy and Kwang Lee for their good advice and
bright ideas.
Finally I thank my family who 'suffered' and supported me in the last eighteen month of my
graduate studies at MIT, my wife Elana, and my daughters Yarden and Noa.
Shiran Purvin
Cambridge, Massachusetts, December 23, 2002.
4
Hydrodynamic Design of High Speed Catamaran Vessels
Table of Contents
Abstract.......................................................................................................................................
3
A cknow ledgm ents ......................................................................................................................
4
Table of Contents .......................................................................................................................
5
List of Figures.........................................................................................................................8
List of Tables ........................................................................................................................
10
Chapter 1: Introduction.............................................................................................................
11
Overview ..............................................................................................................................
12
Chapter 2: Theoretical Background.......................................................................................
15
2.1 Linearized Boundary Conditions................................................................................
15
2.1.1 Kinem atic Boundary Condition...........................................................................
16
2.1.2 Dynam ic Boundary Condition.............................................................................
17
2.1.3 Com bined Condition ..........................................................................................
18
2.2 Plane Progressive W ave .............................................................................................
19
2.3 Ship Resistance...............................................................................................................21
2.3.1 Frictional Resistance ..........................................................................................
22
2.3.2 Three Dim ensional Ship W aves..........................................................................
23
2.3.3 Calm w ater W ave Making Resistance..................................................................
26
2.3.4 Induced Resistance ...............................................................................................
28
2.4 Ship Response in Sea W aves......................................................................................
29
2.4.1 Sea Spectrum ......................................................................................................
29
2.4.2 Ship Response ......................................................................................................
31
2.4.3 V elocity Potential...............................................................................................
32
2.4.4 Wave exciting Forces ..........................................................................................
34
2.4.5 Ship Equation of Motion........................................36
2.4.6 Hull loads.................................................................................................................37
2.5 Lifting Folls ....................................................................................................................
5
38
Hydrodynamic Design of High Speed Catamaran Vessels
2.5.1 Prandtl's lifting line theory..................................................................................
38
2.5.2 The effect of Lifting Foils on the Equation of Motion........................................
40
2.6 Panel M ethod..................................................................................................................44
2.6.1 Rankine Panel M ethod ........................................................................................
Chapter 3: SWAN -2 Implem entation....................................................................................
3.1 Background and Linearization of the Boundary Value Problem ...............................
45
48
48
3.1.1 The Free Surface..................................................................................................
49
3.1.2 Body Boundary....................................................................................................
50
3.2 Ship Response in Calm Water and in Am bient W aves ..................................................
51
3.2.1 Ship Translating in Calm Water..............................................................................51
3.2.2 Ship Seakeeping with Forward Speed..................................................................
51
3.2.3 Thin Wake Sheets - Transom conditions................................................................51
3.2.4 M otion Control Lifting Appendages ...................................................................
3.3 N um erical Solution......................................................................................................
53
53
3.3.1 Spatial D iscretization...........................................................................................
54
3.3.2 M eshing ...................................................................................................................
54
3.3.3 Temporal D iscretization ......................................................................................
56
3.3.4 N um erical dispersion, damping and stability ......................................................
57
3.4 Radiation Condition....................................................................................................
59
3.5 Flow Solver ....................................................................................................................
59
3.6 H ydrodynam ic Pressure.............................................................................................
60
3.7 H ydrodynam ic Forces .................................................................................................
62
3.8 Equations of M otion....................................................................................................
64
3.9 Applications....................................................................................................................66
3.9.1 W ave m aking Resistance, Sinkage and Trim ......................................................
66
3.9.2 Linear Seakeeping in the time dom ain ...............................................................
66
3.9.3 Evaluation of Frequency D omain RA O 's ..........................................................
66
3.9.4 Wave Induced Loads...........................................................................................
67
Chapter 4: The Steady Problem .............................................................................................
68
4.1 Introduction ....................................................................................................................
68
4.2 SW AN -2 Execution....................................................................................................
69
4.3 Wave Patterns Predictions...........................................................................................
70
6
Hydrodynamic Design of High Speed Catamaran Vessels
74
4.4 D ynam ic Sinkage and Trim ........................................................................................
4.5 Calm W ater Resistance...................................................................................................76
4.5.1 Preface .....................................................................................................................
76
4.5.2 Ideal Fluid Resistance...........................................................................................
77
4.5.3 Wave M aking Resistance ...................................................................................
79
4.5.4 V iscous D rag ...........................................................................................................
82
4.5.5 Total Resistance and V erification ........................................................................
85
Chapter 5: Seakeeping Evaluation.........................................................................................
87
5.1 Introduction ....................................................................................................................
87
5.2 H eave and Pitch Responses at unidirectional, irregular sea......................................
88
5.3 M otion Response - Standard D eviation........................................................................
103
5.3.1 Analysis of results .................................................................................................
108
5.4 The influence of lifting appendages on Heave and Pitch RAO's.................................110
Chapter 6: Structural Loading Analysis .................................................................................
120
6.1 Preface..........................................................................................................................
120
6.1.1 The exciting forces................................................................................................
120
6.1.2 Force RA O 's..........................................................................................................
121
6.1.3 Ship w eight distribution ........................................................................................
122
6.2 Significant Forces.........................................................................................................
123
Chapter 7: D iscussion and Conclusions .................................................................................
128
7.1 Conclusions ..................................................................................................................
128
7.2 Recom m ended Future W ork.........................................................................................
132
Chapter 8: N om enclature........................................................................................................
133
Chapter 9: References & Bibliography ..................................................................................
137
Annex A: Ship characteristics................................................................................................
139
Annex B : Seakeeping Results. ...............................................................................................
141
Annex C : Loading Results......................................................................................................153
7
Hydrodynamic Design of High Speed Catamaran Vessels
List of Figures
Figure 2.1: The three dimensional propagating wave ..........................................................
20
Figure 2.2: Kelvin wave pattern in a ship wake ...................................................................
24
Figure 2.3: Kelvin wave pattern for a monohull at Fn=0.74 .................................................
25
Figure 2.4: Kelvin wave pattern for a catamaran at Fn=0.74...............................................26
Figure 2.5: W ave spectrum illustration .................................................................................
30
Figure 2.6: Ship motions, six degrees of freedom and the incident wave............................
31
Figure 2.7: Lifting foil characteristics .................................................................................
39
Figure 2.8: Rankine Source and an arbitrary point coordinate system.................................46
Figure 3.1: A thin wake sheet...............................................................................................
52
Figure 3.2: Quadrilateral discretization of the fluid domain and the hull surface................56
Figure 3.3: Stability criterion for emplicit Euler time marching schemes. .........................
58
Figure 4.1: wave pattern for s/L=0.3, at Froude Numbers 0.57, 0.74 and 0.90 ...................
71
Figure 4.2: wave pattern for s/L=0.4, at Froude Numbers 0.57, 0.74 and 0.90 ...................
72
Figure 4.3: wave pattern for s/L=0.5, at Froude Numbers 0.57, 0.74 and 0.90 ...................
73
Figure 4.4: Dynamic sinkage for all catamarans and a monohull as a function of Fn ......
74
Figure 4.5: Dynamic trim for all catamarans and a monohull as a function of Fn...............75
Figure 4.6: Ideal fluid resistance coefficient results.............................................................78
Figure 4.7: Wave making resistance coefficient .................................................................
80
Figure 4.8: Wave making resistance as a function of the dynamic trim ..............................
81
Figure 4.9: Wave making resistance coefficient - comparison chart...................................82
Figure 4.10: Form factor correlation line ............................................................................
84
Figure 4.11: Friction resistance coefficient...........................................................................
84
Figure 4.12: Total resistance coefficient ...............................................................................
85
Figure 4.13: Comparison chart of s/L = 0.3 catamaran total resistance results....................86
Figure 5.1: Coordinate system, heave and pitch motion and wave headings.......................89
Figure 5.2: Seakeeping simulation of a vessel advancing in 45 knots (Fn=0.74)................90
Figure 5.3: Heave RAO for bow waves for s/L=0.3 ............................................................
91
Figure 5.4: Heave RAO for bow waves for s/L=0.4 ............................................................
92
Figure 5.5: Heave RAO for bow waves for s/L=0.5 ............................................................
92
Figure 5.6: Pitch RAO for bow waves for s/L=0.3 ..............................................................
94
8
Hydrodynamic Design of High Speed Catamaran Vessels
Figure 5.7: Pitch RAO for bow waves for s/L=0.4 ..............................................................
94
Figure 5.8: Pitch RAO for bow waves for s/L=0.5 ..............................................................
95
Figure 5.9: Heave and Pitch RAO's for Fn=0.99 including monohull performances..........97
Figure 5.10: Heave and Pitch RAO's for 1500 wave headings at six different
Froude
Numbers for catamaran separation ratio of s/L=0.3, s/L=0.4, and s/L=0.5 .................... 99
Figure 5.11: Heave and Pitch RAO's for 1200 wave headings at six different
Froude
Numbers for catamaran separation ratio of s/L=0.3, s/L=0.4, and s/L=0.5 ...................... 100
Figure 5.12: Heave and Pitch RAO's for 900 wave headings at six different Froude Numbers
for catamaran separation ratio of s/L=0.3, s/L=0.4, and s/L=0.5 .......................................
101
Figure 5.13: The variance of the heave response spectrum at Fn = 0.57, bow waves (180')104
Figure 5.14: The variance of the pitch response spectrum at Fn = 0.74, beam waves (900). 104
Figure 5.15: 3D view of the heave and pitch standard deviation of a s/L=0.3 catamaran .... 105
Figure 5.16: 3D view of the heave and pitch standard deviation of a s/L=0.4 catamaran .... 106
Figure 5.17: 3D view of the heave and pitch standard deviation of a s/L=0.5 catamaran .... 107
Figure 5.18: Location of lifting appendages along the hull....................................................
111
Figure 5.19: Lifting appendages at Fn=0.66 at bow waves....................................................
112
Figure 5.20: Lifting appendages at Fn=0.66 at oblique waves...............................................
113
Figure 5.21: Lifting appendages at Fn=0.82 at bow waves....................................................
114
Figure 5.22: Lifting appendages at Fn=0.82 at oblique waves...............................................
115
Figure 5.23: Lifting appendages at Fn=0.99 at bow waves....................................................
116
Figure 5.24: Lifting appendages at Fn=0.99 at oblique waves...............................................
117
Figure 5.25: Large area lifting appendages at Fn=0.99, bow waves, s/L=0.3........................119
Figure 6.1: Simulation of weight distribution along the hull .................................................
122
Figure 6.2: Significant shear forces and bending moments at 1800 wave headings for Fn =
0.66, 0.82, 0.99, for s/L = 0.3 .............................................................................................
124
Figure 6.3: Significant shear forces and bending moments at 1800 wave headings for Fn =
0.66, 0.82, 0.99, for s/L = 0.4 ...............................................................................................
124
Figure 6.4: Significant shear forces and bending moments at 1800 wave headings for Fn =
0.66, 0.82, 0.99, for s/L = 0.5 .............................................................................................
125
Figure 6.5: Significant shear force, bending and torsion moments at 1200 wave headings for
Fn
=
0.66, 0.82, 0.99, for s/L = 0.3.....................................................................................
9
125
Hydrodynamic Design of High Speed Catamaran Vessels
Figure 6.6: Significant shear force, bending and torsion moments at 1200 wave headings for
Fn
=
0.66, 0.82, 0.99, for s/L = 0.4.....................................................................................
126
Figure 6.7: Significant shear force, bending and torsion moments at 1200 wave headings for
Fn = 0.66, 0.82, 0.99, for s/L = 0.5.....................................................................................
126
Figure 7.1: Ideal fluid resistance ............................................................................................
129
Figure A. 1: Qualitative description of the examined catamaran............................................
139
Figure B. 1: Catamaran advancing forward in beam waves....................................................141
Figure B.2: Roll motion response at Fn = 0.99 ......................................................................
142
Figure B.3: Heave and Pitch motion response at Fn = 0.90 ...................................................
143
Figure B.4: Roll motion response at Fn = 0.90 ......................................................................
144
Figure B.5: Heave and Pitch motions response at Fn = 0.82 .................................................
145
Figure B.6: Roll motion response at Fn = 0.82 ......................................................................
146
Figure B.7: Heave and Pitch motion response at Fn = 0.74...................................................147
Figure B.8: Roll motion response at Fn = 0.74 ......................................................................
148
Figure B.9: Heave and Pitch motion response at Fn = 0.66...................................................
149
Figure B.10: Roll motion response at Fn = 0.66 ....................................................................
150
Figure B. 11: Heave and Pitch motion response at Fn = 0.57.................................................
151
Figure B.12: Roll motion response at Fn = 0.57 ....................................................................
152
Figure C. 1: Shear force and bending moment RAO for 1800 wave heading, at Fn = 0.66... 154
Figure C.2: Shear force, torsion moment and bending moment RAO for 1800 wave heading,
at Fn = 0 .6 6 .........................................................................................................................
15 5
Figure C.3: Shear force and bending moment RAO for 180* wave heading, at Fn = 0.82... 156
Figure C.4: Shear force, torsion moment and bending moment RAO for 1800 wave heading,
at F n = 0 .82 .........................................................................................................................
157
Figure C.5: Shear force and bending moment RAO for 180* wave heading, at Fn = 0.99... 158
Figure C.6: Shear force, torsion moment and bending moment RAO for 1800 wave heading,
at F n = 0 .9 9 .........................................................................................................................
List of Tables
Table 5.1: Lifting appendages alternatives.............................................................................110
Table A . 1: Ship C haracteristics..............................................................................................140
10
15 9
Chapter 1. Introduction.
Chapter 1: Introduction
Semi-displacement sea vehicles and particularly fast, slender catamarans became
recently a popular way of transportation at sea. The broad deck enables the catamaran to
transport larger number of travelers and commerce in comfort and faster than ever. Naval
applications of these hull shapes are also considered, especially in light of their capabilities
to carry more man, support and armor into harms way.
Catamarans were used throughout history as a favorite way of transportation over seas.
Primitive catamarans were found in the South Pacific and off the coast of India. In some
regions this vessel was made of three logs lashed together (a trimaran design). The middle
log was longer than the other two and was pointed to form a prow. In other areas the
catamaran consisted of two canoe hulls attached parallel to each other.
In recent years the catamaran design was built out of composites or aluminum in demand
to lighter vessels with superior speed at sea. The catamaran has larger deck area compared
to the traditional monohull and has a better transverse stability features. Its main
disadvantage is the need to withstand a large torsion moment, caused due to loads created
over the demi hulls; and acting as a moment on the vessels low cross section area around
the centerline. This area suffers consequently from strains, cracks and breaks.
Computational naval-architecture methods were developed in the last two decades along
with the development of personal computers. This development allowed more ship
hydrodynamic and structure work to be done on computers and less by experiments and
model testing, although the stage of eliminating these completely was not reached yet.
Advance marine vehicles as the catamaran, possesses different characteristics than the
conventional monohull and by that present new technical challenges. These challenges
include operating in high speeds, geometrical complexities (as transom stems) and
11
Chapter 1. Introduction.
multiple hulls. Experimental data for these unique vessels is limited and typically
proprietary. Numerical and computational techniques that can evaluate the performance of
this unique vessel in calm water and in waves are thus invaluable to the designer.
Before computational method appeared, hydrodynamic problems were solved analytically
using approximations like the two-dimensional strip theory algorithm. Strip theory utilizes
two-dimensional boundary value problem solutions, and integrates them over the ship
length, to find the three-dimensional solution. This method is limited to slender bodies
advancing at relative slow speeds. A more accurate solution to the ship response problem,
require solution of the fully three-dimensional problem. This can be done solely using
powerful computer platforms utilizing fluid dynamics - panel methods algorithms. Such
methods discretize the boundaries of the fluid into elements with associated singularity
strength; impose appropriate boundary conditions; and most of them use linear, potential
flow theory. In recent years a code based on the Rankine Panel Method was developed at
the Massachusetts Institute of Technology by the name SWAN (Ship Wave Analysis). At
first the frequency domain was used to ease calculations, but later the time domain was
developed and included in recent upgrades of SWAN by the name of SWAN-2. The
Rankine Panel Method is a subgroup of the aftermentioned panel methods, which employ
the Rankine Source as the elementary singularity.
Overview
The research presented in the following paper, invokes computational methods in
predicting ship responses while advancing with forward speed in calm water and in waves.
The responses predicted were the seakeeping heave and pitch (and roll) motions and the
body loads, shear force, bending moment and torsion moment of the vessel about its
center-line. A generic catamaran was created and was examined as a case study in this
thesis. Three different separation ratios between the demi-hulls of the catamaran were
evaluated in order to establish a pattern of response and to analyze the influence of the
interference between the hulls on their behavior at sea. The sea spectrum chosen to test the
ship was the ISSC (International Ship and Offshore Structures Congress) spectral
12
Chapter 1. Introduction.
formulation for fully developed sea. A significant wave height of 4 meters and a modal
period of 12 seconds were used with this spectrum. The vessel analyzed was a generic,
slender hull, round bilge, deep transom catamaran. A slender, round bilge, deep transom
monohull with similar length and displacement values was examined as well for
comparison purposes. All vessels were evaluated in the semi-displacement regime (Froude
Number = 0.5 - 1.0).
The catamaran examined in this paper was evaluated as if it was tested in a towing tank.
The benefits of the three dimensional, time-domain, computational fluid dynamic codes are
obvious. The vessel evaluated here was developed on a spreadsheet using polynomial
equations, and was never built, yet a set of 'numerical' model tests were conducted upon it.
Not only the vessel behavior at sea was evaluated, but also an analysis of alternatives of
improving its seakeeping characteristics by attaching to it lifting appendages, was
conducted.
The monohull that was analyzed for comparison purposes had the same overall length and
displacement as the catamarans. The length and displacement of a vessel determine a great
deal of its measure of effectiveness. Characteristics of its effectiveness could be the weight
of the payload, the vessel ability to maneuver in harbors, or its cost. It was interesting to
compare, for instance, between two ways to deliver the same payload across the Atlantic
Ocean. It turned out that the catamarans are much more efficient then the monohull in
wave making resistance. This result is off course expected because of the slenderness of
the demi-hulls in contrast of the 'bulkiness' of the monohull. SWAN, as a potential flow
code, could not simulate viscous effects; hence the total resistance of the vessels was
estimated manually.
Chapter two of the thesis lays the theoretical background for the physical problem at hand.
The chapter presents the boundary conditions for the problem and the mathematical model
that simulates the physical world. The influence of the lifting appendages on the hull is
also presented. Some sections deal with the method used to solve the mathematical model
- Rankine Panel method.
13
Chapter 1. Introduction.
Chapter three presents the time domain, three-dimensional-code, SWAN-2. It elaborates
the of implementing the mathematical model over the physical problem by the code (for
example by linearization of the boundary conditions).
Chapter four elaborates in details the results for the steady problem. The Kelvin wake
pattern and the ideal fluid resistance of the vessels were evaluated in calm water. The
results are compared with a model test. The dynamic trim and sinkage of the vessels were
evaluated and analyzed as well. The ideal fluid resistance of the catamaran is compared
with twice the resistance of one demi-hull.
Chapter five presents the results of the seakeeping simulation predictions for all
catamarans and monohull. The standard deviation of the response was found and analyzed.
The effect of several alternatives of lifting appendages over the seakeeping response of the
vessels was examined.
Chapter six examines wave load effects on the catamaran hull. All three catamarans were
evaluated at bow waves and oblique waves. The significant one third highest shear force,
bending moment and torsion moment along the center line of the vessels were calculated.
Chapter seven is a short summary and discussion of all above. Several suggestions for
future work are elaborated.
Chapter eight is a nomenclature table.
Chapter nine is a list of references that were used while writing this paper.
Three annexes are attached to this paper: (1) Annex A elaborates the characteristics of the
vessels evaluated; (2) Annex B contains the seakeeping RAO's for heave, pitch and roll
modes obtained by SWAN-2; and (3) Annex C includes the wave loads non-dimensional
RAO's calculated by SWAN-2.
14
Chapter 2. Theoretical Background
Chapter 2: Theoretical Background
The following paper investigates the hydrodynamic behavior of a catamaran,
namely, its calm water resistance, seakeeping performances and wave load characteristics.
The research is using a time domain Rankine Panel Method CFD, SWAN-2. The hull used
for demonstration purposes is a generic catamaran, slender with a deep transom stem,
typical for a semi-displacement vessels. Several catamaran types with different separation
ratio were studied. Also tested a monohull with similar displacement and length. The
following chapter lays the theoretical background for the physical problem to be solved
further on.
2.1 Linearized Boundary Conditions
The following paragraph deals with the boundary conditions defined to solve the
flow regime around a vessel with a forward translation speed U. A potential flow model is
introduced and utilized to establish the required boundary conditions. The flow thus is
assumed 'linear', i.e. ideal and irrotational. Viscosity and non-linear effects can be
neglected. Satisfying conservation of mass, the velocity potential flow can be written as:
82
(2.1)
V2#=a
2
+
aX 2
a2
2+
aZ2
ay2
= 0
This is known as the Laplace Equation. It provides the governing partial differential
equation to be solved for the function#, where
The fluid velocity vector can be defined as:
(2.2)
Vxt)= V#
15
#(i,t)
is the total fluid velocity potential.
Chapter 2. Theoretical Background
X = xi + yj + zk is the translation vector with respect to the vessel's local coordinate
system. By integrating conservation of momentum equations, an explicit formulation for
the pressure, known as the Bernoulli Equation, is introduced. The pressure around a ship
with forward speed U is defined as:
(2.3)
Pa
P-Pa =-p
C
at
+-V.V#iU2 +gZ
2
2
is the atmospheric pressure and g is the gravity acceleration.
Two different types of boundary conditions must be discussed: kinematic condition
corresponding to the velocity of the fluid on the boundaries and dynamic condition
corresponding to the forces act on the boundaries. Unlike viscous fluid there are fewer
conditions to impose since no shear stresses exist in the inviscid potential flow.
2.1.1 Kinematic Boundary Condition
The physically kinematic boundary condition for a fluid flow at the boundary of a rigid
body moving at speed U is that the normal component V -n of the fluid velocity must be
equal to the normal velocity U - n of the boundary surface itself, meaning no fluid particle
can flow through the boundary surface (no flux). In terms of velocity potential this
condition becomes:
(2.4)
LO=U -n for X e SB
an
an is the derivative of the potential in the direction of the normal to the fluid and X is a
point on the body surface, SB . The sea floor is subjected to the same no-flux condition for
all time:
16
Chapter 2. Theoretical Background
LO=0 for z = -H
az
(2.5)
where H is the sea depth.
The fluid domain is bounded by the free surface which can be defined by its elevation
z
=
i7(x, y, t). The kinematic boundary condition on the free surface states that a fluid
particle on the free surface remains there for all times. Mathematically this condition
requires that the substantial derivative of the quantity z -
i7
will vanish on the free surface:
DD(z
- 7) = 0
Dt
(2.6)
(U -
_at
VO)- V r7 =
_
az
aO aO aO aI aO aI7
az at ax ax y ay
->
0
The last two terms are of second order and thus can be neglected in the linearized
kinematic boundary condition which therefore can be reduced to:
(2.7)
(2.7)
aIat
_
aO
az
This approximate boundary condition states that the vertical velocities of the free surface
and the fluid particles are equal; hence a fluid particle on the surface stays there for all
times.
2.1.2 Dynamic Boundary Condition
The dynamic boundary condition on the free surface is obtained from integrating equations
of momentum conservation, known as the Bernoulli's Equation, assuming that the pressure
17
Chapter 2. Theoretical Background
of the fluid must be equal to the atmospheric pressure. The dynamic -condition for a ship
advancing in a speed U is expressed as:
(2.8)
(pP- p)=
p
at
IU2 +-V#-V$+gz
2
2
=0
For the free surface, z = 7 so:
(2.9)
1 (#
1
g( at
2
=
O
21
U2+- v -v
2
As before, the second order terms can be neglected.
The linearized equation for the free surface dynamic boundary condition is then:
1 a#
(2.10)
77 g at
2.1.3 Combined Condition
Both conditions for the free surface can be combined for a single formulation and can be
written as:
(2.11)
a20 +
at
=on=
onzz =0
+ =0at2
2
The last condition is a general case of condition (2.5) and requires that the velocity
potential gradient decays to zero when z approaches infinity for all finite t,
(2.12)
IVoA
-).0,IzI --+cx
18
Chapter 2. Theoretical Background
2.2 Plane Progressive Wave
The free surface elevation defined by a plane progressive wave formulation is
simple yet significantly practical representation. The propagating wave has amplitude, it is
sinusoidal in time with a radian frequency and it moves in phase velocity. The wave
elevation can be described in the following form:
(2.13)
q(x,y,t) = A cos(kx cos/J + ky sin P
wt +c)
or in the complex form:
(2.14)
q(x,y,t) = A - 9Ie
i(kxcosfl+ky sinfi wt+e)I
By using linear theory, a long-crested irregular sea can be written as sum of wave
elevations:
(2.15)
(x,y=
j=1
Acos(kjxcosg + kjysin
where A is the wave amplitude, w is the wave frequency,
p
w t + e)
is the wave direction relative
to the x-axis, e is a random phase angle and k is the wavenumber defined as the number
of waves per unit distance or:
k=2
%
(2.16)
where A is the wave length. The wave speed is defined as the velocity of the wave crests
(i.e. the number of crests which travel per unit time from a specific point). It is signed by
V, and is equal to the relation between the wave frequency and the wavenumber.
19
Chapter 2. Theoretical Background
The velocity potential can be described as:
(2.17)
$(x,y, z, t) = A 9e{-i. ei(kxcos P+ky sin fl wt+c) . cosh[k(z+H)]}
W
cosh[kH ]
In order to satisfy (2.11) the wave frequency should be:
(2.18)
w 2 = gk tanh[kH]
This relation is known as the dispersion relation since it relates the wavenumber with the
wave frequency. For deep water tanh[kh]
(2.19)
-+
1 so,
w 2 = gk
The following figure, borrowed from Lloyd [15], describes clearly several characteristics
of the propagating wave.
IX
Figure 2.1: The three dimensional propagating wave
20
Chapter 2. Theoretical Background
It should be noticed that the frequency w, is an absolute 'global' frequency, different from
the encounter frequency which is the frequency that a moving vessel with a forward speed
U 'feels'. The encounter frequency is given by:
(2.20)
We
=w --
w2
U cos3
g
Figure 2.6 demonstrates clearly the advancing vessel (with local coordinates) and the
unidirectional propagating wave.
2.3 Ship Resistance
The resistance of a ship advancing with a forward speed U, is the force required to
tow the ship at that speed in calm water, assuming no interference from the towing ship. In
case the ship has no appendages, this magnitude is called 'The bare hull total resistance'.
The power required to overcome this resistance is called the effective power and is usually
notated as EHP (Effective Horse Power). The total resistance is a sum of several different
components, which are caused due to a variety of factors. These factors also interact with
each other; hence changing one factor affects several resistance components. In order to
simplify this problem, the total calm water resistance is usually considered a sum of four
major components:
1) Frictional (viscous) resistance caused due to the movement of the ship through a
viscous fluid. Contribute mainly a tangential component to the resistance.
2) Wave making resistance. Caused mainly due to the energy 'wasted' by the ship,
generating surface waves. This quantity is closely related to the amplitude of the waves
the ship leaves behind in its wake, as will be outlined below.
3) Induced resistance that is caused due to eddies shed from the hull or appendages and
due to special and unique hull forms. Streamlining the hull and appendages can reduce
this component.
21
Chapter 2. Theoretical Background
4) Air resistance is experienced by the above water part of the hull and the superstructure
of the ship. This part of the total resistance will not be discussed here. Details can be
found in PNA [1].
The wave making resistance and the induced resistance are commonly taken together and
are known as the residuaryresistance.
2.3.1 Frictional Resistance
Frictional resistance is the largest single component of the total resistance of a ship. It
varies from 80%-85% of the total resistance for slow speed ships to 50% of the total
resistance for high-speed ships. Frictional resistance is basically the result of the tangential
fluid force exerted from the ships movement in water. Frictional resistance depends on the
roughness of the surface, on the overall wetted area of the ship and on the ships speed. Any
roughness of the wetted surface will increase the frictional resistance appreciably over that
of a smooth surface. Any increase of wetted surface (mainly due to over loading the ship)
will cause an increase in the frictional resistance component. The frictional resistance force
is usually presented by:
(2.21)
D=
2
Cf -P-SB U 2
When D is the frictional resistance force, Cf is the frictional resistance coefficient, p is
the mass density of water, SB is the wetted area of the ship and U is the ships speed.
Over the years, lots of model and full-scale experiments were performed in order to assess
the magnitude of this component and the frictional resistance coefficient. The International
Towing Tank Conference (ITTC) in Madrid in 1957 adopted the following formula for the
frictional resistance coefficient for a flat plate:
(2.22)
Cf =
0.075
(log 10 R, -2)2
22
Chapter 2. Theoretical Background
where R, is a non-dimensional value and is known as the Reynolds Number:
(2.23)
R
-L
= U
U is the ships speed; L is the ships length and v is the kinematic viscosity of the water.
This line is known as the "ITTC 1957 model-ship correlation line".
2.3.2 Three Dimensional Ship Waves
The most general wave elevation distribution in three dimensions, transformed to a
reference system, moving with a vessel in the positive x-axis direction with velocity U, is
given by:
(2.24)
y0
21r
0
0
]
77(x, y,t )= 9e fdw fdO -A(W,O)-e [-ik(xcosO+y sinO}+i(w-kUcosO)t
This formulation represent oblique wave (with angle 0) distribution in the ships wake. For
a steady motion the second exponential term vanishes, hence:
(2.25)
kUcos9-w = 0
the wave phase velocity becomes:
(2.26)
VP
W
- =Ucoso
Sk
The expression in (2.24) for a steady wake pattern behind a moving ship becomes:
23
Chapter 2. Theoretical Background
(2.27)
=
x~y/2
p
y = 93e
.7(x,
d - A(O).e -ik(XCOSO+YSifO)I
This expression is known as thefree-wave distribution.
For a very large distance downstream from the location of the ship, this expression can be
simplified and the classic ship-wave pattern can be obtained as derived by Kelvin in 1887.
The following figure describes the Kelvin wave pattern in the wake of a moving vessel.
TRANSVERSE
WAVE CRESTS
-DIVERGING
WAVE CAESTS
Figure 2.2: Kelvin wave pattern in a ship wake
The waves generated in the wake of a ship are confined to a symmetrical sector about the
negative x-axis, which includes semiangles of
Y=
19028'. At this maximum, the wave
is directed with a corresponding angle of 0 = 35*16' above or below the x-axis. For all
other
angles in the wake between - 19*28' to + 19"28' there are two corresponding
values of the wave angle, 0. One represents a transverse wave pattern, the other represents
a diverged wave. More Details and the complete formulation of the pattern can be found at
Newman [6].
24
Chapter 2. Theoretical Background
Figure 2.3 demonstrate the Kelvin wave pattern in a monohull wake. The vessel translates
in a uniform speed U in calm water. The figure was produced by SWAN-2 for a 100 m,
7500 ton monohull at FroudeNumber 0.74.
z
x
Y
-2.6
Figure 2.3: Kelvin wave pattern for a monohull at Fn=0.74
Figure 2.4 describes the Kelvin wave pattern at the wake of a 7500 ton, 100m catamaran
with a separation ratio of 0.4 at Froude Number 0.74. This figure is a realization of the
fluid and body domains produced by SWAN-2.
On each demi-hull wake a Kelvin wave pattern develops. Understanding these wave
patterns and measuring their energy is the key to accurate predictions of the vessels wave
making resistance.
25
77.
Chapter 2. Theoretical Background
-1.49
-2.03
-2.56
-3.10
-3.64
-4.18
-4.72
Figure 2.4: Kelvin wave pattern for a catamaran at Fn=0.74
2.3.3 Calm water Wavc Making Resistance
Wave making resistance of a ship is the net fore and aft force exerted from the fluid
pressure, acting normally on all parts of the hull. For a surface ship, movements of the ship
through water generate waves, which change the pressure regime along the hull. The
resultant net fore and aft force along the hull is the wave making force. The magnitude of
this resistance component energy must be equal to the energy required to maintain surface
waves system.
The waves contain energy both potential and kinetic. Considering the wave energy per
wavelength, per unit breadth, per period,
T
(2.28)
potential energy:
P.E. =
A
fJf
00
26
Yq2ddt
Chapter 2. Theoretical Background
TA
(2.29)
K.E.=
kinetic energy:
0 11OV
p12
dzdx
0 0 -H
The x-axis coincides with the direction of the waves so the y-dependence is dropped. After
integration the following results are obtained:
(2.30)
P.E. = _LpgA 2
and
K.E. = - pgA 2
and the total energy density is given by:
(2.31)
E =.jpgA
2
The traveling energy is best expressed as wave power per unit width averaged over a wave
period and can be found to be:
(2.32)
1 I
2kH
1
2
P =-pgA2V
2
P2(
sinh(2kH)
1
2kH
where VP is the wave phase velocity as defined earlier and the term -V 1+
1
2 " si~h(2kH))
is known as the group velocity, V,. Group Velocity is defined as the velocity by which the
energy of the wave travels. In a fixed reference frame, the energy of each plane wave
component moves in direction 0 with a group velocity Vg. The velocity of energy transfer
across the control surface, which moves through the fluid with velocity U, is given by
Vg cos 0 - U. Multiplying by the energy density and integrating along the width of the
control surface, the total energy flux can be expressed as:
27
Chapter 2. Theoretical Background
(2.33)
dt
=
2
pg A 2V cosO
_
g
-
Ulz
Using the stationary-phase approximation the total wave resistance of a moving vessel can
be derived, as shown in Newman chapter 6 [3]:
(2.34)
1
;/2
D =27rpU2
2
(1
cos 3 0 -dO
A
A thin-ship theory was developed by J. H Michell in 1898. The essential assumption is
that the hull is thin, i.e. its beam is small relative to other length characteristics of the ship.
Wave resistance of a thin ship can be expressed in terms of a distribution of sources along
the center line of the hull, where the local source strength is proportional to the
longitudinal slope of the hull. Using this theory, the wave resistance of a thin-ship can be
expressed in the form of Michell's integral:
(2.35)
D=
4
Yg2
T
I
where y =
3
fsec 0.
f
ff
a
(U2sec
2o(z-ixcoso)]
e
2
z dO
0
(x, z) defines the local half-beam of the hull surface.
2.3.4 Induced Resistance
The turbulent frictional belt around a ship consists of eddies caused due to changes of
form, appendages or other projections. This interference to the flow around the hull is the
component of resistance known as eddy or induced resistance. This component is usually
combined with the wave making resistance and called 'residuary resistance'. Semidisplacement vessels experience eddy resistance mainly due to transom stems and
separation of flow there. At high speeds, the fluid leaves the transom dry and so the
pressure there is practically atmospheric. The bow of the ship is submerged so an
28
Chapter 2. Theoretical Background
unfavorable pressure gradient is generated and a force is acting in the opposite direction to
the ship movement. This force can be evaluated as the integrated hydrostatic pressure over
the transverse projected dry area of the transom.
2.4 Ship Response in Sea Waves
The simplest way to discuss ship response at sea waves is to assume regular, small
amplitude, sinusoidal type waves. Actual ocean waves are highly irregular, but even so,
they can be related to regular waves by a linear superposition of sinusoidal components as
shown in equation (2.15). In that case the irregular sea can be described by a sea spectrum,
S(wj):
(2.36)
1Ai
2
=2S(w)W
where A is wave amplitude, and Aw is a small, constant increment, between successive
frequencies. Another important assumption is that the ship behaves as a Linear, Time
Invariant System. This definition provides the designer the opportunity to treat the ship
response as linear and by that to find the total response of a ship by superposing all single
responses. By virtue of linearity, the response due to each component of the wave
spectrum can be analyzed separately and the total response would simply be a linear
superposition of all individual responses. The response is also time-invariant, hence
depend on past responses. This definition is correct for all ship responses; motions (heave,
surge, etc...) and body loads (shear force, bending moment, etc...). More details can be
found in PNA [2], Faltinsen[6], Hughes [8], and Sclavounos [9].
2.4.1 Sea Spectrum
The wave spectrum is estimated from wave measurements. It assumes that the sea can be
described as a stationary random process and is referred as short term statistics. For an
open sea conditions the 15th ITTC (International Towing Tank Conference) recommended
29
Chapter 2. Theoretical Background
the use of the ISSC (International Ship and Offshore Structures Congress) spectral
rw T <
27r
2r
SW)= 0. 11.e
(2.37)
e
-0.44
-
0.11
2T
-
formulation for fully developed sea:
-2-
2
HX .-T,
where HX is the significant wave height, defined as the mean of the one third highest
waves and T, is the mean wave period defined as:
T, = 2rm
(2.38)
where the 'moments' are defined as:
mk
(2.39)
=
fWS(w)dw
0
Figure 2.5 illustrates a wave spectrum which is actually a statistical representation of its
wave components.
~LA
A~A4
S AEFEC
~
4Oaf
SCALE OF
rrT% C
O$TAt4CE
(b) COuPorExrw W*E5
Figure 2.5: Wave spectrum illustration
30
Chapter 2. Theoretical Background
The sea spectrum defined above is used in the following paper and utilized to analyze the
ship motion responses.
2.4.2 Ship Response
For the problem discussed in this section, a plane progressive wave of amplitude A and
direction 8 is incident upon the ship as illustrated in figure 2.6. The ship in response is
free to move in six degrees of freedom, three translational motions: surge, sway, heave and
three rotational motions: roll, pitch, and yaw.
Z
IC
3 (HEAVE)
U(t)
6(YAW)
y
6" (SWAY I
2 0
4-5[PITCH)
6' IfSURGE)
X
S(ROLL)
4
A,w
Figure 2.6: Ship motions, six degrees of freedom and the incident wave
31
Chapter 2. Theoretical Background
The corresponding velocities U (t) are sinusoidal in time oscillating with the frequency of
encounter as defined on equation (2.20):
(2.40)
U, (t) = Re(iw,4je''),
j =1,2,...6
(, is the j' response motion. (g is defined as:
(2.41)
- = 4j{ e'x ''
j =1,2,...6
(;j is the amplitude and X is the phase angle of the j's mode of the response.
Here gg
represent the total response in each direction. The ratio of the response amplitude,
|{ , to
the incident wave amplitude, A, is known as the Response Amplitude Operator(RAO) and
is one of the ship characteristic. The response spectrum of the ship to the incident wave is
then a function of the ocean wave spectrum and the ship's RAO.
2
(2.42)
j=1,2,...6
where H(w) is the Transfer Function. It should be noticed that the sea spectrum is given
in absolute frequency terms, while the ship's transfer function is sometimes given in
encounter frequency terms. The variance of the response can be found by:
(2.43)
2
a,=
IAJ(W)I1jQ'VAUW,
j=1,2,...6
0
and the standard deviation, o- , is the square root of the variance.
2.4.3 Velocity Potential
The total three dimension velocity potential can be presented by:
32
Chapter 2. Theoretical Background
(2.44)
S6
#(x, y, z,t)= [-Ux +#s (x,y, z)]+
(Z4AR (x,y, z )+#D (x, y, z)+ # (x,y, z) ewet
Sj=1
where the terms in the first parenthesis are steady, U is the ships speed and
#s
and is the
perturbation potential due to steady translation.
The boundary condition concerning the steady potential on the ship hull are similar to
those obtained by (2.4):
(2.45)
a s =
--
an
on SB
where n = (n, , n 2 , n3 ) is the unit vector, normal to the ship hull, pointing out of the fluid
domain. The terms on the second parenthesis in (2.44) are the unsteady part of the
OR
represent the radiation potential of a rigid-body motion due to unit amplitude
motion in calm water (in the absence of incident waves) in the j' direction.
#D
+
potential.
represent the potential due to the incident waves and their interaction with the body.
Assuming linearity this motion is independent of the body motion and can be defined as if
the body is fixed.
#,
is the incident wave potential and
OD
is the diffraction potential
represents the disturbance of the incident waves by the fixed body. Boundary conditions
for these potentials are:
(2.46)
an+0
=00 -> --
an
=-
an-,a0
on SB
Each potential has to satisfy also the Laplace equation, stated in eq. (2.1). On the sea
bottom,
# -+ 0
for z -+ -oo
as defined in eq. (2.5) and (2.12). On the free surface the
boundary condition follows equations (2.11) and (2.44) to be:
33
Chapter 2. Theoretical Background
(2.47)
w2
+ -- = 0
on z = 0
az
g
for j =1,2,...6 and #IID
2.4.4 Wave exciting Forces
The general form of the basic Linearized equations in six degrees of freedom using the
local ship coordinates is given by:
6
(2.48)
j~t
j =1,2,...6
t
;
=~k
k=1
where
Ajk
is the general inertia matrix of the ship containing components of mass and
mass moments of inertia. Fj (t) represent the total exciting forces or moments acting on
the hull in the j' direction.
4
are the hull accelerations in the k' mode.
k
Fj (t) can be expressed in terms of the j' direction hydrostatic and hydrodynamic fluid
forces:
6
(2.49)
Fj (t)
= LAjk
k(t) = FHS + FHD
k=1
FHS is the net hydrostatic force and includes gravitational and hydrostatic forces. It can be
expressed by:
6
(2.50)
F HS
6
jk
eIt
j=1
where Cjk is the hydrostatic restoring force coefficient. Exact values of Cjk can be found
in PNA [2]. ke'"' are the six motion responses of the ship as defined in equation (2.41).
34
Chapter 2. Theoretical Background
The hydrodynamic force due to unsteady translations is given by:
(2.51)
Fj HD = FX
+F R
where Fj EX is the excitationforce in the j'b direction and is equal to:
(2.52)
Fj
=
eiwt
i +F D
F.' is the complex amplitude of the exciting force due to incident waves, known also as
the Froude-Krylov exciting force. It results from the pressure integration over the body
surface, which would exist in the wave system if the ship were not present. It equals to:
(2.53)
F,' =
#>dS
fnj iWe -U
D
FD is the complex amplitude of the exciting force due to diffracted waves, called also the
diffractedforce. It is caused by the diffraction of the incident wave due to the presence of
the vessel. It equals to:
(2.54)
-PPJfrWe
FjD
-U
SB\
-
1x)
#ODdS
FFR is the hydrodynamic force in the jh direction due to forced motion. It results from
the radiation of waves away from a vessel that is forced to oscillate. It is given by:
(2.55)
FR
nj
_-Lp
=
k=1
'We
sB
35
-U
kdS 1kei"'
Chapter 2. Theoretical Background
(;ke''' are the six motion responses of the ship as defined in equation (2.41). Solving for
the integral in equation (2.55) can simplify the radiationforce expression to:
6
(2.56)
FR
= EWe2
jk
e'e
weBjkf
k=I
where
AJk
is the added mass in the j'h mode due to unit motion in the k'V direction and
BJk is the damping coefficient in the j' mode due to unit motion in the k'h direction.
2.4.5 Ship Equation of Motion
Taking into account the various components of forces as introduced above, the total fluid
force on the ship becomes:
(2.57)
F(t) =
F/
FHS +FEX
jF +F
+FR
jF
Plugging in equation (2.57) the components of the forces as were defined above:
(2.58)
-We2Ajk + Ajk )+
B, +Cie
= FJ' ++FC.,
1,2,...6
Equation (2.58) is the linearized governing equation of the three dimensional motions for
an unrestrained vessel in sinusoidal waves. Using complex number analysis, this equation
can be used to find, for example, the heave motion RAO of a vessel:
IF
(2.59)
3
3
3
A
+ FD
-tC
33 -w,
(A 33 +A
33
)] 2 +We B 3 3
}2
The three dimensional added mass and damping coefficients can be found using
experimental results or analytical calculations. Using strip theory (for slender bodies) this
36
Chapter 2. Theoretical Background
characteristics can be derived from the two-dimensional coefficient describing crosssection elements of the hull. For example, in order to find A 33 (the added mass value of
the hull in the '3' direction due to force in the '3' direction) of a transom stern vessel the
following formulation can be used.
u
Xb
(2.60)
As
33 -
a33 (x)dx
-
b 33 s
or B 3 3 of a transom stern vessel (the damping coefficient of the hull in the '3' direction due
to force in the '3' direction) can be found using the following equation:
Xb
(2.61)
B33 = Jb33(x)dx + Ua3 s
where a33 and b33 are the two-dimensional
respectively, and a33
added mass and damping coefficient
and b 3 3 s are the two-dimensional added mass and damping
coefficient of the ships stern (in case of transom stern). The curios reader will be able to
find more details describing the three-dimensional components in PNA [2], Faltinsen [6],
and Lloyd [ 15 ].
2.4.6 Hull loads
All floating objects are subjected to loads that result from the body weight, its buoyancy
and the environment; wind, current and waves. As discussed above, waves excite ship
motion in three translational and three rotational directions. Waves also excite three
'translational' and three 'rotational' body loads.
A way of classifying loads is according to how they vary in time: static, slowly varying or
rapidly varying. Three ways of calculating loads effect correspond to these loads: static,
quasi-static and dynamic. An example to static loads is all the stillwaterloads, internal and
37
Chapter 2. Theoretical Background
external pressures (buoyancy and weight) and thermal loads. Examples to slowly varying
loads are the wave induced dynamic pressure distribution along the hull as a result of wave
encounter and ship motion, green seas, sloshing of liquids in ship tanks and inertia loads
especially on masts or other elongated structures. Some examples for rapidly varying loads
are slamming and mechanical vibrations.
Almost any irregular dynamic or quasi-static loading can be represented as a combination
of regularly varying loads especially if the force-displacement ratios are linear or almost
linear. The problem of calculating the load effects can then be solved 'in the frequency'
domain with frequency as the principal independent variable instead of time. The
frequency based distribution of loads or load effects is presented in a shape of spectrum, so
wave spectrum causes a load response spectrum.
2.5 Lifting Foils
As part of the research, several alternatives of flat lifting foils were adjacent to the
vessel, trying to reduce heave and pitch responses. The following section lays the
theoretical fundamentals to evaluate this problem.
2.5.1 Prandtl's lifting line theory
The three dimensional lifting and induced drag forces for a foil with span s, chord c(y)
and advancing speed U are given respectively as:
(2.62)
L=--p.CL.S.U
2
2
D=- p.-CD
2
*S.
U2
where S is the planform area of the foil, CL is the three-dimension lifting coefficient and
CD
is the three-dimension drag coefficient. The next figure, borrowed from Newman [3],
explains clearly the variables and the coordinate system concerned with the foil.
38
Chapter 2. Theoretical Background
Z
LIFTv
oo RAG,
D
Y
Figure 2.7: Lifting foil characteristics
The lift and drag coefficient are difficult to evaluate. Newman chapter 5 [3] derives a
general expression for these quantities under the assumptions that the foil circulation F(y)
and its derivative F'(y) are continuous along the span and that F(y) vanishes smoothly at
the foil tips
s (Kutta-condition).
(2.63)
CL
(2.64)
CD
usY
=
2
Kdc) dK dK
dK y-K
S
2,rU2 S _JEIY;dj
2
2
where k is a dummy variable utilized to determine the integrated circulation at the
transverse position, y . In order to solve for the coefficient introduced above, it is essential
to solve Prandtl'slifting line equation to determine the circulation distribution along the
span of a lifting surface.
39
Chapter 2. Theoretical Background
1
(2.65)
where
12D (y)
)J
]F(Y) = ]F2 D (Y) +-4C(y) 2
4_Y2
du(K) dK dK
dK
K-y
is the two-dimension circulation of the foil given as the point vortices along
the chord.
(2.66)
r
2
D (Y)=
_2
where g is a dummy variable.
Using the circulation distribution and the lift and drag coefficient defined above, simple
expressions can be evaluated for an elliptic circulation distribution along the span of a
symmetric foil.
(2.67)
CL=27r
AR
AR+2
C 4
AR
(AR+2)2
2
CL 2
7rAR
where AR is the three-dimension aspect ratio of the foil and is given by the ratio of the
span to the mean chord. a is the angle of attack relative to the x-axis.
Prandtl's lifting line theory is valid mainly for large aspect ratios but has proven to give
good results also for relative small aspect ratios. It is valid for 'thin' foils for small angle of
attack. For angles greater then 12-13% stall phenomena occurs and lift is lost.
2.5.2 The effect of Lifting Foils on the Equation of Motion
A vessel with lifting foils has different damping and restoring forces then those obtained
without these foils, hence different motions and forces response. The lifting force as a
result of a foil attached to a vessel can be defined as:
40
Chapter 2. Theoretical Background
(2.68)
L(t) = Re{F3 Le iwle
While the vessel and the attached foil advance on sea with ambient waves, oscillatory
displacements occur and the foils effective angle of attack alters with time. The time
dependent change in the angle of attack is given by:
a#,
(2.69)
+
a(t)=-
3z fi5
Ox
where g;3, g, ; are the surge, heave and pitch total responses respectively and their doted
values represent the motion velocity in this direction,
#, = iweg;
e'.
x10 i, is the
longitudinal distance between the location of the foil and the origin of the coordinate
system located at midship.
#,
is the incident wave potential. Combining equations (2.41),
(2.68) and (2.69), assuming surge speed,
4,,
and wave speed in the x-direction,
' , are
Ox
negligible relative to the ships speed, gives:
(2.70)
F 3 L=F
3L
g3 +I
"iWl'J -(U
)
5
ao,
U
3U
az
e ''' canceled on both sides, where F is given by Prandtl's Lifting theory (2.62), (2.67):
(2.71)
F =rSU 2 AR
AR+2
these expressions lay on the assumption of slowly varying flow around the lifting foil. This
assumption is correct for small reduced frequencies, 0, where:
41
Chapter 2. Theoretical Background
weC
(2.72)
U
for hydrofoil vessels this assumption is almost always correct since the vessels speed is
much higher then wec.
The vertical lifting force obtained in (2.70), can be combined in the heave equation of
motion (2.58):
(2.73)
We2(A 33 + M)-
F
3
- we (A35 + M)-
X101.
UW
-U
-1
g,
5
+ iwB 33g3 + iWB 3 5 g5 + C 33q 3 + C 35 q5 =
1 8#i
iwe
UWe
"
-
U+
U
U azj
similarly, the pith moment contribution from the foil can be expressed as:
(2.74)
F
_X
F
this result can be combined in the pitch equation of motion:
(2.75)
(A 55 + 155 )- -
F' + F5 D -xf0
F
We
(A 53 + 153 )3 + iwe B55g + iweB5 3 g 3 + C55 ; 5 + C5 3;g
fiW
U
- 1 g
)
- we
-
g3
U
-__
+I
U az
=
I
In equations (2.73) and (2.75) heave and pitch responses are coupled and surge responses
are neglected. Collecting terms from (2.73) and (2.75) relative to the ships heave and pitch
motions give:
42
Chapter 2. Theoretical Background
(2.76)
+ M)
3
WeF
- We (A 35 + M)- 5 + iweB 33
U
+
F);, =
F,'
+ FD'
+
+ (C3
+
We B3
-
efoil
)U
F;
5
+ C33 ; 3
)
33
+
--We2(A
FF1 a 1 1
U azj
-We
(A55 +I55)5 -We (A 53 + 153)g3 +
-xF
F| +
+
D
WeB 55 +
xfig F[
-
U
"
2
U
+ (iWe
B53
"e Foil
U
0
az_
The damping and restoring coefficients of the ship with the foil can be presented by the
following relations:
(2.77)
B33foil = B3
B
F
U
xf011F
foil = B35
B53 foil = B53
B55 foil = B55 +
U
xfoil F
U
foil
F
U
C foil = C3
C "" =C + F
C
"" = C,
C55 foil
=
C55 -Xfo, 1
F
The exciting forces with the foil can be presented by:
43
Chapter 2. Theoretical Background
(2.78)
FI F D
F
=
FF +Fs D )+ = F I +FF D F . -kTTiikxf i kCOS
+F Dril F
+F D)
x5FD
Xfoil
U
z
=
1 + F D)
+oil
-kT0"-ikx pijCOS1
F
- iwe f
where T, is the foils' draft, 8 is the wave heading, F is given by (2.71) and U is the
vessels speed.
2.6 Panel Method
Panel methods are developed to solve complex, open form, three-dimensional fluid
dynamic problems where greater accuracy is required. Panel methods are based on
potential flow theory where oscillating amplitudes of the fluid and the body are small
relative to the dimensions of the body cross-section.
Panel method relies on the assumption that any irrotational, incompressible flow can be
represented by a proper distribution of sources, sinks or doublets over its bounding
surfaces. A source is defined as a point from which a fluid is imagined to flow out
uniformly in all directions. A sink is a 'negative' source, where fluid is 'sucked' in
uniformly. A doublet is a combination of source and sink. G represent the potential of a
source at an arbitrary point inside a control volume such that:
(2.79)
G=1
41r r
r is the distance from the source to the arbitrary point where the potential is to be
evaluated and p is the 'strength' of the source, defined as the total flux outwards (or
inwards) across a small closed surface surrounding the arbitrary point.
44
Chapter 2. Theoretical Background
The governing mathematical identity utilized to solve fluid hydrodynamic problems is
called Green's second Identity. This identity is derived from the divergence line.
(2.80)
fn-U -dS
JV.U-dV
control
surface
control
volume
where U is a vector. U can be replaced by the vector #VG - GV#0, where
#
is the
velocity potential inside the control volume such that V = V0 (V is the body velocity)
and G is the source potential. The Green's second Identity becomes:
fJ
V.vG - GV)dV= fn* (OVG - GV0)dS
control
control
i'olutne
surface
(2.81)
C
JG -G
fV 2G-GV20 PV
control
control
volume
a
dS
n
surface
This identity basically relates the governing equation of the physical problem to the
velocity potential on the bounding surfaces of the boundary value problem. On the left
hand side of the identity the term
-
JGV 2VdV turns into zero due to Laplace Equation
control
volume
(2.1). This fact indicates that an infinite control volume problem in space is reduced to a
finite closed form problem over the bounding surfaces of the body.
2.6.1 Rankine Panel Method
Rankine Source potential with unit strength takes the form of:
45
Chapter 2. Theoretical Background
4;r r
(2.82)
1
I1 -
4;r x - g
1
1)2 + (X2 -
2)2 + (X3 -
x is the vector aiming to an arbitrary point and
'
3 )2r
is the vector aiming to the source point.
Figure 2.7 demonstrates the coordinate system and the distances from the origin satisfying
equation (2.82). The simple Rankine Source is used to model free surface flows since it is
difficult to evaluate the wave Green Function. The penalty for using this source is the
necessity to discretize also the free surface in addition to the body surface.
y
A
x
Arbitrary
point in
space
z/
x -
=r
source
Figure 2.8: Rankine Source and an arbitrarypoint coordinate system
Using the Rankine Source as the Green Function in equation (2.81), the Green identity
becomes:
(2.83)
aG
-+sF
46
)d
Chapter 2. Theoretical Background
I(A, t) is the unknown velocity potential of the problem stated in section 2.1, SB is the
surface plane of the body, SF is the free surface and j is the position of the Rankine
Source.
The contribution from a closing surface at infinity vanishes due to the decay of i(Y) and
G(Y , fas
X -+ x for fixed values of f and t. Over SB I On is known and can be found
using the body boundary conditions. Over SF the linearized boundary conditions establish
D, =
I,. Substitution of these boundary conditions reduces (2.83) into two integro-
differential equations for (D over SB and ((D, ) over SF which is solved by SWAN and
described on the next chapter.
47
Chapter 3. SWAN-2 Implementation
Chapter 3: SWAN-2 Implementation
SWAN-2
(Ship Wave Analysis) is computational
fluid dynamic software,
developed in the Ocean Engineering department at the Massachusetts Institute of
Technology in recent years. A potential flow, Rankine PanelMethod for a body with or
without uniform speed is used to solve the three-dimensional time-domain ship-wave
interaction problem. SWAN-2 finds the wave making resistance of a translating body,
solves its seakeeping problem in ambient waves and calculates the different induced loads
acts on it. The following chapter elaborates the theory and assumptions behind the
computer code. More details regarding the code can be found in Sclavounos [4] and in the
theory and user manuals of SWAN [11], [12].
3.1 Background and Linearization of the Boundary Value Problem
The linearized boundary conditions defined in section 2.1 are solved by SWAN to
find the total velocity potential of the free flow around the hull. The program uses the
coordinate system defined in figure 2.5. As elaborated on section 2.4.3, the total potential
of the flow is constructed of three components:
1) A steady potential due to steady translation of the ship in calm water,
2) An unsteady potential due to unit amplitude motion in calm water (in the
absence of incident waves). Arises from steady dynamic sinkage and trim of the
ship and wave radiation.
3) The potential due to the incident waves and their interaction with the body.
The steady flow past a ship and its positive image above the free surface is selected to be
the basis-flow and is referred to as the double-body flow. The advantage of this choice
over the uniform stream, Ui, is that the effects of ship thickness are better modeled by the
basis-flow potential, hence better results are obtained.
48
Chapter 3. SWAN-2 Implementation
Linearization of the boundary conditions (as demonstrated in section 2.1) is justified if the
next two conditions hold: (1) the ambient wave slope is small and (2) the hull is
sufficiently thin, slender or flat. For the zero speed case, linearization exists by dropping
all quadratic terms. For the forward speed case linearization assumptions follows that the
fluid disturbance and wave amplitudes (caused by the ships translation) are much smaller
compared to the ships speed U. This also means that the free surface elevation due to the
disturbance is much smaller relative to the free surface elevation due to the double-basis
flow.
3.1.1 The Free Surface
Using the Bernoulli Equation stated in (2.3) the basis flow surface elevation can be
expressed as:
(3.1)
r
=
Ua-ks
1
ax
2g
g
VOs VOs
bs is the steady basis flow potential or the linear first order component of the potential..
Substituting equation (3.1) in the non-linear form of the free-surface boundary conditions,
(2.6) and (2.9) leads to the following formulation of the unsteady radiated potential and
elevation.
(U -
(3.2)
(3.3)
#R
at
v~s)- V ? =
(U-VLS).VR
I12
2
77+
on z=O
S
on z = 0
is the radiated unsteady potential, or the second order non-linear potential. It is assumed
that the basis wave elevation, 17s , is sufficiently small to be transferred on the z = 0 plane
with small error.
49
Chapter 3. SWAN-2 Implementation
3.1.2 Body Boundary
Assuming small radiated amplitudes, the boundary conditions for the steady and unsteady
potentials on the mean vessel position are given in section (2.4.3). The total unsteady
potential can be expressed as:
6
(3.4)
where
#,
and
$(x, y, z' t)= I#
j=1
#D are the incident
+ ,+
D'
and radiated potentials respectively.
Using (2.46) the radiated potential can take the form:
a#R
(3.5)
where g, (t) ,
h = (n ,
,n2n
3)
6
1
j
=
tni + g*ymj
on S
on
SB
1,2.. .6, is the ship oscillatory motions in six degrees of freedom,
is the unit vector, normal to the ship hull, pointing out of the fluid domain
(into the hull) and
(3.6)
(n4 , ,n
5 n)
5 )i
(MI M2,M3
0
(M4 , Mn), = ( v X (U
U
-
s
VA)
The formulation presented here along with the conditions presented in sections 2.1 and
2.4.3 complete the boundary value problem calculated by SWAN.
50
Chapter 3. SWAN-2 Implementation
3.2 Ship Response in Calm Water and in Ambient Waves
3.2.1 Ship Translating in Calm Water
The basic case of zero speed is treated by assigning U = 0 and
, = 0. In the case of a
ship starting to translate with a forward uniform speed U, the steady and unsteady
potentials obey the conditions (2.11), (2.45). The transient wave flow induced by the ship
forward speed is subject to the free surface conditions stated in section 3.1.1. The vessel
motions,
, (t) =0 except for ;3(t) which corresponds with the ship steady dynamic
sinkage and
,(t) which correspond with the ship steady dynamic trim. The steady
motions are produced after the transient effect die out.
3.2.2 Ship Seakeeping with Forward Speed
This is the most general case where ambient waves exist (e#,
w 0 as opposed to the case
in section 3.2.1). It involves solving the complete set of boundary conditions, namely
(2.11), (2.45) for the steady potential and (2.46), (3.2)-(3.5) for the unsteady, time
dependent potential. Solving the potential flow enables finding values of interest as the
ship motions in six degrees of freedom. More details and a test case are examined in Kring
[16].
3.2.3 Thin Wake Sheets - Transom conditions
The majority of the modem ship have transom stems. For such stem shapes, flow
separation at the sharp lower edge of the transom is triggered by viscous effects and
requires implicit enforcement of the correct behavior of the potential flow.
SWAN-2 allows modeling of thin sheets in the wake of vertical and horizontal objects as
struts or transom stems. Lifting surface theory methods are utilized. The velocity potential
is forced to satisfy the Kutta-Condition at the vessels transom: the fluid is enforced to
detach the stem in a smooth manner. As an outcome, the fluid velocity is finite and the rest
51
Chapter 3. SWAN-2 Implementation
of the transom is dry and causes induced drag as introduced in section 2.3.4. The total
potential at the wake should satisfy the Laplace equation.
Figure 3.1 illustrates the free surface wake shed by a transom stem vessel at Froude
Number 0.74. On the horizontal wake surface, the free surface conditions (3.2) and (3.3)
are enforced. The wave elevation right after the transom is enforced to be equal to the
vessels draft (flow separates at Trailing Edge,).
The Trailing Edgeo is defined to continue with the z =0 plane. The wave slope at the
transom must be equal to the slope of the hull at the transom and the velocity potential at
the transom and its normal derivative are equal on the vessel and in the free surface wake.
TRAILING
EDGE I
TRAILING
EDGE 0
Figure 3.1: A thin wake sheet
52
Chapter 3. SWAN-2 Implementation
The dynamic Kutta-Condition imposes a dynamic pressure on the free surface that
balances the hydrostatic pressure due to a given transom draft. Along with the kinematic
Kutta-Conditionthis ensures continuity in pressure along the detaching streamlines of the
transom.
3.2.4 Motion Control Lifting Appendages
SWAN-2 allows modeling of motion control lifting appendages attached to the ship in
order to improve its attitude at sea. In most cases the appendages dimensions are small
relative to the hull, hence the flow around the device does not change significantly the hull
potential. SWAN-2 ignores the effects of the hull disturbance potential due to the presence
of these appendages. The lift force achieved by these devices is appreciated using the ship
forward speed and angular displacement and the incident wave velocity vector. The overall
force is calculated by summation of the steady force (as stated at equations (2.62) and
(2.67)) and the time dependent lifting force (as stated at equations (2.70), (2.71) and
(2.74)).
(3.7)
where
Ltota
LTotal
=pU2
A
(ead,
A+2
+ a(t))
is the total lifting force acting vertically, ate
and a(t) are the steady and
time dependent angles of attack. The total lifting force is assumed to act vertically at the
center point of the lifting surface. Its contribution to the total force and moment acting on
the hull is included in the derivation of the equation of motions introduced at (2.75)-(2.78).
3.3 Numerical Solution
Green's Second Identity (equation 2.81) is being solved by SWAN-2 using the
boundary conditions introduced in sections 2.1 and 3.1 for the velocity potential and its
normal derivatives over the fluid domain boundaries. A Rankine point source (equation
2.82) is invoked as the unit strength Greens function and leads to the integral equation
53
Chapter 3. SWAN-2 Implementation
introduced in equation (2.83). This equation is utilized to solve all linearized boundary
value problems over all surfaces: body surface, free surface (including the surface between
the hulls of a twin hull vessel) and wake sheet.
3.3.1 Spatial Discretization
The vessel boundary and the free surface within well defined boundaries are divided into a
large number of quadrilateral panels. Over the surfaces the velocity potential,
#
and the
wave elevation, t7 are approximated by a bi-quadratic spline variation:
(3.8)
$(Y, t)~
$j(t)Bj (Y)
where the basis functions, B, (Y), is centered in the
j'
panel and provides continuity of
the potential and elevation and their tangential gradient. The coefficients $, and
m, are not
equal to the potential and wave elevation in the center of each panel but are related to them
linearly through the spline coefficients of the relevant panel.
3.3.2 Meshing
SWAN-2 utilizes panel shapes elements through a well defined fluid domain and hull
boundaries. The surfaces are discretized using rectangular topology within cartesian or
polar coordinates. The shape of the body is crucial in determining which coordinate system
is to be used (for example: offshore cylindrical platforms are modeled using polar
coordinates, slender ship hulls are models using cartesian coordinates). Transom stern
vessels are modeled in SWAN-2 using three different types of panel sheets.
1. A free surface panel sheet. Includes all the free surfaces around the vessel. The
boundary conditions introduced in equation (3.2) and (3.3) together with the
Greens Identity (2.83) are applied for the velocity potential and the wave elevation.
54
Chapter 3. SWAN-2 Implementation
At the boundaries of this sheet the second derivative of the potential and the wave
elevation are enforced to zero. The free surface can be modeled using rectangular
or oval panels. The free surface panel discretization includes several components:
a. A thin wake sheet behind the vessel where the transom stern KuttaConditionsare also imposed.
b. An artificial beach on the fluid domain boundaries to absorb (damp) the
wave energy. Damping forces are measured and are the wave making
resistance of the vessel.
c. A thin wake sheet behind a vertical solid boundary as strut.
2. A submerged body panel sheet. The submerged part of the hull or components of it
are modeled by this type of sheet. Body boundary conditions are applied (2.45),
(2.46) and (3.5). It also include panel sheets used to discretize thin vertical shapes
as struts.
3. A transom stern panel sheet. Imposes the Kutta-conditions on the trailing edges to
resume continuity with the leading edges of the wake behind it.
SWAN-2 utilizes each sheet automatically based on input data entered by the program
user.
A common feature of all types of sheets is that all of them have a rectangular topology.
More details can be found in SWAN Theory Manual [11].
The next figure demonstrates a discretization of the fluid domain and of the left side of a
twin hull vessel in a cartesian coordinate system. Only half the vessel in modeled (in case
of a monohull - half the hull) for symmetry reasons. 'Cutting' half the problem saves
calculation time.
55
Chapter 3. SWAN-2 Implementation
FREE
SURFACE
SHEET
SUBMERGED
BODY
TRANSOMPAEL
STERN
PANELS
Figure 3.2: Quadrilateral discretization of the fluid domain and the hull surface
3.3.3 Tcmporal Discrctization
In the time domain, a time-marching scheme is selected to approximate the time derivative
of the state variables: the potential,
#
and the wave elevation, t7 or equivalently their
spline coefficients as defined by equations (3.8) and (3.9). The Euler time scheme as
defined here is used by SWAN-2.
(3.10)
At
( t
At
56
Chapter 3. SWAN-2 Implementation
where n marks the time step. Equation (3.10) is substituted in the free surface condition,
equations (3.2) and (3.3), and in the Green's Identity, equation (2.83). The remaining terms
in (3.2) and (3.3) which does not contain time derivatives of the state variables can be
enforced either by the nth or the (n+1)st time step. SWAN-2 enforces the kinematic
boundary condition to be satisfied by the explicit Euler discretization for the past solutions
at time t = tn, and enforces the dynamic boundary condition to be satisfied by the implicit
Euler discretization at present time where t = tn 1 .The resulting is a mixture of both
explicit and implicit discretization methods and is referred to as the emplicit Euler
discretization.
3.3.4 Numerical dispersion, damping and stability
A discretization error throughout the free surface mesh could distort the propagating wave
pattern to the point where the solution becomes inaccurate. This error is quantified by the
numerical dispersion and damping. These quantities measure the discrepancies in the
phase and amplitude between the discrete and the continuous wave patterns respectively.
The condition required for a numerical error to vanish in the limit as the panel size goes to
zero is known as the numerical stability criterion.
The free surface discretization is characterized by the panel aspect ratio, a and the grid
Froude Number, Fh where
h=
(3.11)
h,
F
-
(3.12)
g
The panel dimensions in the stream-wise and transverse direction are presented by hx and
h, respectively.
57
Chapter 3. SWAN-2 Implementation
The stability analysis of marching schemes in time domain utilizes a criterion which
restricts the choice of the non-dimensional time step:
V
(3.13)
g
At
The stability properties of a broad range of marching time schemes were studied by Nakos
[17] and Vada & Nakos [18]. The result, borrowed from Sclavounos [4], illustrates the
critical values of 8, is presented in the following figure.
".0
I
I
I
I
I
I
I
/
/
'75
12.5
-
15.0-
10.0
-l
7.5
5.0
a=0.1,1..24.A,10.1
2.5
0.0 t
0 2
I
0.5
I
1.0
I
1.5
I
2.0
I
2.5
I
3.0
I
3.5
I
I
I
4.0
45
5.0
5.5
F,= UI(gh,)"
Figure 3.3: Stability criterion for emplicit Euler time marching schemes.
The diagram plots the values of 6 as functions of a and Fh. It elaborates the upper
boundary of the time step, At. It is noted that for higher speeds a smaller time step is
required, thus longer execution time.
58
Chapter 3. SWAN-2 Implementation
3.4 Radiation Condition
The proper enforcement of the radiation condition is essential for the success of
Rankine PanelMethods in simulating ship flows.
SWAN-2 enforces radiation condition by the design of an absorbing dissipative beach
located at the outer end of the free surface mesh. Over the dissipative beach the following
conditions are enforced:
(3.14)
(3.15)
U -r
atc x az g
(at
2v7 +---#
-- 0=
# -gr7
&x
0 is the wave disturbance potential, and v is the strength of the damping parameter. v is
chosen such that its value increases towards the outwards of the absorbing beach. Over the
inner edge of the beach its value is zero for continuity reasons. At zero speed, these
conditions lead to the dispersion relation:
(3.16)
w=iv
gk
More details about the construction and performances of the dissipative beach can be
found in Nakos, Kring & Sclavounos [ 19].
3.5 Flow Solver
Enforcing the free and body surfaces boundary conditions into the Green'sIdentity,
leads to an integral equation in space and a system of ordinary differential equations in
time. The unknowns are the basis flow spline coefficients,
59
( 0 ), (t)
for i =0,1.. .N, the
Chapter 3. SWAN-2 hplementation
wave disturbance floe spline coefficients, Ok(t) for i
elevation spline coefficients, 1b (t) for
j
=
0,1.. .N and the free surface
= 0,1.. .Nf . N is the over all number of panels
over the domain and Nf is the number of free surface panels.
The velocity potential and wave elevation are then solved for each time step of the emplicit
Euler scheme described in section 3.3.3 at each panel according to equations (3.8) and
(3.9).
3.6 Hydrodynamic Pressure
The hydrodynamic pressure follows from the application of Bernoulli's Equation
(2.3) for the total potential (2.44). SWAN-2 decomposes the total potential into two
components:
(3.17)
$(Y,=
t (Y)+ (Y, t)
I(Y) is defined as the impulsive potential and has the following boundary conditions:
(3.18)
(3.19)
V(0)= 0
on the free surface
on the body surface
On
Since V/ vanishes at the free surface, all wave effects are included in the second
component, the residual wave potential, qp(y, t). This component satisfies the following
condition:
(3.20)
On
On
0
on the boy surface
60
Chapter 3. SWAN-2 Implementation
This condition is obtained by substituting (3.17) into equations (3.2) and (3.3). This
decomposition is essential to ensure the stability of the time marching scheme, invoked to
treat the vessel equations of motion, discussed later.
For a fixed point in the fluid, the hydrodynamic pressure can be expressed in terms of the
Bernoulli Equation for a coordinate system translating with the ship in speed U as
follows:
(3.21)
d(
where
pkX, t)= -p
"
_Bt
is the basis-flow potential and
(U #
VOO)- V]#1# 2
#g
is the total potential.
The total potential can be decomposed, hence the pressure can be obtained in terms of the
potential components. It is noted that the impulsive potential (or the basis-flow potential)
are functions of space and independent of time.
Calculating the pressure over the hull surface is more difficult, since the hull position
changes in time due to a time dependent sinkage and trim, and oscillations caused by
waves. 5(t) is denoted as the displacement vector of a fixed point on the body surface
where:
(3.22)
g(t)
'(t)= -(t)+
(t)xx
is the vessels linear translation motion,
G(t)= (gg2,,
Id(t)
is the vessels
rotational motions defined as Z(t)= (;4, g5 ,g) and i is the position of the point, or as
defined by SWAN-2 - the center of a particular panel.
For small values of 5(t) a Taylor series expansion is used to evaluate the time dependent
pressure of the center of each panel on the hulls surface as:
61
Chapter 3. SWAN-2 Implementation
(3.23)
p(, t()
= p(y,t- +3(t). VP|
+0(32)
The spatial gradient of the hydrodynamic pressure, defined in (3.21), is calculated by first
and second spatial gradients derivations of the velocity potential. The potential is evaluated
by the B-Spline approximation, equation (3.8).
Expression (3.21) is the fundamental form of Bernoulli's Equation used by SWAN-2 to
evaluate the steady state as t
-> co
and the time dependent forces acting on the vessel.
3.7 Hydrodynamic Forces
The hydrodynamic forces exerted by the surrounding fluid on the ship hull are
necessary for the derivation of the vessels equation of motion and for evaluation of the
structural loads exerted on the body.
The vector h represents a unit normal vector pointing into the hull. By virtue of Taylor
Series, the instantaneous position of the hull surface is given by:
(3.24)
nw(t) = - x ii+ 0(52)
The total hydrodynamic force acting on the hull is expressed by:
(3.25)
F(t)= fp(Y, t)(t)dS
SB)
where SB (t) is the instantaneous wetted surface of the hull, p(Y, t) is given by (3.23) and
i(t) by (3.24). The corresponding moment about the reference coordinate system is given
by:
62
Chapter 3. SWAN-2 Implementation
(t)=
(3.26)
p(, t)( +5)x h(t)}S
SB)
where xz= xi+yj+ zk and J is defined in (3.22).
Linearization of equations (3.25) and (3.26) about the mean position of the ship hull SB is
required for two reasons:
1. All quantities evaluated by SWAN-2 are known over panels located on SB.
2. Linearization will reveal the steady state, linear and quadratic components of the
force and moment as individual effects.
The time dependent wetted surface can be expressed similarly as the hydrodynamic
pressure about its mean position.
(3.27)
SB W)=B
+dS(t)+0(32)
dS(t) is the differential wetted surface arises near the water line and can be expressed by:
(3.28)
dS(t) = d1 r7(t) - Y -k leriine
where dl is a differential length of a segment along the vessels mean water-line. r7(t) is
the wave elevation along the water line, accounts for all wave elevation effects such as the
incident wave disturbance. 3 is defined by (3.22), and accounts for all the vessels
displacements as sinkage and trim, and the wave oscillations. k is the unit vector pointing
in the z-direction.
The resulting time dependent, linearized expressions for the induced force and moment on
the body are:
63
Chapter 3. SWAN-2 Implementation
Jp(Yt)*(t)[(t)5
-6 kil
F(t)= fp(Y,t)i(i)dS +
(3.29)
water-line
SB
(3.30)
M(t)=
Jp(Y,tX+ 5)x f(t)dS+ fp(Y,tX
+)x
h(t
r/(t)-,
i21g
water-line
SB
SWAN-2 Invoke Taylor's Series expansions for p(Y, t) and h(t) from equations (3.23)
and (3.24), in order to evaluate the force and moment as expressed above. It employs them
to evaluate the vessels small amplitude displacements, structural loads, wave resistance
and wave added resistance.
3.8 Equations of Motion
The vessel undergoes time-dependent motions, due to the hydrodynamic forces and
moments presented in the last section. Applying Newton's law results in the following set
of equations for dynamic equilibrium.
(3.31)
[Wygj(t) + C(t)] = F, (f g g t)
i=1...6, j=1...6
MY is the vessels inertia matrix and Ci are the vessels hydrostatic coefficients.
As time increases surface waves are generated. Theses waves propagate outward from the
body, but still continue to affect the fluid pressure around the hull and the body force for
all subsequent times. Memory Effects are introduced. Under the assumption of small vessel
motions and wave slopes, linearization of the equation of motion is justified. The forces
acts on the hull can be decomposed into a local component due to the instantaneous
motion, and to a time dependent component due to memory effects. For an arbitrary timedependent motion, g1 (t), the time-dependent component of the pressure force acts on the
hull can be expressed in terms of a convolution integral.
64
Chapter 3. SWAN-2 Implementation
6
(3.32)
F()
X, (t) -
t
K(t --r-#(t) -dr
6
j=1 ...
j=1
X, (t) is the hydrodynamic exciting force and the kernel K(t -'r)
can be interpreted as the
force, at time t, due to a delta-function body velocity at an earlier time r . In short it can
be referred to as the velocity impulse responsefunction.
The following equation of motion is accepted.
(3.33)
+a, ",(t)+ b, j(t) +(Cj + cij ;j(t)]= Fj,,, yt)
+M
i = I... 6, j =I... 6
ay , b, and c4 represent the local force coefficients. Solving this equation requires
determining the values of these coefficients and the time-dependent force. This is achieved
by solving the boundary value problem stated at section 3.1 for the instantaneous force.
The left hand side includes all inertia, 'hydrostatic' and "effective" impedance forces
induced by the impulsive potential (as defined in section 3.6). The right hand side includes
all wave effects, namely calm water radiation, ambient wave induced radiation and
diffraction effects, wave induced excitation and the wave induced lifting forces and
moments on motion control devices considered in sections 2.5 and 3.2.4.
SWAN-2 solves equation (3.33) for each time step, yielding the vessels kinematics and
forces. The solution of the boundary value problem and evaluation of the hydrodynamic
forces and responses happen simultaneously for each time step. The result is a time record
for the vessels response, g (t), and corresponding forces and structural loads.
65
Chapter 3. SWAN-2 Implementation
3.9 Applications
3.9.1 Wave making Resistance, Sinkage and Trim
SWAN-2 calculates the vessels ideal fluid resistance, sinkage and trim motions while
translating at calm water, by invoking the boundary conditions presented in section 3.1 and
solving the equations of motion introduced by equation (3.33). The solution of these
equations, provides the time record of the sinkage and trim time-dependent displacements
of the vessel.
The wave resistance is evaluated by direct pressure integration, as introduced in equation
(3.29), for the positive x-direction. As time grows, the force converges and gain a steady
value, which is extracted by SWAN-2 as the vessels wave resistance. Sinkage and trim
affects the vessels wave resistance appreciably. This is particularly true for high speed
semi-displacement vessels at Froude Numbers 0.5 and higher.
3.9.2 Linear Seakeeping in the time domain
The ambient wave potential, given by equation (2.17), is included in the seakeeping
boundary value problem described in section 3.1, and in the solution of the equations of
motions, equation (3.33). The resulting quantities of interest include the vessels response
and structural loads in the time domain, excited by a wave spectrum, defined by the
software user. The solution of the seakeeping problem can not be separated from the
sinkage and trim of the vessel due to its speed. However these quantities are small
compared to the response due to the incident waves, hence can be neglected.
3.9.3 Evaluation of Frequency Domain RAO's
SWAN-2, as a time domain seakeeping program, solves for the vessels response for all
degrees of freedom, excited by an incident polychromatic wave record, with velocity
potential defined as:
66
Chapter 3. SWAN-2 Implementation
(3.34)
1(xt)P t) =
Re
"ek'
e-k"XcoS/--kmY'rn/J'i'''''6m
All wave components are assumed to have the same heading, P, but their amplitudes, A,,,,
wave number, kin, wave frequency, w,, and random phase angle, E, may be different.
SWAN-2 simulations for the response provides a polychromatic signal for large time
interval. This signal is analyzed by a Fourier Transform to extract the amplitude and phase
of each response.
3.9.4 Wave Induced Loads
The local hydrostatic and hydrodynamic pressure distribution over the hull is given by
expression (3.23) evaluated over the mean ship position. This pressure distribution is
required as an input for structural analysis calculations, to evaluate structural loads. These
loads are often needed for preliminary design of a ship. These loads include mainly the
bending moment, shear force and torsion moment. The time-dependent load records are
obtained corresponding to a steady forward translation in calm water or in waves.
67
Chapter 4. Calm Water Resistance
Chapter 4: The Steady Problem
4.1 Introduction
The time domain, Rankine Panel Method code, SWAN-2 (Ship Wave Analysis)
was used to evaluate the vessels calm water resistance. This was done by specifying no
ambient waves, hence calm water sea - steady state solution. The extent of the interaction
between the two demi-hulls of the catamarans was examined for three separation ratios
between these demi-hulls, at various forward speeds. All the catamarans examined had
symmetric demi-hulls, and a deep and wide transom shape, where the draft of the transom
was equal to the draft of the keel, and its beam was equal to the maximum beam of the
demi-hull.
Transom stern vessels often experience hydrostatic induced resistance at high speeds, as a
result of flow separation at the transom. The pressure over the transom in these cases is
atmospheric, and since the rest of the vessel is half-submerged, there exists an un-favorable
net force against the direction of ship advance. If the transom stern is deep and wide, the
drag penalty has a significant magnitude. In cases of transom stern vessels, SWAN-2
always assumes dry transom. This assumption is correct for the high speed vessel
evaluated in this paper.
SWAN-2 calculates the total resistance of the vessel by direct integration of the pressure
along the hull, as stated in section 3.9.1. This value is thus includes the wave making
resistance and the induced resistance component over the dry transom. All resistance
results presented in this chapter are the summation of these two quantities and are referred
to as the ideal fluid resistance.
68
Chapter 4. Calm Water Resistance
SWAN-2, as a potential flow code, does not predict viscous resistance values. The ideal
fluid resistance calculated by SWAN-2, is thus only a part of the total resistance of the
vessel. Friction drag can be approximated by solving the boundary value problem
including viscous effects, by performing model tests, or as will be shown below, by
estimating the friction drag on the basis of the flat plate 1957 ITTC line and an estimated
form factor.
4.2 SWAN-2 Execution
The vessels ideal fluid resistance is sensitive to the hull geometry. It is important
therefore, to establish a spatial mesh, which is as dense as possible, to assure close
similarity to the real physical problem. SWAN-2 allows the user to establish a number of
panels to be used in the analysis, in a prescribed domain, hence to establish the size of the
domain. Several sensitivity checks were made over the catamaran hull. It was found that a
combination of 28 panels along the hulls length, and 7 panels along the hulls width, gave
the optimum solution. This combination gave good and reasonable results on one hand,
and saved computer calculation time (comparing to other more dense combinations) on the
other. This panel combination over the hull surface, and the free surface domain, resulted
in about 3700 elements. This combination was left constant for all ideal fluid resistance
simulations.
The size of the fluid domain is important. It is required to choose a domain that is large
enough, to avoid wave reflections from the artificial beach. A large domain ensure square
panels in the vicinity of the free surface, which is important to the correct representation of
the steady wave pattern, downstream the vessel.
The domain chosen for all catamaran
simulations included 0.5L,, upstream from the bow to the free surface boundary, 1.5L,
from center-line of the port demi-hull to the transverse free surface boundary, and 2LP
downstream from the vessels stern to the free surface boundary. L, represents the length
between perpendiculars of the vessel, or the design water line length.
69
Chapter 4. Calm Water Resistance
SWAN-2 simulations use only the port side of the symmetric hull, in order to save
calculation time. In the case of the catamaran, only the port demi-hull was modeled,
including the free surface between the hull and the center-line of the entire ship. This
parameter was changed for the three different separation ratios.
The selection of the time step is crucial for stability of the steady state simulation. For all
simulations the default value of time step, calculated by SWAN-2, was used. As stated in
section 3.3.4, stability of convergence depends on the size of the panel and the chosen time
step, where the critical value of 6 is given in equation (3.13). SWAN-2 chose as a default,
a time step which coincides with 2/8, such that stability is always ensured with a safety
margin.
4.3 Wave Patterns Predictions
The steady wave patterns for all separation ratios, at all speeds were predicted.
Some of these patterns are presented in the following figures. The flow around each demihull is asymmetric due to interference from the second demi-hull. The interference
increases as the separation ratio decreases due to increasing proximity between the hulls.
The wave patterns are all within the Kelvin Wave Pattern sector as described in section
2.3.2. From the figures it can be noticed that as the speed of the vessel increases, more
diverging waves are seen. For lower speeds, more transverse waves exist.
The following figures present several results of the simulations performed at the semidisplacement regimen. The figures describe Kelvin Wave Patterns for the catamarans
separation ratios of s/L=0.3, s/L=0.4 and s/L=0.5, at three different forward speeds, which
comply with FroudeNumbers 0.57, 0.74 and 0.90.
Green portions of the figure present free surface with zero elevation or a very low
elevation (positive or negative). Yellow to red spots describe positive elevation, where the
darker color stands for higher waves. Light blue and blue present negative elevation, i.e.
wave troughs. As the color becomes darker the trough is deeper.
70
Chapter 4. Calm Water Resistance
A deep trough exists at the wake of all vessels close to the transom, similar to what can be
observed at the wake of ships, traveling at high speeds. This trough is followed by a high
crest at the wake panel sheet and around it.
Figure 4.1: wave pattern for s/L=0.3, at Froude Numbers 0.57, 0.74 and 0.90
71
Chapter 4. Calm Water Resistance
Figure 4.2: wave pattern for s/L=0.4, at Froude Numbers 0.57, 0.74 and 0.90
72
Chapter 4. Calm Water Resistance
17
Bob.,
Figure 4.3: wave pattern for s/L=0.5, at Froude Numbers 0.57, 0.74 and 0.90
73
Chapter 4. Calm Water Resistance
4.4 Dynamic Sinkage and Trim
The dynamic sinkage
(g)
(g) of the vessel influence
and trim
the vessels calm
water ideal fluid resistance especially at high speeds. It is essential that the values of the
sinkage and trim would converge to steady values, when the ideal fluid resistance is
measured. Convergence is achieved by running calm water simulation several times, each
time with the sinkage and trim results of the previous run, until the difference between
input and output values is negligible. Usually 7-8 runs were required for convergence. The
sinkage and trim angles are calculated at the origin of the coordinate system, hence at
midship. Negative sinkage means that the ship draft increases, positive trim means a trim
angle by the stern. Another characteristic of the hull, calculated during calm water
simulation, is the ships wetted surface, which is important for calculating viscous effects.
The following graphs present the converged sinkage and trim for all separation ratio
catamarans, and for a similar monohull (with the same displacement / length ratio).
Dynamic Sinkage
1.OE-01
5.OE-02
O.OE-+OO
-5.OE-02
I
-1.OE-O1
-1.5E-01
-2.OE-01
-2.5E-01
-e
-3.OE-01
-3.5E-01
catamaran s/L=0.3
catamaran s/L=,0.4
catamaran s/L=0O.5
---
X:
monohull
I
-r
).55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Froude Number
Figure 4.4: Dynamic sinkage for all catamarans and a monohull as a function of Fn
74
..
........
. .....
..
........
Chapter 4. Calm Water Resistance
2.5E+00
-
Dynamic Trim
catamaran s/L=0.3
catamaran s/L=0.4
catamaran s/L=0.5
2.OE+00 _
-n-
1.5E+00 -
x
monohull
5.OE-01
-
O.OE+0O
-
-
1.OE+00
-1.OE+00
-
-5.OE-01
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Froude Number
Figure 4.5: Dynamic trim for all catamarans and a monohull as a function of Fn
The monohull has lower dynamic sinkage and trim values then all catamarans. This is
mainly due to a lower wetted surface (the monohull wetted surface is smaller in about 25%
the average wetted surface of the catamarans), hence lower force exists due to pressure
distribution and by that, a lower motion displacement appears.
The dynamic trim for all catamaran types is increasing for higher speeds, i.e. the steady
pitch angle increases for higher Froude Numbers. A negative trim (bow up, stem down)
locally increases the draft of the vessel at the stern, hence increase the induced resistance
(due to dry transom). For large transom stems as exist for the catamarans investigated, an
increase of the draft has a significant impact on the overall ideal fluid resistance of the
vessel. At higher Froude Numbers, the trim angle gets relatively high values, which are not
practical for design purposes. This pattern is caused due to the catamarans round bilge
shape. The catamaran has a displacement hull shape, and it translates at the semidisplacement regime. For this type of hull shapes, at high speeds, the vessels bow is
'sucked' into the water. A very large pressure differential exists in the bow area and causes
this phenomenon.
75
Chapter 4. Calm Water Resistance
The dynamic sinkage of the catamarans does not have a steady pattern at the semidisplacement region. The mean of all sinkage values, is close to zero sinkage. The
catamarans have higher sinkage value then the monohull, meaning that their drafts do not
increase as much as the monohull drafts, i.e. at Froude Numbers of 0.5 to 1.0, the
monohull gain more wetted surface then the catamarans. As will be shown below, the
wetted surface of the monohull, regardless of the large negative sinkage, is still smaller
then these of the catamarans.
4.5 Calm Water Resistance
4.5.1 Preface
The 'numerical' catamaran evaluated in this paper, was examined at speed range of 35-60
knots with a 5 knots step. All results at calm water and in waves are presented as a function
of Froude Number, a non-dimensional quantity which is defined by the vessels speed and
length.
(4.1)
Fn
U
g9L
where g is gravity acceleration. All Froude Numbers evaluated in this research were in the
semi-displacement regimen, Fn = 0.5 - 1.0.
Couser [20] breaks down the resistance of a vessel into the following components: (1) a
tangential stress, due to viscousity and form effects and (2) a normal stress, due to pressure
distribution (wave making resistance, induced drag over the transom and around the hull
and pressure due to viscousity). The induced drag of catamaran ships includes also the
affect of the interface between the demi-hulls. This effect is not treated numerically yet,
but was measured by model tests. A particular problem for catamarans at high FroudeNumbers is that, due to their slenderness, the total resistance is dominated by the viscous
76
Chapter 4. Cahn Water Resistance
resistance component. This fact is highlighted in Couser [20]. For Froude-Numbers at the
range of 0.5-1.0, the ratio between the wave making resistance component and the viscous
resistance component (containing a normal component due to pressure) is about 1:4. This
is unfortunate since the friction is calculated empirically based on the friction of a flat
plate, and is modified to obtain the viscous resistance component.
SWAN-2 measures the normal resistance, obtained by integrating the pressure distribution
along the hull. Since a bare hull was evaluated (with no other appendages), no other
normal components exists. The wave making resistance and the induced drag due to dry
transom are included in this magnitude.
4.5.2 Ideal Fluid Resistance
The component of resistance, calculated by SWAN-2, was referred here (section 4.1) as the
ideal fluid resistance.
Predictions of the ideal fluid resistance for the three catamarans and the monohull, as a
function of their FroudeNumber, are detailed in figure 4.6.
The ideal fluid resistance is calculated by integrating the pressure distribution along the
hull, thus including an induced resistance component at the dry transom. The induced
resistance can be calculated by integrating the hydrostatic pressure over the transverse
projected area of the wetted transom. The draft of the catamarans and the area of the
'wetted' transom changes as a function of the catamarans speed, thus related to alterations
of sinkage and trim. At zero speed, the draft of the transom at calm water is 5.75 m, the
'wetted' area at zero speed is about 31 m2, and so the induced drag is estimated by 1800
KN. This component can be deducted from the ideal fluid resistance in order to get the
wave making resistance results.
Figure 4.6 also include an ideal fluid resistance prediction for 'twice demi-hull'. This
prediction was conducted by measuring the drag over one demi-hull (modeled in SWAN-2
77
Chapter 4. Calm Water Resistance
as a monohull), and doubling the results. This measurement is very much like measuring
.
the resistance of a catamaran with a separation ratio of: s / L -> o
Ideal fluid resistance coefficient
-
2.OE-02
catamaran s/L=0.3
---
C
1.6E-02 -
---
catamaran s/L=0.4
catamaran s/L=0.5
--x--.mono hull
8.OE-03
-
4.OE-03
-
1.2E-02
-
-- E-- dmi-hull X 2
0.OE+00
I
0.55
0.60
I
0.65
I
0.75
0.70
I
I
I
I
0.80
0.90
0.95
1.00
0.85
Froude Number
Figure 4.6: Ideal fluid resistance coefficient results
The separation between the demi-hulls does not affect appreciably on the catamarans'
ideal fluid resistance. For the most parts of the semi-displacement range, the larger
separation ratio has higher resistance values then the smaller separation ratios, but the
magnitude of this difference is small comparing to the absolute value of the resistance. The
drag predicted for 'twice demi-hull' was measured in four speeds, and its values were
larger then the resistance for the widest catamaran. The phenomena of increase in
resistance as the distance between the demi-hulls grows, is caused due to a favorable
interaction between the demi-hulls. The Kelvin wave pattern, exerted from one demi-hull,
'wets' the transom stern of the adjacent demi-hull and vice versa. The induced, adverse
drag force is decreasing, since less area of the stern is dry, hence the overall ideal fluid
resistance is smaller.
78
Chapter 4. Calm Water Resistance
The monohull examined in this paper has the same length and displacement values as the
catamarans. It had larger beam, in order to keep the same displacement. The length and the
displacement of a vessel have a large effect on its measure of effectiveness (for instance,
the amount of payload the vessel is capable to transform, etc.) and cost. It was interesting
to compare different hull forms with similar requirements. The monohull had higher values
of resistance then the catamarans. The main reason is that the catamarans demi-hulls are
much slender then the monohull. Their hulls are more streamlined and surface piercing, the
monohull has larger transom area, higher B/T ratio and higher B/L ratio. It can be
concluded that all the catamarans, evaluated in this paper, are more effective then the
monohull, in a sense that they require less power to gain the same speeds (or less fuel to
gain the same range). It should be noted that although the ideal fluid resistance
component is lower for the catamarans, the friction component of resistance is probably
higher, since the overall wetted surface of the catamarans is larger in 25% from the overall
average wetted surface of the monohull.
The catamaran hull forms developed in this paper, has a superior ideal fluid resistance
characteristic over a monohull with a similar mission. It is also cost-effective relative to its
demi-hulls components due to favorable interaction between the hulls.
4.5.3 Wave Making Resistance
For each catamaran, at each speed, the results of the wave making resistance were isolated
by subtracting the induced drag at the transom.
Calculating the induced drag, due to dry transom, shows that the induced drag component
is significantly large relative to the wave making resistance values, mainly since the
vessels has a 'deep transom'. The values of the induced drag were calculated separately for
each speed, and their magnitudes varied due to the trim and sinkage effects over the
transom draft and 'wetted' area. This component of the induced resistance was easy to
approximate. Other components of the induced resistance (or normal drag as referred
above) due to the shape of the hull, are harder to appreciate, but it is assumed that their
influence is negligible, due to the hull smooth lines. The area of the 'dry' transom was
79
Chapter 4. Calm Water Resistance
calculated as if there were no interactions between the demi-hulls, and the wake does not
'wet' the adjacent demi-hull transom. Calculating this variable is somewhat difficult, and
since the ideal fluid resistance magnitude of all catamarans was similar it was decided to
neglect this favorable force.
Figure 4.7 emphasizes the catamarans wave making resistance coefficient after deducting
the influence of the induced drag at the transom.
1.6E-03
-
--
catamaran s/L=0.3
catamaran s/L=0.4
catamaran s/L=0.5
1.2E-03
1.OE-03
-
I
I
1.4E-03
-
-
1.8E-03
-
2.OE-03
-
Wave making resistance coefficient
4.OE-04
-
6.OE-04
-
-
8.OE-04
-
2.OE-04
0.OE+0
I
I
--
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Froude Number
Figure 4.7: Wave making resistance coefficient
In figure 4.8, the resistance coefficients were analyzed as a function of the vessels dynamic
trim. As the dynamic trim increases, the vessels resistance coefficient decreases, hence as
the bow is submerged deeper in the water, the values of these coefficients are decreasing.
80
Chapter 4. Calm Water Resistance
Wave making resistance coefficient
2.0E,03 1 .8E-03 -
catamaran s/L=0.3
1.6E-03 ---
catamaran s/L=0.4
catamaran s/L=0.5
I-
-
8.0E-04
4.OE-04
-
2.OE,04
-
6.O-04
-
1.OE-03
-
-
I1.4E-03
O.OE+00
-0.4
I
I
I
-0.8
0
0.8
0.4
Dynamic Trim
1.2
1.6
2
Figure 4.8: Wave making resistance as a function of the dynamic trim
The trend of the wave making resistance curves is similar to the trend of the ideal fluid
resistance curves. Generally when the speed increases, the resistance decreases. Although
the vessel has a displacement hull, it behaves at high speeds as a planning hull. Its draft
values decreases, (an indication for planning) so its induced drag at the transom is lower
and the vessel 'invests' less energy in producing waves. As noticed above, the separation
between the demi-hulls has a minimal effect on the wave making resistance coefficient
values.
The wave making resistance results were compared to several model tests conducted for
catamarans with similar separation ratios. Figure 4.9 presents the results of the comparison.
Model test results were obtained from Steen [21] and from Bruzzone [22]. Steen [21]
presented resistance results for several catamaran models, tested by MARiNTEK. The
curve shown in the graph above is a mean value of the wave resistance coefficients,
measured by MAR7NTEK.
81
...............
..
Chapter 4. Calm Water Resistance
Wave making resistance coefficient - conparison chart
-
Cw - swan results for s/L=0.3
-
-
1.4E-03 -
-
- [22] Exqperiment for s/L=0.3
A
1.2E-03
-
8.OE-04
6.OE-04
4.OE-04
2.OE-04
-
-
I
1.6E-03
-- Cw [21] Model tests
-
I
I
1.8E-03
-
2.OE-03 -
O.OE+00
I
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Froude Number
Figure 4.9: Wave making resistance coefficient - comparison chart
Bruzzone [22] presented several separation ratio results of a 40m catamaran. The curve
above presents a separation ratio of s/L=0.3. The resistance coefficients predicted by
-
SWAN-2 are slight higher then these obtained from the tests at the range of Froude
Numbers 0.55 to 0.85. At high speeds, Fn = 0.85-1.0, SWAN-2 predicts lower values of
wave making resistance. The differences from model tests are due to differences in hull
forms and demi-hull dimensions. Despite these differences, the trend of the lines is similar
and their values are in the same order of magnitude.
4.5.4 Viscous Drag
The ITTC 1957 line presented in equation (2.22), is valid for viscous drag over a flat plate.
A form factor was needed to be established in order to estimate the ratio between the flat
plate viscous drag and the viscous component of the resistance of the vessel in hand.
(4.2)
C
+k =
""al
it C lat-
82
Chapter 4. Calm Water Resistance
k is the form factor and the Cf 's are the viscous drag coefficients. The difference
between the real frictional resistance and the flat plate resistance is mainly due to the
curvature of the hull.
Steen [21], suggested an empirical method to find the form factor, k, based on several
model tests of high speed, displacement, catamaran hulls. The form factor was found to be
a function of the length-displacement ratio of the ship between values of 6 to 12. The
length-displacement ratio of the catamaran examined here is less then 6, but with no other
better way to estimate friction drag, this was the method used.
According to Steen [21], the form factor can be approximated as follows:
(4.3)
1+ k = 3.4275 -V
L
-0.443
Y3
Length displacement value for the catamaran is:
L
V"3
100
5.108, the form factor is
7500Y
then: 1 +k =3.4275-5.108-0443 = 1.6641. This result was verified by the suggested form
factors table for a high-speed, round-bilge catamaran forms, presented in Couser [20]. The
form factors in that table were defined as functions of the vessels length over cubic root of
the underwater volume coefficient. The results of this representation were extrapolated to
find the form factor corresponds to the catamarans
L
L
V3
coefficient.
Figure 4.10 presents the correlation between the vessels
L
V3
coefficient and its form
factor, and the extrapolation using a second order polynomial trend line. This form factor
was used to 'normalize' the flat plate viscous drag approximation and to calculate the
viscous drag coefficient as presented in figure 4.11.
83
NNW=
Chapter 4. Calm Water Resistance
Form factor prediction
2
1.8
p 1.66-
Correlation line
1 .6
-
1.4
1.2
-
.
Polinomial trend line
0.6
-
0.8
-
0.4
-
0.2
0
5
S.1
4
3
10
9
8
7
6
LIV 3
I [-]
Figure 4.10: Form factor correlation line
-
2.36E-03
-I
2.32E03
-
2.34E-03
-
2.38E-03
-
Friction Resistance Coefficient Prediction
2.30E-03
2.24E-03
-
2.22E-03
-
2.26E-03
-
2.28E-03-
2.20E-03
i
0.55
0.6
0.65
0.75
0.7
0.8
0.85
froude number
Figure 4.11: Friction resistance coefficient
84
I
I
I
0.9
0.95
1
.
........
. ............
.....
Chapter 4. Calm Water Resistance
The friction drag coefficient, in figure 4.11, was estimated as constant for all separation
ratios. Its values were determined as functions of the ships Reynolds Number and an
estimated form factor, without any influence of the separation between the demi-hulls.
4.5.5 Total Rcsistance and Verification
A summation of the resistance components evaluated so far, leads to a total drag
coefficient of the ship. It should be noted that the total drag coefficient evaluated here does
not take into account the wind or air components of the drag. Figure 4.12 presents the total
drag coefficient.
Total resistance coefficient
-
4.50E-03
-
--- *-Ct
4.25E-03-
-- E--Ct - catamaran s/L=0.4
4.OOE-03 -
Ct - catamaran s/L=0.5
-
3.75E-03
catamaran s/L=0.3
3.25F-03
-
-
3.50E-03
3.00E-03
2.50E-03
-
2.25E-03
-
2.75E-03
2.OOE-03
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Froude number
Figure 4.12: Total resistance coefficient
As stated above, there is no significant difference in the drag results, between the three
separation ratios vessels evaluated here, although the smaller separation ratio shows
slightly better performances.
85
....
....
......
Chapter 4. Calm Water Resistance
Very few model tests and experiments regarding the behavior and resistance of catamarans
at sea were available. The numerical results obtained here were verified by comparing
them to the few results found, most of them in the 'FastShip Proceedingspapers' from
recent years.
0.3 catamaran and a
Figure 4.13 present the total drag coefficient results for the s/L
comparison with a routine model test conducted for a similar s/L
=
0.3 catamaran for the
semi-displacement range of speeds. The results were obtained from Bruzzone [22]. There is
a good agreement between the results obtained by SWAN-2 and by the model test.
4.72E,03
0.3 total raseistance coefficient
-
Comparison of s/L
4.5 1F,03 -
---
Ct - catamaran s/L=0.3
-
4.3 1E3
--
4. 1 OE03
routine model test-ITTC78,
s/L=0.3
3.89E-03
3.68E-03
I
3.47E-03
3.26E,03
3.05E-03
2.84E-03
2.63E-03
2.42F,03
2.21E-03
2.00E,03
0.55
I
I
I
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
Froude number
Figure 4.13: Comparison chart of s/L = 0.3 catamaran total resistance results
86
Chapter 5. Seakeeping Evaluation.
Chapter 5: Seakeeping Evaluation
5.1 Introduction
In recent years, greater attention has been given to vessels'
seakeeping
performances due to several factors: (1) the increase in use of high-speed semidisplacement vessels; (2) increasing demand for comfort (especially for passenger ferries);
(3) the development and use of a delicate, sophisticated systems onboard (especially on
naval ships); (4) growing pressure from the public and regulatory bodies for safer vessels;
(5) fast advancement in computational, predictive, simulation and analysis tools for
seakeeping.
As the speed of vessels increases, their seakeeping characteristics become more important.
This is especially true for the growing passenger ferries market, and also for several naval
applications. When the speed grows, the momentum of the advancing vessel grows, hence
every small diversion of the vessels course has the potential of structural damage to the
ship and a risk to the operating personal and to the passengers. Predicting the vessels
motions and particularly suppressing them has therefore a great importance during the
vessels design process.
The seakeeping problem can be separated into three parts: (1) estimation of the
environmental conditions encountered by the vessel (i.e. sea state); (2) prediction of the
vessels response characteristics; (3) comparison of the vessels response with specified
motion criteria. For passenger vessels this criteria would consider mostly the comfort of
the passengers, while for naval vessels this criteria would probably consider equally the
87
Chapter 5. Seakeeping Evaluation.
wellbeing of the instruments and weapon systems onboard. This kind of comparison was
not in the scope of this paper.
The seakeeping heave and pitch motion responses of the generic, 'numerical' catamaran,
introduced in Annex A, are presented in the following chapter. These characteristics were
obtained by the time domain, computational fluid dynamic code, SWAN-2. Seakeeping
computations were carried out in order to evaluate the effect of different separation ratios
between the demi-hulls of the catamarans heave and pitch motions, in a unidirectional,
irregular sea spectrum. The frequency dependent added mass and damping coefficients of
the vessel were derived from the forced heave and pitch oscillations at some prescribed
frequencies, starting from rest at t =0. The resulting forced records converged to a
harmonic signal, which upon Fourier Transform, lead to heave and pitch frequency
dependent, hydrodynamic coefficients. The ship motion response standard deviation was
measured invoking equation (2.43) and the ISSC spectral formulation, introduced in
section 2.4.1. Lifting appendages in various sizes were attached to a constant location of
the ship, and their influence on the ships' motions was examined as well.
5.2 Heave and Pitch Responses at unidirectional, irregular sea
The seakeeping characteristics examined in this section were the heave and pitch
motions of a variable separation ratio catamaran. Three different separation ratios were
tested: s/L=0.3, s/L=0.4 and s/L=0.5. A similar monohull with the same displacement and
length characteristics was investigated as well. The monohull is wider then the demi-hulls,
but is narrow compared to the overall breadth of the catamarans. The characteristics of the
monohull and the catamarans are elaborated in Annex A. The vessels were evaluated at six
different speeds, at four different wave headings: bow waves (waves that advance towards
the bow of the ship with an angle of 1800, as defined by SWAN-2, with respect to the
negative x-axis), oblique waves (waves that approach the ship from the first quarter,
towards the port or starboard sides of the ship, with an angle of 120' and 1500 with respect
to the negative x-axis) and beam waves (waves that approach from the side with an angle
of 900 with respect to the direction of the ship advance). The ships speed were at the semi88
Chapter 5. Seakeeping Evaluation.
displacement range (Froude Numbers = 0.5-1.0). The following figure is a remainder of
the heave, pitch and roll motions of a vessel. It also defines the ships coordinate system
and the wave headings angle as defined by SWAN-2.
z
I
3 (HEAVE)
U(t)
Y
4",(PITCH)
X
S(ROLL)
4
,8= 180* -+ Bow waves
,8= 150
-*
Oblique waves
,8= 120
->
Oblique waves
6= 90
-+
Aw
Beam waves
Figure 5.1: Coordinate system, heave and pitch motion and wave headings
Most of the figures in this chapter present the motions RAO. As discussed in chapter 2, the
RAO (Response Amplitude Operator) is the ratio between the response modulus and the
incident wave amplitude. For the vessels examined in this paper, thirty different waves
(different in their frequencies) were generated and invoked thirty different ship responses.
The response amplitudes were isolated and the RAO of the ship heave and pitch motions
was derived. SWAN-2 measures ship response amplitude at the encounter frequency.
89
Chapter 5. Seakeeping Evaluation.
Response motions are measured at the origin of the coordinate systems, i.e. at midship, at
the centerline, on the design water line of the ship.
The incident wave amplitude was chosen to be 2 meters. The incident wave frequencies
were chosen at the range of 5 to 25 [rad / sec]. The frequency step was usually a constant
at the size of 0.04 [radlsec]. At the peak response of each motion, more measurements
were taken to verify results and to achieve 'fine tuning' of the response. At this range,
usually smaller frequency step was used. The abscissa of the graphs presents the nondimensional magnitude of A IL, wavelength over ships length. The y-axis presents the
heave or pitch RAO in units of [m/m], non-dimensional units in case of heave and
[deg/m] in case of pitch.
The following figure present a seakeeping simulation sample of bow waves, 1800, and an
incident wave with time period of 7 [sec] and amplitude of 2 [m] applied upon a catamaran
advancing forward with a speed of 45 knots.
Figure 5.2: Seakeeping simulation of a vessel advancing in 45 knots (Fn=0.74)
90
-71-
Chapter 5. Seakeeping Evaluation.
Figure 5.3 present the heave motion RAO of a catamaran with separation ratio of 0.3, i.e.
the ratio of the distance between the centerlines of the demi-hulls to the length of the vessel
is 0.3. The catamaran response is to bow waves. Six different speeds were examined. The
peak response decays (for the lowest speeds), as the forward translating speed decrease. At
the high end of the ship speed range (Fn = 0.82-0.99), the peak response increases as the
speed increases. The natural frequency of the response for s/L=0.3 increases as the ships
Froude Number increases. The responses at Fn = 0.90-0.99 have similar peak values and
peak frequencies.
Heave RAO for heading waves, catamaran s/LO.3
2.2
2.0 ------------ ------------
-----------
---
------
-----------
-------------
-a-Fu=0.90
------ - --------------------------- - - ----------
1.8-
FB4.99
Fn'0.82
1.6- ----------- ------------ --------- -- - ----1.4 ------------ ------------ -------- --
----
----- - --
------
---
---------- ~
-
------ -
-x--Fn=0.74
--
------------ ----------- ----- -- - -------- --- -
~
-m -tFn=0.66
-n46
Fn0.57
----
0.
-
1.0
------------ ------------ -------
---------- ------------------------ ------------ ------------
-
0.6
0.40.2 ------------ ----------0.0
0. 00
0.50
1.00
-----I------------ ----------- ------------ T----------- -----------
1.50
2.00
2.50
3.00
3.50
4.00
/L
Figure 5.3: Heave RAO for bow waves for s/L=0.3
Figures 5.4 present the heave response for bow waves for a catamaran with separation ratio
of 0.4. For this catamaran the peak heave response decays smoothly as the speed of the
vessel increases.
/L values of the peak response (at each speed) grow as the Froude
Number of the vessel grows. For this catamaran, as for the s/L=0.3, Fn = 0.90-0.99 have
similar peak response, this peaks have similar A / L values.
91
...............
Cbapter 5. Seakeeping Evaluatiom
Heave RAO for heading waves, catamaran s/L=0.4
2.2
Fn=0.99
-------- ------------
2.0
----------- ------------ ------------- -- --- --
1.8
------------ ------------ I------------ - ---- -- --
1.6
------------ ------------- ------
1.4
------------------------ I---------- ----
-4&- Fn=0.90
-------------
Fn=0.82
--- --- --- - ---------
----
- - ----- --- ----
Fn=0.74
Fn=0.66
-----Fn=0.57
-
a
------------------------ -------- - - ---- -- --- --- -----
-E 1.2
E
0 1.0
-------------------------------- - - ---- - ------ -------------------------
0.8
------------ ------------------------ ------------------- - - ---- - --------------------- ------------
0.6
----------- ------------
0.4
----------- ------------ -- ----
0.2
------------ ------------
----------- ----------- i------------ ------------ -----------t -------------I ----------- 1 ------------ ------------ ---------------------- -------------------------- ------------ ------------
0.0
0.0
0
0.50
1.00
2.00
1.50
2.50
3.00
3.50
4.00
X/ L
Figure 5.4: Heave RAO for bow waves for sAL=0.4
Heave RAO for heading waves, catamaran s/L=0.5
2.2
Fn=0.99
------------ - --- -- -- ---- --------I ----------- .
2.0
Fn=0.90
- - -- --- -----------1.8 ------------ ------------------------ ---- -- --1.6
Fn=0.82
Fn=0.74
----------- ------------ T---------- -
Fn=0.66
1.4
Fn=0.57
1.2
----------- ------------ r------
0 1.0
----------- ------------ -----
0.8
----------- ------------ -----
- --- - ------- ------------ ------------------------- ------------
0.6
-------------r---------------!
--- - ---------- ------------ ------------
0.4
----------- --------------
0.2 ------------ ------------
----------
-
------------------------
-----------
------------ ------------ ------------ -------------I---------------- ------------ i ---------------------------------------------------
0.0
0. 00
0.50
1.00
2.00
1.50
2.50
3.00
3.50
X/ L
Figure 5.5: Heave RAO for bow waves for sAL=0.5
92
4.00
Chapter 5. Seakeeping Evaluation.
Figures 5.5 present the heave responses for bow waves, at the same six ship speeds, for the
same hulls, with a separation ratio of 0.5. The peak heave response of this vessel increases
for Fn = 0.57-0.74 and then decreases for higher value of Froude Number. Heave
responses of Fn = 0.82-0.90 has a similar peak values and similar natural frequencies.
Several conclusions for the bow wave heave motions can be made after a brief analysis of
the graphs.
1. Slow catamaran response in bow waves is higher then a fast catamaran responses
for the same wave heading. This rule is exact for the s/L=0.4 catamaran.
Exceptions are seen and elaborated for the s/L=0.3 and s/L=0.5 catamarans.
2. Peak responses for the semi-displacement catamaran are at A IL range of 1.5 to
2.5.
3. Higher catamaran speeds produce a peak response at higher frequencies.
4. At low A /L (up to 1.5) slower catamarans gain higher heave response. At high
A/ L (above 2.5) faster catamarans gain higher heave response.
5. Peak response occurs usually at the same A IL value for all separation ratios.
6. As expected, for large wavelength the ship response amplitude equals the incident
wave amplitude, i.e. the RAO curve goes to 1. The ship gains the exact wave
motion. For infinitely large frequency (low values of A /L), the excitation is so
quick so that is no time for respond, hence the ratio between the response amplitude
and the wave amplitude (the exciting amplitude) is vanishes.
7. At small wavelength numbers the response vanishes.
Figure 5.6 present the pitch motion response of an s/L=0.3 catamaran at the semidisplacement speed range for bow waves. The heave and pitch responses for all separation
ratios were measured at the same frequencies.
For the s/L=0.3 catamaran, slower catamaran provokes higher pitch responses. An
exception is at Fn = 0.99. Peak response is decaying for the range of Fn = 0.57-0.82 where
the highest response is for the slowest speed. At Fn = 0.90-0.99 the peak grows as the
speed increases.
93
Chapter 5. Seakeeping Evaluation.
Pitch RAO for heading waves, catam aran s/L=0.3
3.5
-- Fn=0.99
-
Fn=0.90
--
------ ----------- ------------
------------------------ -------- ----
-
3.0
Fn=0.82
-x-Fn=0.74
-
2.5
------------ -----------
------ -- -
-- - -
----
- -
----
F = .6
- -- ~ -
-Fn=0.57
-
2.0
--- --- ---- ----------
- - ----
-------------------- --1.5-
------------ ----------- ------- - ------- - --------- -------
------------
1.0
-----------
- --------------
-""--"4 ------
0.5
0.0
0.00
0.50
2.50
2.00
1.50
1.00
4.00
3.50
3.00
X/L
Figure 5.6: Pitch RAO for bow waves for s/L=0.3
Pitch RAO for heading waves, catamaran s/L=0.4
3.5
-oFn=0.99
-eFn=0.90
--------------- -- --------
3.0
Fn=0.82
2.5 1 ----------- i ---------
-------
-------------- -
-
Fn=0.66
o-
-
--- ---- - ---------- -S1.5
Fn=0. 57
-------- - - ------ --------------------
2.0
0.
-
-N-Fn=0.74
---
---
4
---- - - -- - -- ---- - --- - ------------- -- --------- ---
10
-- - -- ------ -- ----- - ------ ----
----------- ----------
0.5
1
------- ----
----------------L------------------
0.0
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
Figure 5.7: Pitch RAO for bow waves for s/L=0.4
Figure 5.7 presents the pitch response of an s/L=0.4 catamaran for bow waves.
94
=402
Chapter 5. Seakeeping Evaluation.
For the s/L=0.4 catamaran, as before, slow speed invokes high response. For a response of
Fn = 0.74 and higher, two peak responses are clearly spotted. Peak response is decaying
for Fn = 0.57-0.66. At Fn = 0.74-0.82 peak response is similar (although in different
frequencies). For Fn = 0.90-0.99 the peak response increases as the speed grows.
For this
Figure 5.8 presents the pitch response of an s/L=0.5 catamaran for bow waves.
separation ratio, the pitch peak response decays for Fn = 0.57-0.82 as the speed grows and
increases for Fn = 0.90-0.99. For the former Froude Numbers, the peak response is similar.
Pitch RAO for heading waves, catamaran s/L=0.5
3.5
+-Fn=0.99
-----------
2.5
-----------
--------
--- --------
--
----
--
-------- --------- --
--
---------
Fn=0.90
F
Fn='0.82
-x-
Fn=0.74
-
FnO.66
4
--
-
3.01 ----------- ----------- ----------
---
-a -Fn=0.57
2.0
------- ------- ------------
----
--- - ---
-
---- ------ ----- -
1.0-
0.5-
nn i
0.00
0.50
1.00
2.00
1.50
2.50
3.00
3.50
4.00
/L
Figure 5.8: Pitch RAO for bow waves for s/L=0.5
Several conclusions can be made from the pitch RAO curves.
1. The pitch response is not a smooth curve as the heave response curve. Two peak
responses are often seen. The second natural frequency is due to the hydrodynamic
influence of one hull over the other.
2. Pitch peak responses of a catamaran at the semi-displacement speed range for bow
waves is at the A/ L range of 1.5 to 2.5.
3. Peak responses of higher speeds are usually generated at higher frequencies of
encounter.
95
Chapter 5. Seakeeping Evaluation.
4. Peak pitch response at Fn=0.57-0.66 is generated at higher AIL values for higher
separation ratios.
5. At lower A/L ratios (up to 1.5), low speed generate higher response then high
speed catamarans. For high A /L ratios (above 2.5) high speed catamarans generate
the higher response.
6. The pitch response vanishes for short and long waves. For the long wave extremes,
the slope of the incident wave tends to zero hence no pitch RAO.
Figure 5.9 present seakeeping heave and pitch motions of the three catamarans (s/L=0.3,
s/L=0.4, s/L=0.5) and a monohull at four wave headings at Froude Number of 0.99. All
other monohull / catamaran comparison charts are attached in Annex B. The monohull
dimensions are elaborated in Annex A.
Several conclusions can be made from figure 5.9:
1. Heave motion:
a. Monohull heave response has approximately the same natural frequency as
the catamarans for wave headings of 1200, 1500 and 180'.
b. At wave heading of 900 the shapes of the response are different. The
catamaran are oscillating with peaks at several natural frequencies were the
s/L=0.5 has the highest peak response as expected. The monohull has a
smooth, non-oscillating response shape which peaks at higher frequency
and has larger response amplitude value.
c. Catamaran natural frequencies are getting smaller as the wave heading
advance towards beam waves.
d. All responses start from zero at very short waves and go to one at long
waves.
e. The monohull amplitude of response is lower then this of the catamarans for
wave headings of 1500 and 1800.
96
Chapter 5. Seakeeping Evaluatiom
P itch - 180 deg. wave heading
Heave - 180 deg.wave heading
2.8
2.0
1.8
-------------- -------------- -
1.6
-------------- -------------
1.4
-------------
------------
-----
I
---- ----- --------------
------------- --------- ----------------
<
OC 0.8
0.6
------------- -------- --
--------------
-------------
'& 1.6 --------------1.2 --------------- ------
--------------
------------- ------------- --------------
cc
---------
----------
- - - - - - -wave spectra/2
monohuU
0.8 --------------- ---- --------
catarn arans/L=0.3
0.4 ---------------
- -----------------------2.0
1.0
X/ L
-------------
------------ --------------
0.0
0. 0
--------------------------
----------------------------------- --------------
----------
0.2
E
- -------------
--------------------- ----
------------- ------
--- --- ------------- --------------------------
------------- ----------
2.0
----------------------------
61.0
0.4
2.4 --------------- ------------
------
------ -I ---
-----------
-------------- ----------
E 1.2
--------------
-----------------
-----
0.0
I
4.0
3.0
5.0
c a t a rn a ra n, s / L= 0 .4
catarn aran. s/L=0.5
-------WL 2.0
1.0
0.0
4.0 _
3.0
0]
_75
P itch - 150 deg. wave heading
Heave - 150 deg. wave heading
1.8
------------
1.2 --------------- --------
T
E
E Lu - ------------- -------
--
t------------- --------------L -------------
--------------- ------
0.8 - ------------- -- --------0.4
1.0
----------------------- --------------
----
1.6
-----------------
1.2
- -------------
E
-ILI
0.4 - ------------
------------------
-------------------------------------
---------
0
QC 0.8 - ------------
- ------- -- ---- - -----------------------------------
------------- --------------
-------------
------------
- ----------- r------------- r---
0.4 ------------------------------------- ------------- -------------I
0.2 - ----------
0.0
X/L
M
1.0
0.0
0.0
5.0
4.0
3.0
-- -------
0.6
------- I- - ----
0.4
----
0.2
--- - -- ---- -- -- ----
--- -- -
4.0
3.0
5.0
2.0 ---------------- ---- -------------------------------- ------------------------------
__, ------
0.8 - -----------
7 /L
-
2.4
1.0 ---------------------------------------------------
2.0
1.0
Pitch -90 deg.wave heading
Heave - 90 deg. wave heading
L2
E
E
-----------------
----
--- ------ - -------------------------------------------
0.6 - ------------- - - ------------------------ ------------- --------------
-
5.0
4.0
3.0
-------------------
1.0 --------------- - --- -- -- ---- -------------------------
7 /L
-
2.0
1.2 --------------- -- --
2.0
P itch - 120 deg. wave heading
Heave - 120 deg. wave heading
0.0
-----------
-------- ------------- ------------- -------------
-------------
0.0
5.0
4.0
3.0
1.4
0.8
---- ------
- -----
0.0
2,0 X / L
1.0
0.0
E
E
0
--------------------------------
---- ------------ --- ---------- --------
1.2 - ------------0
<
cc
-
0.2
1.6
- ----
-------------
Op
--------------- --- -------- T__ ----------- ------------- ------------: ----------------------- ----------------------------- -----------
0.4
0.0
--------------------
--------------
-------------------- ------------- -------------
- --------------------- ------------- ----
2.0 -------------------------
------------- -------------
---------
----------
-
1.4 ---------------
00.8
-cc
10C
0.6
2.4 --------------- ---------- --
----------- ---------
----------
1.6 --------------- ---------- -1
---------
E
- ---------- i-----------
-- ----------- ------------------------- ----------- ---------- -----------
0.0
M
0
1.6 --------------- -- ----
----------- ----------------- ----------------
1.2 - ---------- - - - ----- - ------ I------------------ ----------------
ir 0.8 - --------
-- ----
----------------- ----------
0.4
0.0
0.0
0.5
1-0
L
1.5
2.0
2.5
3.0
0.0
0.5
X
L
i.o
1.5
Figure 5.9: Heave and Pitch RAO's for Fn--0.99 including monohufl performances
97
2.0
Chapter 5. Seakeeping Evaluation.
2. Pitch motion:
a. The response of the s/L=0.3 catamaran has the largest amplitudes of the
catamarans for the bow and oblique waves. At beam waves it has the lower
response amplitude.
b. All the catamarans at 1200 and 900 wave headings have several natural
frequencies.
c. Pitch motion of the monohull has similar response at wave headings of 1200
and 180' as the catamarans. It has similar natural frequencies although at
180* it has lower response amplitude and in the 1500 heading it has higher
amplitudes then the s/L=0.4 and s/L=0.5 catamarans.
d. At 1200 heading, the monohull has the highest response amplitude but only
one peak. The catamarans have several natural frequencies.
e. At 900 heading the response amplitudes of the catamarans are larger
significantly then these of the monohull. The catamarans response has
several natural frequencies.
Figures 5.10, 5.11 and 5.12 presents the heave and pitch RAO's for 1500, 1200 and 900
wave headings for all three different separation ratios.
98
Chapter 5. Seakeeping Evaluation.
P itch RAO for 150 deg waves, catamaran, s/L=0.3
Heave RAO for 150 deg waves, catamaran s/L=0.3
3.00
2.00
180 ---------------------
---------------- --------------
--------------
140 - -------------
E 120 - ------------- -.E 100 -------------0
-- --- -
E
----------
- ------- ------------- -------------- --------- ------------- ---------------
------------ -------------------------X/
1.00
3.00
2.00
-
0.50 - -----------
------------ -------------- -------------
0.8
-------
- - - - -- - - - -
---
--------------
-------------- --- - --
0
-----------
--------------
-------
-------------- -
-- - -- - - - - - - - - --
--------------
0.6
-------------- --------------
0.4
---------------- -- ---------- --------------- --------------
0.2
-------------
-------------------------- ---------------
-
0.0
f
L
1.00
0. )o
E
-
14
E
E 1.2
0 10
---------- --------------
- -- ----
I
2.00
1
3.00
1
4.00
3.0
2.8
2.6
2.4
2.2
2.0
1.8
1.6
1.4
12
1.0
0.8
0.6
0.4
0.2
0.0
---------------
-------------- ----------------------------- --- - --------- -------------- ---------------
-- - - - - - - -- -- -
16 ----------------
- - - -- - - - - - - - --
----------- ---------------
E
E 1.2 -------------------- - -- ------ -- --- -------------------E LO ----------------- -- -0
0.8 - -----------------
-
C%
4F
M
0
Cc
--------------------- --------------
0.6 - -------------- - -- ------------------------ -------------------------I--------------- --------------
0.4 - --------------
----------- --------------- --------------
0.2 --------------0.0
100
0.00
1
9
L
Fn-0.99
2.00
M
3.00
Fn7-0-90
---------- ---------------
--------------
---------------
- - - - - -- - - - - - - -
--------------
------- ------- - - -- ------------ - --------- -- -------- ------------------- -------------- ----------- -------------- -------------
---------------------------------------------------- - ------------ -------------- -------------------------------------- ----------------I- - - -- - - - - - - - -- - - - - -- - - -
--------------- --------------
-------------I
--- I
0
100
L
3.00
2.00
4.00
Pitch RAO for 150 deg waves, catamaran s/L=0.5
-- --- ------ ---------------
1.4 --------------------- -
--------------
--------------
Heave RAO for 150 deg waves, catamaran s/L=0.5
-
--------------
------------------- - ----
0.0
2.0
1.8 ---------------- T- - - - - - -
---- -------------- --------------
------
--------------------- ------ -----------------
-
-------------- --------------
-------------- ---- - --
4.00
P itch RAO fo r 150 deg waves, catamaran s/L--0.4
-
16
------
3.00
X/L 2.00
too
0.00
-------------- --------------- -------------------------------------------
----------------------- --
0.00
Heave RAO for 150 deg waves, catamaran s/L--0.4
18
- I - - - --------
1.00 - -------------
4.00
2.2
2.0
---- -------------
-
0.00
0.00
cc
X
- --------- -------------- --------------
------------
2.00 - ------------- - - -- - --- ---------
0.60 - ---------------
0.20
-- ---------------------------
--- ----150--------------
0.80 - ----------------
0.40 - -------------
-
2.50 -------------------
- ---- -- ---------- --------------
160 ---------------------
2.8
16
2.4
2.2
2.0
1.8
16
14
1.2
10
0.8
0.6
0.4
0.2
0.0
---------------- ----------- - - -------------- ------------------------------- --------- ---- ------------------------------------------------- -------- ------ - ----------- --------------- -------------- ---- --- ---- ---- --------- ---------------- ---- --------------------------------------------------------------------- 4-- ---------------------------------------------- -- - ------- ------------ -- - ----------------------------------------------------- : - - ---------- -------------- --------- -------------- --------------- ------------------------------------------- ----------0.00
4.00
Frr-0.82
x
Fir-0.74
--------------- ------------------------------------------------------------------------------100
X / L 2.00
w
Fn7-0.66
3.00
0
FrL=0.57 I
Figure 5.10: Heave and Pitch RAO's for 1500 wave headings at six different
FroudeNumbers for catamaran separation ratio of sAL,--0.3, sAL--0.4, and sAL--0.5
99
4.00
. .
...
...........
.......
...
...
. -- ----- ...
...
Chapter 5. Seakeeping Evaluatio.
Heave RAO for
120 deg waves, catamaran s/L=0.3
P itch RAO for 120 deg waves, catamaran s/L=0.3
18
1.4
---------- --------- -- ----
16
-------------------
12
--- ------ ----
------ --- ---- - --- -1.4 -10
----------- -
- - -- - -- - --
E 0.8
E
I
4,
------------------ ----------------
------------------
1.2
-----------------
1.0
---- ---
0
C 0.6
--
0.6
---------
0.4
---------
0.4
------------------- --------------------
0.2
-------------
---
-----------------
- ---
- --
----------------- ----- -------------
---- -- -- ---------- --------
0.2
----------
---- ---- - -- -- ---------
---- - ----- -------
0 0.8
-------------- - ------------------ -- --------------- ----
---------- ---- ---
-
- --------------- ,
----------
---- ---
-- - - - -- - - - - T
0.0
0.0
1.00
0.00
)'/L
2.00
0.0 0
3.00
16
1.6
--------- --------------------------------
1.2 ----
E
E
10
0.6
- ------------------- - ---------------- --------------------
0.4
-------------------
------------------
--------------------
0.4
----- -- --------
0.2
- ----------------
-------------------- --------------------
0 .2
---
-
---------------------
- - -------- - --------------------- - --------------------
-- ------
-
----
----
0.0
-
0.0
0.00
X/L
100
2.00
3.
0
0.00
X/ L
100
2.00
3.00
P itch RAO for 120 deg waves, catamaran s/L=0.5
Heave RAO for 120 deg waves, catamaran sI/L=0.5
16
1.4- -- --------------------
12
-- - ---------------------
S1.0
--------------- - --- --- -
- -
- ------------------
- --
------ --------- -- --- -- - --- ----
4,
- - - - -------------
0
0.4
0.0
100
0.00
-4-
X/L
--------
2.00
Fn=0.99 -W-- Fn=090
3.00
-
- --------
------- --- --
-----------------------------
0.2
0.0
- --- - -----
-
- - - - --
-
0 .2 -- - - - - - - - -
-----------
-------------- ------------
-
0 0.6
-
- - - - - - -- - - - - - - - -
-
C 0 .6 - -- - - - - - - - - --
------
-
--- - -- - - ----- ---- --
-E 1.0 -------------------- ---- --
--- --------- ---------
-
--------------------- ---------
-
16
0.8
0*
- --- - - -
- ------ --- - -- - -- --- M, 0.8 - - -- - --0
- -- ----------- - - - - - - - --S 0.6 - ---- -- - - ----
- --------------------- - - --------------
E
- ---
------ ----- -------
4P
0.8
12
---
-
E
-
- --
---- - -------- -----------
-- --
---------------------
12 - -------------------- - -L -
0
3. 00
2.00
Pitch RAO for 120 deg waves, catamaran s/L=0.4
Heave RAO for 120 deg waves, catamaran s/L=0.4
12
X/ L
.OO
1-
0
Fn=0.82 -x-Fn=0.74
I
I
X/ L
-u-Fn=0.66 --
2
Fn=0.57
Figure 5.11: Heave and Pitch RAO's for 1200 wave headings at six different
FroudeNumbers for catamaran separation ratio of s/L=0.3, s/L=0.4, and s/L=0.5
100
3
Chapter 5. Seakeeping Evaluation.
Pitch RAO for 90 deg waves, catamaran s/L=0.3
s/L=0.3
I
1.80
E
E
0
-------------- - .---------------------------
0.50
o
-
0.60
- - -- - ------------------------- ~---
----- - - -
-
- ---
X/L 100
0.50
-
-
---
- ------
---
-- --
- ---
-- ---
---+
1.50
100
X/ L
0.50
0.00
2.00
----
-
---
0.40 - --- --- - - --
0.00
150
------ - -- - - -----
--
0.20 0.00 -0.00
- ---- - ---
------
0.80----------
9C
0.25
- - ---- ----
-----
-----
----
-
----
1.20 - ------ - -too - ----- ----------
,
-- -
- - - -- - - -------
---- -- ---- -- -------
140
---- ----
--------- ---- ---------- -------------
0.75
-----
-
1.60
-
catamaran
Heave RAO fo r 90 deg waves,
100 -
P itch RAO for 90 deg waves, catamaran s/L=0.4
Heave RAO fo r 90 deg waves, c atamaran s/L=0.4
10
.
2.5
-
0.8
---- -
---
---- - - - - -- - -
+- -
------------
---------- -----------
--------- -
----- -- - --- -- ---- ---
1.0 -------- -- ----
-
0
-
E
0
a 0.4
2.0 -------------------
-
E 0.6
--- ---- ------- ----- ---- ---- -------- ---- ------------------
------ ----- ---- ------- -r--------------
- - - -----
0.5
---- --
-- --
----
--
---
- -
-
--
--
-
0.2
-
------------------0.0
0.0
/L
0.50
0.00
o
150
P itch RAO for 90 deg waves, catamaran s/L=0.5
Heave RAO for 90 deg waves,catamaran s/L=0.5
12
--------:--------- - - - - -------------
0.8 - -----------
----
--
-
2.0
----
-- ------
----------
--------------
- --- -- -
-
---------------------
2.5
-
10
150
100
X/ L
0.50
0.00
2.00
15
0
0 .6
-- --- -- - ------ - ---
- - ----- ----- ----
0
0.2 ---
--
-
0.4 - ----
-
------
-
----
---
----
to
--
--
-
0.5
--
--
- ----
------- --------
--
----
00
0.0
.
E
0.00
0.50
t/OL
o.
- 9 Fn=0.99 -
150
Fn=0.90
2.00
0.00
Fn=0.82 -(
0.50
X/ L
Fn=0.74 -x-Fn=0.66
100
150
-+-Fn=0.57
Figure 5.12: Heave and Pitch RAO's for 900 wave headings at six different
FroudeNumbers for catamaran separation ratio of s/L=0.3, s/L=0.4, and s/L=0.5
101
Chapter 5. Seakeeping Evaluation.
Some observations can be made from these charts.
Heave:
1. One natural frequency exists for all heave motions for the separation ratios at wave
headings of 150' and 1800.
2. Heave response at these waves is higher for the low ship speeds. At Froude
Number 0.57 and 0.66 response amplitudes are maximal.
3. Hydrodynamic interaction between the hulls is observed at oblique waves (120'
wave headings) and beam waves (900), two separate modal frequencies are noticed
at the oblique waves at higher frequencies. Their response amplitude decays as the
distance between the demi-hulls diminishes.
4. At beam waves, several heave response peaks are shown. At separation ratio of
s/L=0.3, the forward speed of the ship, has small impact on its heave RAO, almost
all frequency ranges (except A/L between 0.4 and 0.6) has the same response
amplitude. At separation ratio of s/L=0.4 a distinction exists between the heave
motions at different forward speeds. At low Froude Numbers (0.57, 0.66) three
peak responses are observed. At higher Froude Numbers another peak exists. For
the s/L=0.5 separation ratio four response peaks are noted for the fastest speeds (Fn
= 0.90, 0.99). at lower speeds only three peaks are observed.
Pitc h:
5. Hydrodynamic interaction between the hulls is noticed at the pitch response for
bow waves. At Fn = 0.74 there are two peak responses for the s/L=0.3 catamaran.
As separation ratio increases, two peaks are seen for higher FroudeNumbers.
6. At oblique waves this interaction is well observed. At 1500 headings, at least three
Froude Numbers, has double peak response. At 1200 headings all speeds has at
least two peaks. The small peak exists at high frequency range similar to the
situation for heave motion at oblique wave.
7. For beam waves the pitch response pattern for each separation ratio repeats itself
for all FroudeNumbers. Same number of peaks exists for all speeds. At s/L=0.3 the
main peak appear at about A/L = 0.5 and is maximal for Fn = 0.74, 0.82 and 0.90.
At s/L=0.4, two distinct peaks are observed. The highest response is generated by
the fastest speed, the lowest response by the lowest speed. Same thing happens at
102
Chapter 5. Seakeeping Evaluation.
s/L=0.5. At this separation ratio the highest response appear in two different
frequencies. The high speed vessels (Fn
=
0.82, 0.90, 0.99) peak at different
wavelength from the low speed vessels (Fn = 0.57, 0.66, 0.74).
5.3 Motion Response - Standard Deviation
The heave and pitch responses variance and standard deviation were developed.
The wave spectra used to analyze the responses was the ISSC (International Ship and
Offshore Structures Congress) spectral formulation for fully developed sea. The
formulation stating this sea spectrum is elaborated in section 2.4.1. The formulation as
stated in section 2.4.2 was used to calculate the variance and standard deviation of the
responses:
(5.1)
a-
=
S(w)Hj(w
2
dw,
j=1,2,...6
0
where c-, is the standard deviation.
Figures 5.13 and 5.14 are examples of the calculated variance (>
2).
The heave motion has one peak amplitude response for all catamarans at the same natural
frequency, where s/L = 0.3 catamaran has the highest peak, and s/L = 0.5 catamaran has
the lowest peak. Theoretically they should have had the same response for all frequencies,
yet the growing distance between the demi-hulls damps the response and each hull behaves
as a single monohull.
The pitch response for beam waves has two obvious natural frequencies for the
catamarans. The peak response decays as the distance between the demi-hulls grows. As
for the heave motion, it is shown that the response variance decays as the distance between
103
Chapter 5. Seakeeping Evaluation.
the demi-hulls grows. The monohull has much superior response variance then the
catamarans examined here.
heave response spectrum @ Fn=0.57,
wave heading 180 deg
-
14.00
-
12.00
s/L=0.4
s/L=0.3
s/L=0.5
monohull
-
-
10.00
8.00
-
4.00
-
2.00
-
-
6.00
0.000.2
0.3
0.25
0.35
0.45
0.4
freqency
Figure 5.13: The variance of the heave response spectrum at Fn = 0.57,
bow waves (1800)
Pitch response spectrum a Fn = 0.74,
wave heading 90 deg
-s/L=0.3
-
0.25
-
0. 20
-s/L=0.4
s/L=0.5
-m on ohull
0.10
-
0.05
-
0.15
0.00
-0.0
0.2
0.4
0.8
0.6
frequency
1.0
1.2
1.4
1.6
Figure 5.14: The variance of the pitch response spectrum at Fn = 0.74,
beam waves (900)
104
Chapter 5. Seakeeping Evaluation.
The following figures present a three-dimensional surface view of the standard deviation
values, for the heave and pitch responses, at four wave headings and six Froude Numbers
(heave in [m] units, pitch in [deg] units).
catamaran, s/L=0.3, heave st.dv.
-
1.00
U 0.9000-1.0000
*0.8000-0.9000
E3 0.7000-0.8000
* 0.6000-0.7000
* 0.5000-0.6000
M 0.4000-0.5000
o 0.3000-0.4000
o 0.2000-0.3000
.0.1000-0.2000
* 0.0000-0.1000
0.90
0.80-0.700.60
st.dv. [m] 0.50
0.40
0.30
0.20
0.10
0.00
90
0.57
wave heading
[deg]
0.99
catamaran, s/L=0.3, pitch st.dv.
o 1.4000-1.6000
*1.2000-1.4000
*1.0000-1.2000
* 0.8000-1.0000
o 0.6000-0.8000
1. 60
1. 4 0
00.4000-0.6000
1.*20
. 0.2000-0.4000
* 0.0000-0.2000
1.00
st.dv. [deg] 0.80
0.60
0.57
0.40
0.20
0.00
180
wave heading
[deg]
90
Figure 5.15: 3D view of the heave and pitch standard deviation of
a s/L=0.3 catamaran
105
Chapter 5. Seakeeping Evaluation.
catamaran, s/L=0.4, heave st.dv.
*0.9000-1.0000
*0.8000-0.9000
00.7000-0.8000
U0.6000-0.7000
00.5000-0.6000
M0.4000-0.5000
00.3000-0.4000
00.2000-0.3000
*0.1000-0.2000
0.0000-0.1000
1. 00 T7
0.90
0.80
0.70-,
0.60st.dv. [m]
0.50
0.40
0.30
0.10
0.00
0.57
90
20
0.90
'qq
wave heading
[deg]
180
0.99
catamaran, s/L=0.4, pitch st.dv.
1.4000-1.6000
*1.2000-1.4000
1.0000-1.2000
.0.8000-1.0000
[30.6000-0.8000
1.60
1.40
00.4000-0.6000
1 0.2000-0.4000
S0.0000-0.2000
1.20
st.dv. [deg] 0.80
0.60
0.57
0.40
06
0.20
0.00 -0.74
1
Fn
09
150
0.99
wave heading [deg]
90
Figure 5.16: 3D view of the heave and pitch standard deviation of
a s/L=0.4 catamaran
106
....
....
...
Chapter 5. Seakeeping Evaluation.
catamaran, s/L=0.5, heave st.dv.
m 0.8000-0.9000
c 0.7000-0.8000
* 0.6000-0.7000
* 0.5000-0.6000
N 0.4000-0.5000
o 0.3000-0.4000
0.2000-0.3000
m 0. 1000-0.2000
M 0. 0000-0. 1000
0.90
,.800.70
0.60
0.50
st.dv. [m]
0
0.30-.
0.20
0.10
0.00
90
0.57
120
wave heading
[deg]
0.99
catamaran, s/L=0.5, pitch st.dv.
'
1. 60
0
*
*
*
3
0
*
*
1. 40
1.201.00
1.4000-1.6000
1.2000-1.4000
1.0000-1.2000
0.8000-1.0000
0.6000-0.8000
0.4000-0.6000
0.2000-0.4000
0.0000-0.2000
st. dv. [deg] 0.80
0.60
0.57
0.40
0.20
0.00180
15
wave heading [deg]
90
Figure 5.17: 3D view of the heave and pitch standard deviation of
a s/L=0.5 catamaran
107
Chapter 5. Seakeeping Evaluation.
5.3.1 Analysis of results
The standard deviation of the response is the most practical and useful parameter when
analyzing a vessels response to a sea state. Twice the motion's standard deviation is a very
good approximation of the vessel's significant one third highest responses. The graphs
shown above, present a comprehensive and thrall view of all the standard deviation values
for the semi-displacement regimen, for a round bilge catamaran, at all wave headings
between bow and beam waves. The standard deviation three-dimensional surfaces were
analyzed. Six polynomial equations of the forth power, were developed, one for each
surface using the least square method. The independent variables were the ships Froude
Numbers and the waves heading (7/2 ->Z). The polynomials enable calculating a good
approximation of the heave or pitch standard deviation, at a prescribed wave heading, for
the semi-displacement range, without the use of any fluid dynamic code. The following
equations present a polynomial representation of each standard deviation plane. x
represent the Froude-Number (0.57 - 0.99), y represent the wave heading (7/2 -+ Z).
Equation 5.2 presents the heave response standard deviation for a catamaran with a
separation ratio of s/L=0.3.
(5.2)
st.dv= 3.1218-7.2017. x-5.1223-y+21608-x-y-1&6892-10.2314-
x. y
2
+0.8961-y -8.3601.
3
x4
+3.5165.
x2
x 3 -y+0.4021.
+211994. x3 -9.1842.
2
2
x .y +1.3856-x.y
3
x2
.y
-0.1861-y
4
Equation 5.3 presents the pitch response standard deviation for a catamaran with a
separation ratio of s/L=0.3.
(5.3)
st.dv =12.2972- 59.3273- x -9.0814- y +52.4701. x -y+ 50.2794-26.9204-
x. y 2
+2.7084-y 3 +14.1437-x4+3.3513.
108
x2
-51.8678- x 3 +6.6776. x 2 . y
x3 -y -2.4701. x 2
.Y 2 +4.2914.x.y-0.6145-y4
Chapter 5. Seakeeping Evaluation.
Equation 5.4 presents the heave response standard deviation for a catamaran with a
separation ratio of s/L=0.4.
(5.4)
st.dv =z 6.3698-5.8148- x-10.324-y+43.3073- x-y-43.12- x 2 +50.66. x3 -18.6816. x2 . __
13.4193.x y2 +1.4684. y3 19.4171.4+6.2381-
x3
y+1.0069-
x2 .2
+1.6854-
xy
3
-0.29- y
4
Equation 5.5 presents the pitch response standard deviation for a catamaran with a
separation ratio of s/L=0.4.
(5.5)
st.dv=-9.0454+47.1918- x-5.5041-y+31.6508- x-y-123.997-15.4472. x-y 2 +1.8328-y 3 -37.0484-x 4 + 4.7096. x3 -y -0.8175.
+106.7237- x3-3.6441- x2 .Y
4
3
x 2 .y 2 +2.4313- x. y -0.4165. y
x2
Equation 5.6 presents the heave response standard deviation for a catamaran with a
separation ratio of s/L=0.5.
(5.6)
st.dv=128538-30.0638- x-12.3850-y+45.9155. x-y -0.1218- x 2 +14.1992- x 3 -19.0007. x 2 .y
-14.2509-x - y2 +1.7879- y 8.0792- x4 +6.5843. x 3-y +0.8646. x2 .2 +1.8141. x -y3 -0.3537. y 4
Equation 5.7 presents the pitch response standard deviation for a catamaran with a
separation ratio of s/L=0.5.
(5.7)
st.dv=-1.0906+10.6090- x -5.3467. y +27.6056- x -y -52.1748. x 2 +51.2927. x 3 -7.0362. x 2 .Y
-11.8827- x-y 2 +1.4853-y 3 -17.4245- x 4 +0.6019- x3 -y +1.8545. x 2 .y 2 +1.3138. x-y 3 -02908. y 4
109
Chapter 5. Seakeeping Evaluation.
5.4 The influence of lifting appendages on Heave and Pitch
RAO's
Reduction of heave and pitch motion RAO was one of the objectives of this paper,
especially at the vessels natural frequency. After determining the shape of the response,
passive lifting appendages were 'attached' to the hull for extra 'damping'. The appendages
were attached to each catamaran at similar longitudinal positions at their bow and stem, at
a constant vertical position under the keel, with a zero angle of attack. Two foils were
located at the bow of the vessel (each foil under the keel of a demi-hull) and two foils at
the stern (same configuration). The foils had a rectangular shape with a constant span and
chord, they were not cambered. The catamarans were evaluated at two different wave
headings: bow waves (180') and oblique waves (120') at three forward speed conditions
(Froude Numbers: 0.66, 0.82 and 0.99). The total area of the foils was 80 M 2 , which was
about 4.7% of the water-plane area of the ship and about 2.5% of its wetted surface. Eight
options of foils were evaluated; aspect ratio and area of the foils were changed although
the total area of the foils, as already stated, was kept 80 M2 . For symmetry reasons, two
adjacent foils had the same area and aspect ratio. The next table summarizes eight foil
combinations.
Option #
Fwd foils
Fwd foils Area
Aft foils aspect
Aft foils area
Aspect ratio
[M 2 ]
ratio
[M2 ]
1
1.25
20
1.25
20
2
0.8
20
0.8
20
3
1.67
15
1
25
4
0.6
15
1
25
5
1
25
1.67
15
6
1
25
0.6
15
7
1
30.25
1
9.61
8
1
9.61
1
30.25
Table 5.1: Lifting appendages alternatives
110
Chapter 5. Seakeeping Evaluation.
The following figure presents the location of the foils along the hull.
z
Figure 5.18: Location of lifting appendages along the hull
The curves in the following figures, present reduction in the heave and pitch motion
responses of all catamarans, to an incident, unidirectional sea state.
Figures 5.19 and 5.20 presents the affect of the lifting appendages over the three different
separation ratio catamarans, for bow and oblique waves, at Fn = 0.66.
Figures 5.21 and 5.22 presents the effect of the lifting appendages on the ship at Fn = 0.82.
Figures 5.23 and 5.24 presents the effect of the lifting appendages on the ship at Fn = 0.99.
111
Chapter 5. Seakeeping Evaluatiom
P itch-180 deg. wave heading - s/L=0.3 with foils
He ave-180 deg. wave he a ding -s/L=0.3 with foils
3
2.8
2.6
2.4
2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
1.8 ------------------- ----------- ------- ----------------------------------I
------- J: -- ---------------- -------------------
1.2 - ---------------- ------
-----
---------------- ------
-----------
I - ----------
0
CC 0.8
----------------
--- -----
1.4 - ---------------- -------
-
1.6 ------------------- --------
----------------- -----------------
----------------------
0.6
-----------------
0.4
----------------
-----------------------------I------------------
0.2
----------------
------------- ----------------- -----------------
-;-------------------------
---------------
0
X/ L
2.0
1.0
0.0
Heave-180 deg. wave heading - s /L=0.4 with fo ils
P itch-180 deg. wave he a ding -s/L=0.4 with foils
2.2
2
-----------------
1.8
- -- - - - - - - -- - - - - - - -- - - - - - - - -
1.6
----------------- --------- ------- -- ----------------- ---------------
-1.4
E
E
--------------------------- ---- ------------------
1.2
0.8
0.2
-
-- - - - - - -
----------------- -------- ----- -- ---
-----------------
------------------------ --y
-------------------- ----
---------------------------------
------------
-----------------------------------
---------------------- -A
0.6 1 ---------------------- 0.4
- -- - - - - - - -- - - - - -
-----
-
--
-----------------
-----------------
------------- ----------------- -----------------
---------------
0
----------------- -------------------------- -----------------
M.rm x
0.0
2.0
1.0
X/ L
3.0
3
2.8
2.6
2.4
2.2
2
E 1.8
GP 1.6
"a
0 1.4
CC 1.2
CE
I
0.8
0.6
0.4
0.2
0
4.0
------------------------------ ---- ----------------- -----------------
1.8
----------------------------
1.6
----------------- ---------
1.4
--------------------------- ---- ----- ------ ae-----------------------
E
E 1.2
CC 0.8
0.6
-------- -----------------------------------------------
---------------- ------ ---
--!-; -----------------------------------
----------------------- --:5
-----------------------
----------------- -----------------
OA
----------------- ------------------------------------ - ---------- -----------------------------------
0.2
-----------------
------------- -----------------------------------
0
0.0
1.0
2.0
X/L
3.0
4.0
catanwan no foils ------- mave spectm/2
foA3
fbA7
E
---- -----------------
------------------------- ---- - -
.0
---------------- -----------------
-- ------------- -----------------
------------------------- ---------
----
- - - - - - -- - - - - - - - -
----------- -----------------
----------------- ------ ---------- ------ --------- ------------------ ------- ---------------------------------- ------ ------------------------------- --------------------- ---- -- ------------------ r ---
-
----------------- -
----
------------------------------
---------
---
--------
-------------- ---------------
-------------------
----------------
-------------
---------
-----------------------
-----------------
-----------------
---------
-----------------
-----------------
-----------------
-----------------
-----------------
-----------------------------
---- -------------
------- --
-------------------
------------------
-------------
-----------
2.0
1.0
X/ L
3.0
4.0
P itch-180 deg. wave heading -s/L=0.5 with foils
2.2
-------
- --------- -------
-------------------------- --------
0.0
Heave-180 deg. wave heading - s /L=0.5 with fo ils
2
4.0
3.0
2.0 )-/ L
1.0
0.0
4.0
3.0
---------------------------------------- ----------------- ---- --------------------- ---------------------------------- ------------------------ L------ --------- - --------------- --------------------------------- ----- ---------------- ----------------------------------------------------- ---------- ------- -------- ------------------: ----- ------ ----- I---------------------- :
-----------------N ------------ ___;I -- ------------I------- -- -------- -------------------------------------------------- -- ----------------- -------------------------------------- ------------------------------------------------------------------------------ ------------------------- --------------------------------- -------r--------- ----------------------------------------------------------------- ----------------- j-------------------
2
foA4
0
CC
(E
3
2 8
2.6
2.4
2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
---------------------------- ------------------------------------------------- ---------- ----- ---- ------------ ----------------------------------- -------- -------- ------ ----------- ------------------------------------------- --------- ------- -------- I------------------ --------------- : ------ ------ ---- 11 --------- ------- ----------------- ----------------- ------ ------------------------ ---------- ----------- --------------- ----------------- ---- --- ---- ------------------------ ------ --------------- ------------------------ -- - ----------- ------- ----------------- r ------------------------ ------------------ ----------- -------------------I--------------------------------- ----------------- ---------------- --------- ---------------
0.0 ----------
------ --------------------------------------------------------------------------------- ----------------1.0
2.0
fbill
N12
M5
fbil6
fOil8
Figure 5.19: Lifting appendages at Fn--0.66 at bow waves
112
X/L
3.0
4.0
Chapter 5. Seakeeping Evaluatiom
P itch-120 deg. wave heading - s /L=0.3 with fo ils
He ave-120 deg. wave he a ding -s/L=0.3 with foils
1.4
1.4
1.2
1
E 0.8
E
0
Kc
cc
---------------- - -----------
----------------- ----------------
0.6
--------------- -- ---------
------------------ ----------------
0.4
--------------
0.2
------------
----------------- ----------------
----------------
------------------- ----------------- ----------------
0
1.0
I
2.0
X/ L
I
3.0
I ---------------
L -------------- - ---------------- -----------------
qm 0.8 - ------------
- ------------ ----------------
0.2 ------------
--- ---------------------------------------------------
----------L---------------- J, -----------------
E
0.6 - --------------- - -------------- ---------------- -----------------
cr
3.0
4. 0
1.2 ------------------ ----- ------
-----------------L ------------------
------------------ ---- ----------- ---------------- ----------------E
OF 0.8 - ---------------- - ------------- --- --------- ---------------V
0
---------------------- c------------< 0.6 - -------------Ir
---- ----------------------------- -----------------
0.4 - --------------
0.4 - ----------------------------- ---------------- -----------------
0.2 - -------------
----------------- ----------------- ------------------------------- ----------------- ------------------
0.2 - -------0
0
1.0
0.0
2.0
X/L
3.0
3.0
4.0
-------------- ----------------
0.8 ------------------
------------------L----------------
0.6 ------------------
---------- ----------------- ----------------
0.4 ---------------
----------- -----------------------------------
T
ir
---------------- ----------------- ----------------
-
1.2
0.2 --------------
X/ L
2.0
P itch-120 deg. wave heading -s/L=0.5 with foils
Heave-120 deg. wave heading - s/L=0.5 with foils
------------------ --- -- --------
1.0
0.0
4.0
1.2
0
2.0 1/ L
1.0
0.0
P itch-120 deg. wave he a ding -slL=0.4 with foils
--------------------1.4 ---------------- -----------
------------- - ------------------
0.8 - ----------------
-------------------------0 0.6 - -----------ir
0.4 - ----------- -------- - ---------------------------------------------
0
He ave -120 deg. wave hea ding - s /L=0.4 with fo ils
----------------
E
4.0
1.2
0
- - ----- ----- ------ - --------- -----------------
----- --------- ----------------
-----------------
0.0
1,2 ----------------
------------ - ---------------------------
--------------
-------------------------------
I ------------------ ----- ---------
------- ------------------- 0.8 ------------------ ---- ---E
9m
GF
----------------E. 0.6 ------------------0
cc
--- --------- ----------------- ----------------OA ---------0.2 - ------ ------
-- -------------- ----------------- -----------------
0
0
1.0
0.0
2.0
X/L
3.0
4.0
caWnmm no foils ------- wave spectra/2
NO
foiI4
foJ17
foiI8
1.0
0.0
2.0
fbill
M2
N15
N16
Figure 5.20: Lifting appendages at Fn=0.66 at oblique waves
113
X/ L
3.0
4.0
Chapter 5. Seakeeping Evaluation.
P itch-180 deg. wave heading - s/L=0.3 with fo ils
Heave-180 deg. wave heading - s/L=0.3 with foils
2.6
18
-------------------------- -- ----
14
--------------
--
E 16
Im 1.4
---------------------------
--------------
0.6
- - ---- - - - - - -- - - - - -
0.4
-------
-----------
-------------
--------------
---
-------------
-------------- --------
- - - - - -- - - - - - - -
0 -i-0.0
- - - - - - - --
X/ L
-------------- -------
--------
-----
--
-------------
-----------
L ---
------
---------------
--
-------
-------
----------
------
---------------
------ -------
---------L -------------
- - - - - -
0.8
0.6
-----T--------------- - -----
---------------I---------------------------------------------------
0.4
--------------
-------------- -------------- -------
0.2
-------------
- - - -- - - -
3.0
4.0
------------ -------------- -------------- ------
-11
0
2.0
10
------------- -----------------------
----- L ----
-
0.2
--------------- r -----
0 12
cc
Ir I
-------
-------------
'P,
--------
--------------- ------- ------ --------- ---- -----------------------------------I------ ------- I------------ ------------- ---------
2
18
-
------
-- -
------
---------------
------------ 4 -------
------
-----------------------
--------------- ---------- ----
2.2
- ------
----
-----------------------
2.4
------------ -------
----- -
---------------
12
E
1
I
0 0.8
--------
-
16
to
0. 0
Heave-180 deg. wave heading - s/L=0.4 with fo ils
2.0
X/ L
4.0
3.0
P itch-180 deg. wave heading - s/L=0.4 with fo ils
2.4
2
----------------------------
1.6
-------------- ---------- -- ------- ------ -------------- -------
14
--------------- --------- --- ----
-E 12
---------------L
E
0
--------------- ------- --
I
0. 8
--------
----
--------
-
--
-------------
-------------- ----
OA
---------------
0.2
-------------
0 1
O'o
---- -------------- ------
--
1.6 - ------------- ------- ------ ----------------- ---------------------- ------- ------- ---14 - ------------- ----- ---
------
---
- --------------
0
cc
ir
------
--- -------------- -------------- -------
I
2.0
L
I
3.0
-------- - ------------------------ -------------
-
0.4 ---------------
----------- ---------------L-------------------------- ---------------
------------- -------------- ------------- -------
0.2 ------------
--------- --------------- -------------- -------
I
1.0
I - ------------- I---0.8
0.6
------
-----------------------------
--------- ------------
12 - --------------
7--------------------------- i-------
---------
0.6
------------------------ ----------- ----- ------------- ---------------------I--------- ---- ------------ --------------- ------2
18 ----------------------- ----- -------------- ------------- -------
2.2
---- --------------------------------
1.8
0
1
2.0
10
0.0
4.0
X/ L
4.0
3.0
P itch-180 deg. wave heading - s/L=0.5 with fo ils
He ave -180 deg. wave he ading - s /L=0.5 with fo ils
2.2
2
2
18 ---------------- -------------- ----- ------- -------------- -------
----------------------
14
---------- -------
-9 12 - -------------- --------- ----
E
- ----------------------- -0
cc 0.8 --------------- ------ _7
cc
---------
-------------- -------
CR
Go
V
0
-------------- L------------- j-------
------- --- -------------------------------- ------ ------------- ------------ ------
1.4
------------- ---- --- - - - -- - --
1.2
------------
1
-
------------- --
0.6
------------- - -----
0.4 --------------- ----- - ----- -------------- -------------- -------
0.4
------------
0.2 ----------------f -
0.2
----------
----------------------
--------- ---------------L ----------------------
0 I0.0
0
to
0.0
2.0
X/ L
3.0
4.0
catanmm no Ms ------- vmve spectm/2
fbi13
fbiX
fbV
fOiI8
-
------
-----
------------- --------------------
------ ------------- ------------------------------- ------------- ------------- r------
i
to
2A/ L
3.0
fbfll
fbi12
N15
N16
Figure 5.21: Lifting appendages at Fn=0.82 at bow waves
114
----- ------
-4 -- --------------------- -- - ----- -I
----------------- ------ i-------------
0.8
0.6 --------------- -----
------------- ------
--- - --------
1.6
1.8
16 ---------------------------- - -------- ---- 14 ---------------------
------------- --------
4.0
. .. ..
.....
....
Chapter 5. Seakeeping Evaluation.
Pitch-120 deg.wave heading -s/L=0.3 with foils
Heave-120 deg. wave heading - s/L=0.3 with foils
14
12 ---------------------- ----- -----
----------------
- - - - -- - - - - - - - --
- ----- ------
----------------- --- -
---------------- ----------------
0.8
---------------- ----------------
0 0.6
cc
-------------
----
-- -------
< 0.6 - --------------Ir
----------
0.2
------ ----------------
-- 4 ---------
----------------------------------------------t ----------0
0
2.0
LO
0.0
0.0
4. 0
3.0
X/ L
X/ L
2.0
1.0
4.0
3.0
P itch-120 dog. wav e hea ding - s /L=0.4 with fo ils
Heave-120 deg.wave heading -s/L=0.4 with foils
1.4
1.4
12 ----------------- ----- ----- ---- ---------------- L ----------------
1.2 ---------------------------- ------ --------------- -----------------
------------- ----------------
I - -------------------- -- ----
0.6
E
0.8
cm
4P
- ---------- ---------------- ---------------
0
cc 0.6
----------------------------- ---------------
0.4
--------------- - ---------------- ---------------- ---------------
0.2
- ----------------
0.2
-------------------------- -------
------------------- ----------------
0.8 - ---------------
0.4 - --------------- -
2.0
1.0
0.0
----------------
------------------ ------- ---
---------------
- --------------- -------------- -----
-------
X/ L
---------
-----------
---------------------------------
---------------- i--------------- I-----------------
1.0
0.0
4.0
3.0
2.0 X/ L
4.0
3.0
P itch-120 deg.wave heading -s/L=0.5 with foils
Heave-120 deg.wave heading -s/L=0.5 with foils
1.4
-
L4
- ----------- ----------------
0
0
- --------------- ----
0.8
- ------------------------ ---------- - ---------- ---------------
------------- ---------------
1 - --------------- :------ -----
--------------- -----------------------------
cC 0.6 - --------------cc
m 0.8
4P
0
-cc
ir
---------------
- --------------- ------
---------
0.6 - ----------------L------ - -- - -
-- --- --------------
-- ------------
--------
------ ---------------- ---------------
0.4 - ---------
------------
0.4 - -----------
------ --------------------------------
1.2 ----------
1.2 ------------------------ --- ---------- --------- ---------------
E
E
--------------
---- - --------- --------------- ---------------
0.4 - -----------
----- --------- ---------------- ----------------
0.2
<
------- r ---------------
r --------------
r
------------
ir
- -------------
0.4
E
7P
0
-------- - ----------------------------
--------------
1
----------------------------
0.8
E
0
--------- ---------------
--------------- ---- - -----
1.2
-
L4
----------------------------- --------------0.
0.2 - ------
----------------
--------------- ---------------
----
0
0
10
0.0
i
2.0
X/ L
3.0
cataniam no foils ------- wave spectra/2
NO
fdiI7
0
to
0.0
4.0
fbiA
2.0 X/ L
fbill
N12
N15
N16
fbilg
Figure 5.22: Lifting appendages at Fn=0.82 at oblique waves
115
3.0
4.0
Chapter 5. Seakeeping Evaluation.
Pitch-180 deg. wave he a ding -s/L=0.3 with foils
Heave-180 deg. wave heading - s /L=0.3 with fo Hs
2
2.8
------18 ---------------- --------------- - -- -------- ---------------L
r
----------------------------- ------- ------ L -------------- L-------
T
12
.E
0
c_ 0.8
0.6
-- - - - - - - -- - - - - - - -- -- - - - - - - -
-
- - - - - - -- - - - - - - -- - - - - - - - --
-
- - - - - - - - -- - - - -
---------
--- - - - - - - --
------
----
-- - - - - - - - - -- - -
- - - --
- -- - - - - - -
-- - - -
--
----
-- - - - - - - -
---
-
-
-
14
----------- ---------
- - -- - -
-
IL6
- - - - - -- - - - - - i -- - - - - - - - --
- - -- - - - - - -
----------- --- -------------- ------2.2 ---------------- ------------ ---------------- ------------------------------------- --------------:
2 ---------------- ----------1.8 ---------------- ---------- ---- --------------- I
- - - - -:- - -.
E
----- ------- f- --------- -----"a 1.6 - -------------- --------- ---2.4 -------
'D 1.4 - -------------- -------- ---- ------- ---0 L2 - -------------- ------- --I - ---------------
----------------
0.4
------------I
-------
-----
--------------
--------------
-------
-------
--- ------------
----
-------
-
---------------------
--------------
0
2.0
1.0
0-0
-------- --------------- ------------------------
Heave-180 deg. wave heading - s /L=0.4 with fo ils
-------------I-------------- -------------- ------2.0
10
0.0
4.0
3.0
X/ L
---
--------------- T-------------- I--------
0.2 f ------------0
----------
-------
---------
0.4 - -------------0.2
------- ---------- ---- -------
0.8 - ---- --------0.6 - -----
-----------------------------
-------- i-------------- -------
----
2.6 ---------------- 1--------------
4.0
3.0
)-/ L
P itch-180 deg. wave heading - s /L=0.4 with fo ils
1.8
2.4 ---------------- -------------- I- -- --------- -------------- ------1.6
1.4
--------------- --------------- ------- ------ ------------- r--------------------- ------------- -----------
2.2 ---------------- -------------
-------------
-------------- -------
2 ----------------------------- - ---------------
--- -------------- -------
------------- -------
1.8 ---------------- ----------- --------------------- ------------------1.2
E
cc
cc
I
-------------- ---------
0.6
-------------- --------
0.4
-------------- ----
----- -------
E
cri
0
cc
T
----------- -------------- r-----------
------------
-
-------------- ---------- -------------------- j- --- --------- -------
16
-------- -----
--------------
0.8
0.2
-------- -------
--------------- ------------ - ---- --
-------- ---------------
L'4
1.2 ---------------- ------- --- -
---
--
--
-------
-- --------- ------
1 ---------------- 4------ -- --------- ----------I---
0.8 ---------------- ---- - ------- --------------------------------
---- -----------------------------
---------------- -- -- ------ -------------- -------------- ------
0.6 ----------------
---------------------------------- -------
0.4 ----------------
--------- --------------- --------------- -------
0.2 ----------------
--------------I------------------------------ -------
0
0
2.0
10
0.0
A./ L
4.0
3.0
2.0
1.0
0.0
X/ L
4O
3.0
P itch-180 deg. wave heading - s /L=0.5 with fo ils
Heave-180 deg. wave heading - s/L=0.5 with foils
10
2.4
L6 ---------
I---------
2.2
---
------------- ----- j-----------
1.2 -----------------------------
------ --- ---------
0.2 ----------------
--------------- -------------- I--------
-----
1.6
--------------
-----------------------------
0.8
--------------
------------
-----------------
----------
------- --------------- --------------- -------
------
--------------------------
--------
--------
-------
---------------------- -----
-- ---- -------- ----- ------- ---------------
-------------- -------
-------------- -
--------------
--------------
----
-------
---
------ ---------------------
0.6
-------------
-------
-------------- -------
------ --------------------
1-8
L2
aGP
0-cc I
T
-------
---------- ----------------- -------
- -----------------------
0.4 ---------------------
------- -------
---------------
-------------- -------------
1.4
- -------------- --------
0.6 - -----------------------
--------------
2
1.4 ---------------- -------------------------- ----------------- --------
E I
i
0 0.8
----
0.4
--------------
0.2
------------
-----------I -------------- -------------- 1 -------------------- --------------- -------------- -------
0
0
10
0.0
2.0
X/ L
3.0
catammn no fbils ------- vmve spectra/2
fbO
fbjI4
fOA7
N18
1.0
0. 0
4.0
2,0
%/L
fbill
M2
fbfl5
N16
Figure 5.23: Lifting appendages at Fn=0.99 at bow waves
116
3.0
4.0
Chapter 5. Seakeeping Evaluation.
P itch-120 deg. wave he a ding -a /L=0.3 with foils
Heave-120 deg.wave heading -s/L=0.3 withfoils
1.4
1.2 -------------------- -------------- -----------------------------------
E
0
r ---------------------------------
---- v
< 0.6 - ---------------- -cr
E
ch
----------------
------------ -------------
1.4 --------------------
1 ----------------E 0.8 -----------------
--------------------V---------------
1.6 ----------------- -- -----
1.2 ------------------ ------- ----------------------- ------------------
------ ---- ---------- ----------------
I - ---------------- --
------------------------------
- --------- ---------------0 0.8 - --------------- - -------------cc
ir
--------- - ----------------0.6 - -------------- ---------
OA ---------------- - ------------- --------------- --------------------- a---------- 1----------------------
0.4 - ------------
0.2
---------------
----------------
---------------
----------------
0.2 -------------
-- 4c -------------------------------- ---------------
0
0
X/ L
2.0
1.0
0.0
3.0
4. 0
X/L
2.0
1.0
0.0
4. 0
3.0
Pitch-120 deg. wave he a ding -s/L=0.4 with foils
He ave -120 deg. wave he ading -s /L=0.4 with fo ils
1.2
1.2
- -------- ----------------
I ------------------ ------ --- ---
I - --------------- --- ------ ---- -- ------------ ----------------
---------------- ----------------
0.8 -----------------T
---------------- ---------- ----------------
'F.6 - --------------0
cc
0.4 - ----------------
E
qm
'M0.6 - ---------------0
- --------- -----------------T- - - - - - - - -- - - - - -
-
-------------
-------------
------------------------------ -- --------
--------------- t---------------- ----------------
0.2 - -------------0
0
2.0
1.0
0.0
X/L
3.0
4.0
2.0
1.0
0.0
1
X/ L
4.0
3.0
P itch-120 deg. wave he ading - s /L=0.5 with fo ils
Heave-120 deg. wave heading -s/L=0.5 with foils
1.2
-
-
1.2
-
0.4 - ---------------
--------------- ---------------- ----------------
0.2 -----------------
--------- ------- ---------------
0.8 ----------------- - -----
I ------------------ -------- ------
0.8 ------------------ ------ -
---------------- ----------------
I ----------------- ------------ ----------------- i------------------------ ----------------
0.8 ------------------ ------
---------------- ----------------
E
T
FV.6 - ---------------- ---0
------- I----------------- ----------------
cr
0.4 - ---------------- -- - ---------- ---------------- ----------------
0.2 ------------------
------------- ---------------- ----------------
--------------
0.6 - --------0
cc
cc
0.4 - ---------------
------- --------------- ---------------
------------- r--------------- I----------------
0.2 --------0
-
0
1.0
0.0
2.0
X/ L
4.0
3.0
camamn no fbAs ------- wave spedrd2
fbO
M4
fIoA7
fbi18
1.0
0.0
2.0 1/ L
fbAI
M2
fbfl5
fbfl6
Figure 5.24: Lifting appendages at Fn=0.99 at oblique waves
117
3.0
4.0
Chapter 5. Seakeeping Evaluation.
From visual inspection of the heave and pitch RAO's, several conclusions can be made.
1. All foils reduce the heave and pitch responses of all catamarans.
2. Heave response:
a.
The lowest reduction of the heave peak response was achieved by the foil
combinations 4 and 8. These combinations consists a large area foil at the
stem. Combination number 4 is the only foil combination where the
forward foil has shorter span then chord. The span of all the foil of
combination 8 is equal to its chord.
b.
The highest reduction of heave RAO was obtained for foil combinations 5
and 7. These combinations reduce significantly the response amplitude for
all catamarans at all wave headings / forward speeds. These foils have
larger foil area at the bow. Combination 5 has a span / chord ratio of 1.67
for the rear foil. The aspect ratio of all the foil of combination 7 is one.
3. Pitch response:
a.
At Fn = 0.66, the best foils combinations were 1 and 5 for bow waves and 1,
3, 5 and 7 for oblique waves. At Fn = 0.82, the best combinations were 1, 3
and 5 for bow waves and 1, 3, 5 and 7 for oblique waves. At Fn = 0.99, the
best combinations were 1, 3, 5 and 8 for all wave headings except for s/L
=
0.5 where combination 8 achieved the lowest reduction of RAO.
b. The pitch response of the catamarans without lifting appendages contained
more then one natural frequency, especially at oblique waves. Some foils
often reduced the number of natural frequencies, examples can be found at
Fn = 0.99, oblique waves, for s/L = 0.4 and s/L = 0.5.
c.
Occasionally, for the same catamaran at the same forward speed and wave
heading, some foils combinations peak at one natural frequency and some at
another frequency. For example, at s/L = 0.3, Fn = 0.66, bow waves: foils 6
and 7 peak at the same frequency as the first natural frequency of the
catamaran without the foils. Foils 4 and 8 peak at the same frequency as the
second natural frequency of the catamaran without foils.
118
Chapter 5. Seakeeping Evaluation.
The standard deviation of the ships' response with all foils combinations was calculated. It
was found that the lowest standard deviation for most of the responses (heave and pitch)
was achieved by foils combination 5. The highest standard deviation was calculated for
combination 4. Upon these findings, it is suggested that hydrofoil catamarans would be
designed with a larger foil at the bow and a smaller, narrow foil near the stem.
A theoretical check of the maximum foil area, where the decrease of the response is
negligible, was made. The s/L = 0.3 catamaran at Fn = 0.99, bow waves was used. All four
foils had equal area and their aspect ratio was one. Total area of foils was raised starting at
160 m2 . Convergence achieved at area of 1200 M2 . This is not a practical foil area from a
design point of view.
The following figure presents the increase of foils area and the decrease of heave and pitch
motion until convergence.
Pitch-180 deg. wave heading -s/L=0.3 with foils
He ave-180 deg. wave heading - s /L=0.3 with foils
2.8
2
2.6
16
2.2 ---
--------
.
0E .
------------------- -------------- - ----
- ---- - -----
0C
.6
0.6
------------
L.2 - - ------- - - 4 - ----
-- ----
- -
-----
--
-
-
--
-
------- - ------
1
020
.
.
0.8
------------------------------- -----------
0.2
-- -----
---;-f--
-
-------------------r
- ----
0.4
- - ---- ----------- - - ---
---
- - - - - --
L.4
---------
--
- --------
- - -
-
1
--- ---- -----
---
- ----
- -------------------------------------
-
--
---
-- -------
-
- -
-
-------
-
------
-
12
-
M 0.8
---
------------ --------------
---------------- ------------ --------
E L2
i *1
0
--------------- -------cc
--- - -----
2 .4 - - - - -
2
-----
---------
----
-----------------------------,-----------------------L4 - ------------- 12-0--
----
--
----- - -- --
--
-- --- - - -
-
-
-- ---------------- ---------L8 ---------------- - ----------- ---
0
0
0.0
L.O
2.0
1/L
4.0
3.0
5. 0
0.0
catamaran, sL= 0.3. no foil
foil2
foil5
foil8
- ---
wave spectral2
foil3
foilS
Foil 1.1 160 m*m
Foil 1.3 500 m*m
Foil1.6 1100m*m
--
foil 1.4 800 m*m
.--
2.0
10
-
/L
3.0
foill
Foil4
foH7
Foil 1.2 300 m*m
foil 1.5 1000 m*m
foil1.7 1200 m*m
Figure 5.25: Large area lifting appendages at Fn=0.99, bow waves, s/L=0.3
119
4.0
Chapter 6. Structural Loading Analysis
Chapter 6: Structural Loading Analysis
6.1 Preface
Wave excited loads can be analyzed in three different ways.
1. An instantaneous local hydrodynamic pressure loads as a result of the ship motions
or the ship interaction with waves.
2. A continues load, obtained by integration of the instantaneous pressure loads over
time. These loads yield the hulls shear force, bending moment and torsion moment.
3. An impulsive, local pressure loads that cause vibratory response (loads like
slamming, whipping, springing etc.).
6.1.1 The exciting forces
The following chapter discusses the continuous shear force, longitudinal bending moment
and torsion moment, applied upon the vessel as a result of ambient waves. The overall
force can be evaluated by equation (2.57), stated again here.
S(t) =FHS
+FEX +FR
FjHs is the net hydrostatic force and includes gravitational and hydrostatic components.
F,
is the excitation force in the j' direction and contains the incident wave force,
known also as the Froude - Krylov force, and the diffracted wave force. F.R is the
hydrodynamic force in the j' direction due to forced motions. It results from the radiation
of waves away from a vessel that is forced to oscillate.
120
Chapter 6. Structural Loading Analysis
The component of interest in this chapter is F,
the wave exciting forces. In order to
exert these forces, it is necessary to solve the equations of motion (2.58), (3.33), to get the
motions response amplitude operators. Then, the incremental vertical force, in excess of or
less then the still water buoyant force, at any instance of time, along the hull, can be
calculated. Evaluation of shear forces and longitudinal moments, and especially transverse
moments, in irregular seas, has important consequences on the structural design of
catamaran hulls. Theoretically, longitudinal bending moments tend towards zero for both
very long and very short waves and often are 'double humped', where one peak
corresponds with the motions resonance and the other corresponds with the wavelength
that cause the largest re-distribution of buoyancy. In practice, longitudinal bending
moment exists also in the steady state, where no interaction with an ambient wave occurs
from two reasons: (1) the interaction between the opposite and coupled weight and buoyant
forces, and (2) the sinkage and trim the ship withstand when translating forward. The Same
theory holds for the shear force along the hull. In long waves, similar to the steady state
situation, the longitudinal bending moment or the shear force are thus tends to a finite
value which is not zero. In case of a port - starboard symmetric ship, where the ship does
not develop any steady heal or list throughout its motion at sea, its torsion moment
vanishes at very short and very long waves..
6.1.2 Force RAO's
Time dependent forces, in all six degrees of freedom, were evaluated by SWAN-2. A
-
Fouriertransform was conducted for three of the forces: F, E, the exciting force in the z
direction, to find the shear force; F
, the exciting moment around the x - axis, to find
the torsion moment; and F17, the exciting moment around the y - axis, to find the
longitudinal bending moment. The results were non-dimensionalized by:
(6.2)
RAO(w,)=
F3
A-p-g-A,
121
non-dimensional shear force RAO.
Chapter 6. Structural Loading Analysis
RAO(w,) =
(6.3)
F
Ap-g-AITT-L'
non-dimensional bending moment RAO.
RAO(w,)=
(6.4)
A.p-g-A, -L'
non-dimensional torsion moment RAO.
Where, A is the incident wave amplitude, Av is the ships water plane area and L is the
ships length. All forces and moment were evaluated at five stations along the center line
of each catamaran.
6.1.3 Ship weight distribution
The weight distribution of the ship was modeled for simulation purposes to comply with
the ships longitudinal center of gravity and it's over all displacement. The following graph
present the weight distribution applied along the ship.
weight distribution
800000
700000
600000
500000
Z 400000
300000
200000
100000-
0
-45
-35
-25
-15
-5
5
15
25
35
location [m]
Figure 6.1: Simulation of weight distribution along the hull
122
45
Chapter 6. Structural Loading Analysis
The weight distribution was constructed such that the ship would not gain a least angle.
Due to the deep transom of the vessel, it was assumed that most of the weight is
concentrated at the aft portion, hence the rising trend of the graph towards the transom.
6.2 Significant Forces
Significant forces and moments are defined as the average of the one third highest
values of forces and moments applied upon the vessel. These values were obtained by
calculating the non-dimensionalized forces standard deviation and multiplying it by two.
Standard deviation of the force was calculated as described in section 2.4.2 and as applied
for the motion responses in section 5.3. The significant forces and moments were evaluated
at two wave headings: 180' and 120', for three ship speeds, or Froude Number values of
0.66, 0.82 and 0.99. All three separation ratio catamarans were evaluated at five stations
along the hull: station 5 - fore perpendicular; station 4 - fore quarter; station 3 - midship;
station 2 - aft quarter; and station 1 - aft perpendicular. The ambient 'wave' was
constructed from 27 waves with different periods between 1 second to 23 seconds divided
into three period intervals: (1) from periods of 1 second to 7 seconds the difference
between the waves was of 1 second; (2) Between 7 seconds and 15 seconds, the difference
was of 0.5 second; and (3) from period of 15 seconds to 23 seconds with period 'step' of
two seconds.
The following charts present the results for wave heading of 1800.
123
Chapter 6. Structural Loading Analysis
Sear force,
Bending moment, 180 deg, s/L=0.3
180 deg, s/L-0.3
0.60
0.90
Fn=0.82
----0
-----------------------
Fn=0.99
Fn=0.99
3 0.40
- - - -- - - ----
----- - -- ------
-
0.20
-
--
--------
0.1
------
--- --- -
----------
~0.3 0
----
- - - - - - - - - - - - - - - - - - - - -- - - - - -
!g 0.20 --------------------------------------------
---------
-
(4.
.
0 ---. -----
-----------
-
0 .60 --- ---- -----0------ ----0.4
----
---
-
0.70 ------
-----
0.50
Fn = 0.66
Fn=0.82
----
-+-Fn = 0.66
-------
----------------
0.10 ------------
--- ---
0.00
0.00
1
2
station
5
4
3
5
4
station 3
2
1
Figure 6.2: Significant shear forces and bending moments at 1800 wave headings
for Fn = 0.66, 0.82, 0.99, for s/L = 0.3
Bending moment, 180 deg, s/L=O.4
Shear force, 180 deg, s/L=0.4
0.60
0.90
0.70
-------------
------------
S 0.60 ---------
.Fn=0.82
Fn =0.99
------------- --------------
- -- ---------
-------- - - ---- -
----------
--------- -
0.40 -- ------------
-- ---------
-----------
- Fn=0.82
Fn=0.99
8
---- - - ----- ------- ---- --- - --- -- --
0.30 - -------
--
0.20 --------
-
--- -- -
-
-
-
0
--- -- ----
-------
------
0.10-----------------
-- -------- -------
--------
0 .10
--
-
-
----------
------- - - -------- - ---- ----------
-
-
0.40
-
4.
------
-
- --------- ---
.
0.50 ----------------
- -----
0.50
0.30
-4-Fn = 0.66
-4--Fn = 0.66
---
-
------------
-
0.80 -----------
0.00
0.00
1
2
station
3
4
5
1
2
station 3
4
Figure 6.3: Significant shear forces and bending moments at 1800 wave headings
for Fn = 0.66, 0.82, 0.99, for s/L=0.4
Torsion moments do not exist for a symmetric port - starboard ship at wave headings of
1800 as stated above. For higher speeds, larger shear forces and bending moments develop
along the hull center line. For each speed the magnitudes of the significant force and
moment, at each station, are similar for all three separation ratios. The shear force and
124
5
Chapter 6. Structural Loading Analysis
bending moment distribution, generally tend to higher values towards the transom of the
vessels as the weight distribution curve.
Bending moment, 180 deg, s/L=0.5
Shear force, 180 deg, s/L=0.5
--------I-s-Fn
S 0.60 - -----
Fn =
0 .4 0
---
0 .3 0
--- - -
0.2 0
- ------------- -- ----- --- --
0.50 -U---
-
- - - - ------ -
t
-
- - - ----- - - - - -----
- - -
--
-
0.30
---
0.20
---------- -
------
-
Fn =0.99
-------
-
----
--------
0.40
=0.82
-Fn
0.99
----------------------------------
-- -
-
---------
------
--
---- ------
0.00
----
- -- -
-
0.10
----- - --- - -- -------------
--
0.66
Fn =
--------------
-- ------- --- - -----
-----
0.66
-
---- - - -----------
-----
0.10
=
-E-Fn=0.82
-
0.70
-- ----
- -- ---------
--------
-
0.80
0.60
-
0.90 -
--
---
0.00
5
4
3
station
2
1
2
1
station
5
4
3
Figure 6.4: Significant shear forces and bending moments at 1800 wave headings
for Fn = 0.66, 0.82, 0.99, for s/L = 0.5
The following graphs present the results obtained for wave heading of 1200.
007
S
-------
--------------
----
---- ---------
-
0.10
-------
0.05
------
.06 -
--------- ---------- ----------
---------
x0.20
0.15
Fn0.99
S=
--------
0.30
0.25
0.09 ----------- -----------0.08 ------------------
Fn =0.66
-Fn=0.82
-------------------
--
-
---
------
---------
035
-------
-- ------
-- ------ --------- ------------------5------------------
0.
-4-Fn=0.66
030
-Fn=0.82
Fn =0.99
0.03
0.02
------
- - ------
0.01
-----------------------------------
0.04
--
1
2
station 3
4
5
--
0.10
------ - --- - - ----------- - -------- ----
0.05
----------- -----------------------
---------
----------
0.00
2
station 3
0
----------- -- --------- -------
0.20
0.00
0.00
Fn =0.66
--.-- - - ...3..Fn
Fn =0,82
Fn =099
0.15
--
-----
0.25
j
------------
--- ---
-
0.40
2 035
0.10
----- --
-- ------
0.45
.7
,
-
--
IBending moment, 120 deg, s/L=0.3
Torsion moment, 120 deg s/L=0.3
Shear force, 120 deg, s/L=0.3
0.50
4
5
i
1
2
station 3
Figure 6.5: Significant shear force, bending and torsion moments at 1200 wave
headings for Fn = 0.66, 0.82, 0.99, for s/L = 0.3
125
4
5
Chapter 6. Structural Loading Analysis
n 099
---------- -------030 ---------- -------------------0.25
---A-----------0.20
-------------------------0.15
------- --------------------0.10 ---------------------
1
2
station
3
4
0.15
-
Fn=099
----------
-----------
-
-
- -
-
------
--------
----------
---------
---
----
-------
0.10
.2
'k,
- -- - - --------
--
--------
----------
---
-------
-
-
-
0.05
-- -- --
5
1
2
0.00
- -- -- -- - -- -- --
-
0 .00
0.00
----
--
---- -- - - - --- - 0.02 - - - - - -1-- - - - - - - - --
--------------------------
0.05
0.20
Fn=0.82
-
-
OA.0-----------------------------------0.0 8 -------- - --------- ----- 70 .06
----------------------
Fn=0.99
-- ------- ---------- ---
c 0.04
.G
0.25
1
5
4
station3
2
station
3
5
4
Figure 6.6: Significant shear force, bending and torsion moments at 1200 wave
headings for Fn = 0.66, 0.82, 0.99, for s/L = 0.4
0.30
--
- ..--
.35
- F = .6
Fn=0.82
Fn=099
T
--------
---------
0.20 -
-.--
Fn=0.66
-
Fn=0.82
0.5
-
0.15
------------------
-- -
60.20
----- -------------
-------
------------ -----
---------
Fn=0.99
a
0.30 ------- - - --- - - ------ - --------- ----------------------------0.0 --------- ------------- --0.25
4
- 2020
----
0.25
-----
-
---------
---------
is
d.
x 0.15
Fn= 0.66
.- Fn=0.82
Fn =0.99
----------
-
0.40.6 ---------------
-
0.25
0.45
0.40
Bendingmoment, 120 deg s/L=0.5
Torsion moment, 120 deg s/L=0.5
Shear force, 120 dog s/L=0.5
----------
-
0.00.........
2
1
5
station
3
0.00
4
---------
-------
2
1
2
----
----3
3
station
---
-4
4
--------
0.10
---------
0.05
-----------
1
---------
-
.15 0.05
0 .0 5 -- - - - - -- - - - -- - - - - -- - -
-
-
-gFn=0992
0.
,
C
Fn=0.66
-. Fn =066
--... - Fn=0.2
-- -------- -----------
-0.12
-
0.35
0.30
-
0.16
0.14
Fn=0.66
--U-Fn=0.82
0.40
Bendingmoment, 120 deg s/L=0.4
Torsion moment, 120 deg, s/L=0.4
Shear force, 120 deg, s/L=0.4
0.45
0.00
5
1
2
station
3
4
Figure 6.7: Significant shear force, bending and torsion moments at 1200 wave
headings for Fn = 0.66, 0.82, 0.99, for s/L = 0.5
At wave heading of 1200, some differences are observed between the magnitudes of the
forces, although the trend of the curves is similar. The trend of the shear forces and
bending moments is similar to this of the 1800 wave heading and follows the weight
distribution curve, larger forces and moments develop at the stern, while smaller forces
develop at the bow.
The catamarans with separation ratios of s/L = 0.4 and s/L=0.5 obtain similar values of
forces at Fn = 0.82 and Fn = 0.99, and the slower speed (Fn = 0.66) gets significantly
lower values of forces. In general, as for the 1800 wave heading, larger forces are
126
5
Chapter 6. Structural Loading Analysis
developed in higher speeds. It is also noted that the torsion moment values are larger at
larger separation ratio.
For the s/L = 0.3 catamaran, the torsion values are approximately in one order smaller from
the bending moment values for the same catamaran at the same station, while for the s/L =
0.5 catamaran the torsion and bending moment values are in the same order of magnitude.
As stated earlier, this is one of the biggest disadvantages of the catamaran hull type, as the
separation between the demi-hulls increases, the torsion moment that acts upon the center
line, is getting higher.
Annex C presents the RAO's curves for each speed, wave heading and separation ratio.
These RAO's were used to construct the significant forces graphs.
127
Chapter 7. Conclusions.
Chapter 7: Discussion and Conclusions
A hydrodynamic 'numerical' model test was conducted upon a 7500 ton, round
bilge, deep transom stern catamaran. The test was developed throughout this paper and
compared with a 'numerical' model test conducted upon a similar length - displacement
monohull. Several catamaran shapes were examined, all had the same demi-hulls, but
different separation ratios. The following chapter summarizes the work done so far and
makes some suggestions for future work.
7.1 Conclusions
The vessels steady state was evaluated. The ideal wave resistance, the Kelvin wave
pattern, and the sinkage and trim were measured when advancing at a constant forward
speed, at calm water. It was shown that for higher advancing speeds, the waves created at
the wake tend to divert, while for slower speeds, more transverse waves are seen at the
wake.
A monohull with the same length (100 meters) and displacement (7500 ton) as the
catamarans was evaluated as well. It was shown that the monohull had lower dynamic
sinkage and trim values then all catamarans. This is mainly due to a lower wetted surface.
The dynamic trim for all catamaran types is increasing for higher speeds, i.e. the steady
pitch angle increases for higher Froude Numbers. A negative trim (bow up, stern down)
locally increases the draft of the vessel at the stern, hence increase the induced resistance
(due to dry transom). For large transom sterns as exist for the catamarans investigated, an
increase of the draft has a significant impact on the overall ideal fluid resistance of the
vessel.
128
Chapter 7. Conclusions.
The following figure was derived from figure 4.6 and presents a comparison of the ideal
fluid resistance component of the three catamarans, and the resistance results obtained
from two times the ideal fluid resistance of one demi-hull.
Ideal fki resistaixe coefficint
7.OE-03
catamaran s/L=0.3
=
-- - aa rns/
s/L=O.4
-4-catamaran
-
.
-- ----- I------- -I- ---------------- -6.0&M 3 ------ -- 6.OE-03
-u-catamaran s/L=0.5
5.OF-03 ---------------
---
--------- -----
--*-dmi-hulX2
I4.0E-03 ----------------
2 .OE -03
--
---- -I --- - --I
-- - - - - --- -- - - - -- - - -- - - -- -I- -- -
-
--- -- -
- - - -- - -- - ---- - - - -- - --
-
--- - - -- - - --I- -
-
3 .0E-03
.0E0-00
0.55
0.60
0.65
0.70
0.80
0.75
Froude Number
0.85
0.90
0.95
1.00
Figure 7.1: Ideal fluid resistance
For the most parts of the semi-displacement range, the larger separation ratio has higher
resistance values then the smaller separation ratios, and the s /L -+ oc (twice the demi-hull
results) has the highest resistance results. The phenomena of increase in resistance as the
distance between the demi-hulls grows, is caused due to a favorable interaction between
the demi-hulls. The Kelvin wave pattern, exerted from one demi-hull, 'wets' the transom
stern of the adjacent demi-hull and vice versa. The induced, adverse drag force is
decreasing, since less area of the stern is dry, hence the overall ideal fluid resistance is
smaller.
The catamaran hull forms developed in this paper has a superior ideal fluid resistance
characteristic over a monohull with a similar mission (they have lower values of ideal
129
Chapter 7. Conclusions.
fluid resistance). The catamarans wave making resistance is also lower relative to the
separate demi-hull wave making resistance, due to favorable interaction between the hulls.
A synergy is obtained when joining the demi-hulls into a catamaran hull since the
catamaran wave making resistance values are better then the values of the separate hulls.
The seakeeping features of the hulls were evaluated in chapter five. The vessels were
evaluated at six different speeds, at four different wave headings: bow waves (waves that
advance towards the bow of the ship with an angle of 1800, as defined by SWAN-2, with
respect to the negative x-axis), oblique waves (waves that approach the ship from the first
quarter, towards the port or starboard sides of the ship, with an angle of 1200 and 150' with
respect to the negative x-axis) and beam waves (waves that approach from the side with an
angle of 900 with respect to the direction of the ship advance). The ships speed were at the
semi-displacement range (FroudeNumbers = 0.5-1.0).
The catamarans were evaluated at the sea states elaborated above with and without lifting
foils. Eight alternatives of lifting foils were examined, all foil in the same location, but
differ in their area and aspect ratio.
Several summary conclusions were gathered during the analysis of the hydrodynamic
features of a catamaran examined here. These conclusions are outlined below.
1. The heave motion of the ship at bow waves includes one peak. For all catamarans,
this peak occurs at the same frequency, where the s/L = 0.3 catamaran has the
highest peak amplitude, and the s/L = 0.5 catamaran has the lowest peak amplitude.
Theoretically they should have had the same response for all frequencies, yet the
growing distance between the demi-hulls damps the response.
2. The interaction between the demi-hulls is obvious when examining the pitch
RAO's for beam and oblique waves. The response at all speeds has two distinct
natural frequencies. As the distance between the demi-hulls grows, it is easier to
observe these two frequencies (also in heave). When the angle of approach of the
waves gets closer to 900, it is also easier to observe two separate natural
frequencies.
130
Chapter 7. Conclusions.
3. The heave response variance decays as the distance between the demi-hulls grows.
4. The monohull has better response amplitudes in pitch than the catamarans at beam
waves, at all speeds. At roll the catamarans has a superior seakeeping, better
then monohull roll.
5. The lifting foils attached to the hull suppressed the heave and pitch motion
amplitude. The foils, in some cases, also reduced the number of peak natural
frequencies of the response. For example, the pitch response at 120' wave heading,
for separation ratios of s/L=0.4 and s/L=0.5 at all speeds included three distinct
peaks without the foils and two pecks after attaching them. In some cases the peak
amplitude is suppressed completely as for the heave response for wave headings of
1200, at Fn = 0.66, for separation ratio of s/L=0.4.
6. A combination of foils, where the rear foils has a smaller area then the bow foils,
achieved the best motion suppression in heave and pitch. It is suggested that
hydrofoil catamarans would be designed with a large foil at the bow and a smaller,
narrower foil at the stern. The unique features of each foil (angle of attack, camber,
thickness, etc.) should be designed separately.
The motions response standard deviation was calculated for each separation ratio, for each
speed at all wave headings. Polynomial expressions were developed to represent the heave
and pitch standard deviation of these particular catamarans.
Wave induced loads were evaluated in chapter six, for bow waves and wave headings of
120*, at three Froude Numbers, for all catamarans. The shear force, bending moment and
torsion moment in five stations along the center-line of each catamaran were measured.
SWAN-2 calculates these forces by direct integration of the pressure over the wetted
surface of the ship. The significant one third highest non-dimensional forces and moments
were calculated at each point. At bow waves, for higher speeds, larger shear forces and
bending moments developed along the hull center line. The magnitudes of the significant
force and moment, at each station, were similar for all three separation ratios. At wave
heading of 1200, some differences are observed between the magnitudes of the forces,
although the trend of the curves is similar. It is also noted at wave headings of 120' that the
131
Chapter 7. Conclusions.
torsion moment values increase in orders of magnitude, as the distance between the demihulls grows.
7.2 Recommended Future Work
The next step following this paper should include validation of results by a model
testing. Resistance, seakeeping and body forces should be measured for the appropriate
ship model, using calm water runs in a tow tank and runs under waves using a wavemaker.
The results should then be compared with those obtained under this paper. A few model
tests were conducted so far for catamarans and even less experiments were done for semidisplacement catamarans; hence their importance in validating the results is significant.
This paper checked one combination of displacement / length of a catamaran. It is
recommended to verify the results and extent the findings to other catamarans with the
same displacement / length ratio. The research can then be expanded for catamarans with
different displacement / length ratios.
Checking following wave headings and their influence on a catamaran behavior at sea, was
not in the scope of this paper. A complete analysis of catamaran in following waves (the
origin of the waves are at the stern of the vessel) and oblique following waves is
recommended. This work is important in order to complete mapping the catamaran
response.
It is recommended to find the optimized shape and contour of the foils that would reduce
the vessel response in the best way. This paper was using flat plate foils without
emphasizing other foil alternatives (cambered foils, angle of attack etc.). It is also
recommended to test the affect of an active control mechanism over the hulls response.
132
Chapter 8. Nomenclature.
Chapter 8: Nomenclature.
sign
description
Units
A
Wave amplitude
[m]
Three-dimensional added mass
[Kg]; [N]
Water plane area
[m2]
Foil aspect ratio
[-1
Two-dimensional added mass
[kg]; [N]
Two-dimensional added mass of the ships' stem
[kg]; [N]
Three-dimensional damping
[Ns/m]
Two-dimensional damping
[Ns/m]
b s
Two-dimensional damping of the ships' stem
[Ns/m]
C,J
Three-dimensional restoring force
[N/m]
Cf
Frictional resistance coefficient
[-]
CD
Drag coefficient on a foil
[-]
CL
Lift coefficient on a foil
[-]
C
Foil chord
[nm]
AR
a s
S
Bj
Frictional resistance
D
force / Drag force / Ship
[N]
resistance
E
Wave energy
[Nm]
F
Foil force
[N]
F,
Froude Number
[-]
Incident force; Diffracted force;
F';FD ;FR;FEX;
FHD ; F HS
Radiated
force;
Excitation force; Hydrodynamic force; Hydrostatic
force
133
[N]
Chapter 8. Nomenclature.
F (t)
Induced force on the ship as a result of ship motions
[N]
F
Grid Froude Number
[-]
g
Gravity acceleration
[m/s 2]
H
Sea bottom depth
[in]
HY
Significant wave height
[in]
H(w)
Transfer function / RAO
[-]
hX
Stream-wise panel size
[in]
h,
Transverse panel size
[in]
I,
Mass moment of inertia
[Kgm 2]
k
Wave number
[m-1]
k
Form factor
[-]
L
Ship length / Lifting force
[m]; [N]
Mi
Sea spectrum 'moment'
[rad/s]
M
Mass
[Kg]
Induced moment on the ship as a result of the ship [Nm]
motions
n, n1
Normal to the surface
[-]
P
Pressure
[N/M 2]
Pa
Atmospheric pressure
[N/M 2]
Rn
Reynolds Number
[.]
Re
Real part of a complex expression
[-]
S
Planform area of lifting foil
[M2 ]
SB
Ships (instantaneous) wetted surface / Ship boundary
[M2 ]
S(w,)
Wave spectrum
[-]
s
Foil span / distance between demi-hulls
[in]
T
Wave period
[s]
T
Modal period
[s]
Tfo01
Foil draft
[m]
134
Chapter 8. Nomenclature.
t
time
U
Ship speed
VP
Wave phase velocity
[m/s]
V9
Wave group velocity
[m/s]
we
Wave frequency
[rad/s]
We
Frequency of encounter
[rad/s]
a
Panel aspect ratio
[-]
a(t)
Time dependent foil angle of attack
[deg]
[s]
[m/s];
[Knt]
Wave angle of encounter,
stability criterion (non-dimensional time
SWAN-2
step)
[rad];
[deg], [-]
General inertia-mass matrix
[Kg]
Ship response phase angle
[rad]
Displacement vector of a fixed point on the body
'5
[im]
surface
# (x, y, Z, t)
#, ; OS
Velocity potential
potential;
Incident
[-]
Diffracted
potential;
Steady
potential
Basis-flow potential
[-]
[-]
77(x, y, t)
Wave elevation
[im]
77o
Basis-flow wave elevation
[im]
Wave length
[in]
Damping parameter strength
[Ns/m]
V
Fluid kinematic viscosity
[m2/s]
0
Ship waves distribution angle
[rad]; [deg]
~1
F;F2D
Three
dimensional
and
dimensional
foil
[K ]
circulation
P
two
[Kg/in 3 ]
Sea water density
135
Chapter 8. Nomenclature.
Ship response motion (translational motions)
[im]
F(t)
Vessels rotational motions
[m/rad]
-R
Variance of ship response
[d
Reduced frequency
[rad/s]
Impulsive potential
[-]
Residual wave potential
[-]
[m2]
/(X)
qp(z, t)
136
2
]
[deg 2
Chapter 9. References and Bibliography.
Chapter 9: References & Bibliography
1) 'Principles of Naval Architecture', Volume II - "Resistance, Propulsion and Vibration"
chapter 5, Edward V. Lewis, 1988.
2) 'Principles of Naval Architecture', Volume III - "Motion in Waves and
Controllability" chapter 8, Edward V. Lewis, 1989.
3) 'Marine Hydrodynamic', J. N. Newman, The MIT press, Cambridge Massachusetts,
1977.
4) 'Computation of Wave Ship Interactions', P. D. Sclavounos, Advances in Marine
Hydrodynamics, 1997.
5) 'The Influence of Lifting Appendages on a High Speed Monohull', Henning Borgen,
MIT Masters Thesis, 12/2001.
6) 'Sea Loads on Ships and Offshore Structures', 0. M. Faltinsen, Cambridge Ocean
Technology Series, 1990.
7) 'Environment Description, Force Prediction and Statistics for Design Applications in
Ocean Engineering', M. S. Triantafyllou, C. Chryssostomidis, Department of Ocean
Engineering, MIT, Cambridge, MA, USA, 1980.
8) 'Ship Structural Design', Owen F. Hughes, SNAME, 1988.
9) 'Surface Waves and their interaction with Floating Bodies', Sclavounos, P. D., course
13.022 class notes, 2002, Massachusetts Institute of Technology.
10) 'Rudder winglets on Sailing Yachts', P.D. Sclavounos, Y. Huang, Massachusetts
Institute of technology.
11)'SWAN-2 2001 Theory Manual', Boston Marine Consulting Inc.
12)'SWAN-2 2001 User Manual', Boston Marine Consulting Inc.
13) 'Low Speed Aerodynamics From Wing Theory to Panel Methods', Kats J., Plotkin, A.,
1991.
14) 'Introduction to Theoretical and Computational Aerodynamics', Moran, J., 1984.
15)'Seakeeping: Ship behavior in Rough Weather', A. R. J. M. Lloyd, 1989.
16)'A Time-Domain Seakeeping Simulation for Fast Ships', D.C. Kring, D.A. Mantzaris,
G.B. Tcheou, P.D. Sclavounos, Fast 97 Proceedings, 1997.
17)'Stability of Transient Gravity Waves on a Discrete Free Surface', Nakos, D.E., MIT
report 1993.
18)'Time Marching Schemes for Ship Motion Simulations', Vada, T. & Nakos, D.E., 8th
International Workshop on Water Waves and Floating Bodies, 1993.
19)'Rankine Panel Methods for Transient Free Surface Flows', Nakos, D.E, Kring, D.C.,
Sclavounos, P.D, 6t International Conference on Numerical Ship Hydrodynamics,
1993.
20)'Calm Water Powering Predictions for High-Speed Catamarans', P.R.Couser,
A.F.Molland, N.A.Armstrong, I.K.A.P.Utama, 'FastShip' conference papers, Sydney
Australia, 1997, volume II.
137
Chapter 9. References and Bibliography.
21)'Resistance Prediction for Fast Displacement Catamarans', S.Steen, H.J.Rambech,
R.Zhao, K.J.Minsaas, 'FastShip' Conference papers, Washington, U.S.A, 1999.
22)'On the Hydrodynamic Characteristics of A high-Speed Catamaran with Round Bilge
Hull: Wave resistance and Wave pattern Experimental Tests and Numerical
Calculations', D.Bruzzone, P.Cassella, C.Pensa, A.Scamardella, I.Zotti, 'FastShip'
Conference papers, Sydney Australia, 1997, Volume II.
23)'Performance of High Speed Multi-Hull Ships', P. Dimitroglou, MIT Masters Thesis,
6/1998.
24)'Seakeeping Analysis for Preliminary Design', P. Couser, Formation Design Systems,
7/2000.
138
Annex A. Ship Characteristics
Annex A: Ship characteristics
A generic, 'numerical' hull forms were used to analyze the resistance, seakeeping
and structural loads of the variable separation ratio catamarans and the monohull. The
catamaran tested had a slender demi-hull shape and a deep transom stern. The following
figure and table describes the main characteristics of the hull.
L
b
;-d
D
L
Figure A.1: Qualitative description of the examined catamaran
139
Annex A. Ship Characteristics
Catamaran
Catamaran
Catamaran
s/L=0.3
s/L=0.4
s/L=0.5
100 [m]
100 [m]
100 [m]
100 [m]
---
11.75 [m]
11.75 [m]
11.75 [m]
22.5 [m]
41.75 [m]
51.75 [m]
61.75 [m]
D (depth)
9 [m]
9 [m]
9 [m]
9 [m]
T (draft)
5.625 [m]
5.75 [m]
5.75 [m]
5.75 [m]
Displacement [ton]
7500 [ton]
7500 [ton]
7500 [ton]
7500 [ton]
-7.19 [m]
-10.19 [m]
-10.19 [m]
-10.19 [m]
1800 [m2 ]
1644 [m2 ]
1644 [m2 ]
1644 [mn]
2250 [M 2 ]
3000 [M2 ]
3000 [M 2 ]
3000 [M2]
-6.43 [m2 ]
-9.00 [m2 ]
-9.00 [m 2 ]
-9.00 [m2
11.25 [m]
15 [m]
20 [i]
25 [m]
26.06 [m]
24.87 [m]
24.87 [m]
24.87 [m]
Characteristic
Monohull
L (length over all)
b (beam of demi-hull)
B (over all beam)
Longitudinal Center of
Buoyancy (relative to
midship)
Water-line area
Wetted-surface
area
(average)
Longitudinal Center of
Floatation (relative to
midship)
Roll
Radius
of
Gyration
Pitch
Gyration
Radius
of
Table A.1: Ship Characteristics
The parallel mid-body of all vessels starts 3 [m] before midship and ends at the transom
stem. the draft and width of the transom stem equals the draft and width at midship. The
cut-off ratio of the hulls is 1, hence no reduction in the hulls cross section at the aft of the
vessel for shafts or propellers.
140
Annex B. Seakeeping Results
Annex B: Seakeeping Results.
Annex B include seakeeping results of the three variable separation ratio
catamarans and the monohull chosen for comparison purposes. The seakeeping results
developed were the heave and pitch RAO's of the vessels. Some roll motion responses
were analyzed too. Roll motions evaluation was not in the scope of this paper, but it was
interesting to investigate the relations between the separation ratio of the catamaran and its
roll response. The monohull roll responses were evaluated too, in order to emphasize the
superiority in roll of a catamaran over a monohull, especially in beam waves. The
following figure presents a s/L=0.4 catamaran, advancing at 45 knots, responding to beam
waves, with wave period of 7 [sec] and wave height of 2 [m].
Figure B.1: Catamaran advancing forward in beam waves
The next figures present the heave and pitch motion responses for the following Froude
Numbers: 0.57, 0.66, 0.74, 0.82 and 0.90. The responses for Fn = 0.99 were presented in
141
Annex B. Seakeeping Results
Chapter 5. All different speeds were evaluated at four wave headings from 1800 (bow
waves) to 90' (beam waves). The roll motion responses are also attached, for all vessels, at
the same FroudeNumbers and also for Fn = 0.99, and for all wave headings.
Roll response - wave headings: 150 deg - Fn = 0.99
2.00
1.80
------------------- ------------------
-----------------
----
1.60 --------------------------------------
00
4
0
9
1.40
- -------------------- - -- - - - - - - - - -- - -
1.20
-----------------------------------
1.00
-----
- -- - -
- - - - - -- - - - - - - --
---
------- -----------------
------------------
------------------
-----------------
--------------------------------------
L
_; ---------
-
-----------------
------------------ ------ I-------------------------------
----------------- ------------------------
---------
- - - - - - -- - - - - -: ---
------------------------------------
--------- ;- --------------------------
---------------- --------------------
----- - - - - - - -- -- - - - - - ------------------ - - - - - - - - - -- - - - - - - - - - - - - -- - - - - - - - -- - ----------- ------
----------------- --- ------------ ----------------- --------------0.90 ------------------- --------------------------- --------------0.60 ------------------- --------- ------ ----------------------------
------- wave spectra/2
monohull
catamaran, 9/L=0.3
-
0.40 -------------------------- ----------------------------- ------------------ -----------s/L=0.4
-catamaran,
----------------- ------------------ ------------------- ------------------1.
--------------0.20
catamaran, a/L=0.5
0.00 -----------8. 0
7.0
6.0
5.0
4.0
2.0
3.0
1.0
0.0
I/L
Roll response - wave headings: 120 dog - Fn = 0.99
4.50
4.00 --------------------------------------------------- ------------------------ --------------------------------------------------- -----------------------3.50 -------------------------- -------
--------- ------------------------ ------------------------ --------------------------------------------------
3.00 -------------------------- ------ ------ab 2.50 -------------------------- I ----
1.50 ----------------------- --
0.50 ---------------------
----------- - ------------------- ----- ------------------------ ------------------------
------------------- - -------------------- ------------------------ ------------------------- ------------------------------------------------------------------------------- -------------------------------- ---------
0 2.00 ------------------------------
1.00 ------------------------
---- ------------------------ ------------------------ --------------------- ---------------------------
------------------------------ -------------------- ----
-------------
--------------------
-
--------- ------------------------ -----------------------I-
------------------------ ------------------------ -------------
0. 0 0
2.0
1.0
0.0
I/ L
3.0
4.0
5.0
6.0
Roll response - wave headings: 90 dog - Fn = 0.99
11.00
10.00 ------------------------- ------------------------ ------------------------ ---------------------- ------------------------------------------------9.00 -------------------------------------------------- ------------------------ ----------------------- ----------------------- ----------------------------------------------- -----------------------8.00 ------------------------- ------------------------ ------------------------ -----------------------------------------------------------------------------------------------------------------------------------------------------------7.00
---------- ------------- - ----------------- ------ ---------------------------------------------- ------------------------ -------------------6.00
0
------------------------
5.00
----------------------- ------------------------ ------------------------ ------------------------ -----------------
4.00
----------------------- ------------------------ ------------------------ ----------------------- ------------------------ ------------------------
3.00 ---------------------2.00 ---------------1.00 -----------
----------------------------------------- ----------------------- ------------------------ ----------------------------------- ------------------------ --------------------------------- ------- ---------
-------
0.00
0.0
0.5
------------------- ------------------------ --------------------------------------------------1.0
X / L 1.5
2.0
Figure B.2: RoH motion response at Fn = 0.99
142
------- --------------2.5
3.0
Amex B. SMweping Results
P itch - 180 dog. wave beading
Heave - 180 deg. wave heading
2.0
L8 ------------------------------------ - - -------- ------------------ ---------------L6 ------------------ --------------- ------L2
------- -- ---- -----------------
-------------
1.4 -----------------
---------------- -------------- ----------------- ---------- ----- -- ------------------------------------- -------- ------------------------- -------------------- elf
-------------------------------- ------------------------- -----------------
1.0
CC 0.8
0.6
0.4
I
----------------I
------------------------- I
----------------;--- ------------------------------0.0 ---------------------4. 0
3.0
1 L 2.0
1.0
0.0
0.2
2.6
2.4
2.2
2.0
1.8
E 1.6
1.4
I
1.2
1.0
0.9
0.6
OA
0.2
0.0
--- -------- ------------ -------------------------- 4----------------------- --------- ------------ ---------------- T--------------------- ---------------------------------------------------------------------------------- 4------- ------------------ - ------------------ --- ----- ----------- 1
--------------------- ..............
-------------- 1----- -------------; ------- wave spectra/2
--------monohuU
--------------catamaran, s/L=0.3
----------- -----------catamaran, s/L=0.4
---------------------------------catam a ran. * /L=0.5
2.5
- - ------------ ----------------2.0
---------------------------- ---- 4-----------------
1.4 ------------------- -----------
1.2 ------------------------------ ------ I---------------1.0 ------------------ -------- ------
00.8 -------------------cc
ir 0.6 ------------------ .....
- ------------------ --------------------- 4----------------- -----------------
-----------------
0.5
-----------------
-----
-----------------
---
-------
--------------- -----
----------
--------
-----------------
----------------
-------------
--
----
---------------- j -----------------
-
. IF
---------- ----------------- -----------------
0.4 --------------------
- -------
----------------- --------
-
1.6 ------------------ - -------------
-------------- ----------------- -----------------------------
0.0
V.V
-------
X/L
LO
0.0
4.0
3.0
2.0
!--------0.0
-
0.2
P itch - 120 dog. wave heading
2.0
1.2
----------------- ----- --
1.0
----------------
E 0.8
-----------
----------
-----------------
-- ------
----------------- -----------------
----------------
0.4
I-----------I----------------- r------------------------------- - - ---
----------------- ----------------- -----------------
-----------------------------------
-----
--- ------------------------------
------------------ ---
1 1.0
0
cc 0.8
(C
0.6
----------------- ------------------------- --------- ----
--------------------------------- --------------------- ------- --------
--------------- -- - --- - ---- ------- --
------------A------------------I
------------ ----------------II --------------------- ---------------- --------------------
----------------
OA --------------- 01
0.0
0.0
X/L
1.0
0. 0
4.0
3.0
2.0
Heave - 90 dog. wave heading
LO
0.9 ------------- ---------- ----------- ------------I--------------------------0.8 --------------- --------- -------0.7 --------
E 0.6
-----
0.5
-----
0
CC 0.4
--------------------------
1.6 -----------
E 1.2
-------
--------- ------------
- -
----------------
-------------
1.9 ------------------ ----------- ------
-1.4
0.6
0.2
4.0
3.0
X / L 2.0
1.0
Heave - 120 dog. wave heading
L4
cc
5.0
4.0
1 L 3.0
P itch - 150 dog. wave heading
Heave - 150 dog. wave heading
1.8
E
2.0
1.0
0.0
-----
0.3
---
0.2
---
------- ------ --------
-, -----
------ -------------- --- ------ --
---
-------- ----- -----------
L----------------- L----------- I
------
- - ----- ------- ------ ----------- ----------- --------------------- L------------L---------- ------ -- ---- ----- ----------- ----------- ---------- ------- ----------- ----------- ----------
-------- - -----
0.1
0.5
1.0
1 / L 1.5
2.0
2.5
P itch - 90 deg. wave heading
2.2
2.0 ----------- I - --------- ---------- ---------- ----------- ----------L8 ----------- - -------- ---------- ---------- ----------- ----------- -------- r---------- r----------------------- r-----------1.6 ----------------- L ------------------------- ;_t
E t4
---- ----- ---------- -------------V L2 - -- - - - - V
C) 1.0
------- ----------r---------- ----------- I
---------- ---------- ----------- ----------4C
cc 0.8
------------------ ---------------------- ---0.6 - -------------- ----------- ----------0.4
0.2
0.0
0.0
0.0
4.0
3.0
X/L 2.0
1.0
0.0
3.0
.0
0.5
"0
1 L 1-5
Figure B.3: Heave and Pitch motion response at Fn = 0.90
143
2.0
2-5
3.0
......
...................
.
Amex B. Seakeeping Results
Roll response - 150 deg. wave heading - Fn = 0.90
2.0
1.8 -------------------- ------------------- - ---------------------------------- ------------------ ------------------- ------------------ ------------------1.6 -------------------- ---------------- - ----------------- - ------------------------ t -------- --------------------------------------- ------------------LA - ------------------ --------------
- -- --- -- ---------- ------------------
------------------- -----------1.0 - ------------------ ----------0
------------------90.8
0.6
------- %wvespectra/2
monohull
--------- z------- -- --L------------------ ------------------ j---------------- ;-
-----
------------------ ------------
------ --------- ------------------ ------ I--------------------z---,
---------------- -r
--------------- I ------------------ I-------------
r ------
-------- ------------
catamaran, s/L=0.3
--------------------- -------------
catamaran, s/L=0.4
catamaran, s/L=0.5
-----.-
-------------------------- ----------------------------- ----------------- --------------- ----- --------------------------------------------------------------------------------------------- -- ------------------ ------------------
0.4 - ------------------ ----
0.2
-------------
0.0
0.0
2.0
1.0
3.0
1/L
4.0
6.0
5.0
7.0
8. 0
Roll response - 120 deg. vave heading - Fn = 0.90
4.5
------- --------------- --------------- ------- ------- ---------------
-------------- L------
----- ----------
4.0 ----------
------------ -------------- -------------- ---------------
3.5 ---------------- ---------
----------------------------
3.0 ---------------- --------------------- - ----- - - -------- -------------- -------------- --------------- -------------- -------------- -------------A
2.5 ---------------- -------------- --- - -------- ------ - ---- -------------- -------------- -------------- --------------- -------------- ------------------------- -------------- --------------- --------------- -------------- --------------..............
------------- - -------------- ------------------- ----1.5 ------------------------------- - ------------ -------------- -----------
2.0 -----------------------
1.0
---------------
---------------------------------
0.5
------------
-------------
-- -
----------
-------------- T --------------
-------------------
---------------
--------------------------
----
-------------- T
-------------- ------ ---------
-------------- -------------- T
----------
0.0
0.0
0.5
1.0
2.0
1.5
1
2.5
3.5
3.0
4.0
4.5
5.0
11.0
-
Roll response - 90 deg. vmve heading - Fn = 0.90
10.0 -------------------------- ------------------------------------------------------------ ------------------------------------------- ------------------------9.0 ----------------------------------------------------- ------------------------ ------------------------- ------------------------ ------------------------8.0 -------------------------- ------------------------ - ------------------------ --------------------------L------------------------ ------------------------7.0 -------------------------- ------------------------- ------------------------ -------------------------------------------------- ------------------------V
6.0 -------------------------- ------------------------ ----- ------------------ ------------------------- ------------------------- ------------------------5.0 -------------------------- -------------------------- ------------------------ ------------------------- ----------------- ------- ------------------------4.0 -------------------------- -------------- -------------------------------------------------------------- ------------------------ ----------------------------------------------------------------- ----------------------------------------"
3.0 ----------------------- --2.0 ---------------1.0
-----------
------ /---
------------------- ------------------------ -------------------------
-------- ----------------
----
-------------------
------------------------
-------------------------
-
-
-- Q---------- T -------------------
0.0
0.0
0.5
1.0
X/L
1-5
2.0
Figure BA RoH motion response at Fn = 0.90
144
2.5
3.0
Am" B. Sa&eeping Results
P itch-180 dog. wave heading
Heave - 180 dog. wave heading
2.0
is
---------------------------------
1.6
----------------- ------------
14
----------------- -----------
---------, ------------- ----------------
----------------- --------
------
-------------------------
a:
-----------------------
0.6
2.0 -------------- ------- --- - ----- - ------------ --------------------- ----- ----- ------------ i---------------1.6 ------------- wave spectra/2
1.2 - ------------- --- ------monohull
--------------------------------
x
---- ------- ----
---------------------------
L2
E to
0
ICE 0.8
-------------
---- _
------ ---
-------------------------
----------- ------------ -------------
2.4 ----------
- - ------- ------------
-----Ir
----
0.4
--------------------
---------------- -------------------------- ---------------- ----------------
0.2
--------------
----------- ---------------- ----------------
4
0.8 - ------------ 1- ------- -------------------- ------
0.4
4.0
3.0
2.0
X/L
to
0.0
to
L 2.0
3.0
4.0
5.01
P itch-150 dog. wave heading
Heave-150 dog. wave heading
I
-
2.5
-----------
-
---------
----
-
-
-----------
-! ---------------
-
-----------------
IA
----------------
-1.2
------
----------------- --------
------
----------------
2.0
-----------------------
------------------------------ -------- ----
----------------------
(COA
----------------
0.4
-----------------4--
0.2
-----------------
------ j-----------------
------
-- --- -----
---------------- ----
-----------------
---------
0cC to
--------- ------
---------------- -- ------------
---------- -------
------------
----
----
-------
-----------
--------
---------------- -------
-------
------ -----------------
1.5
E
E
Oos
---------- ----------------------------------
0.5
-------------------------------------------
---------------
--------------- ---------------- ----------------0.0
0.0
3.0
2.0
1.2
2.0
ff A
1.8 ------------------ --------------- ---------------- ---------
------- I--------- ----------------
------------------
to ------------------ - -
------
--
--- ------ -
---------------
- ---------------------
-----------------
-------- I -------
1.4
-------------------
-----------------
1.2
-----------------
to
0 0.8
<CC
IT
----------------------------- -----------------
0.6
0.4
0.2
--------------
---------------------------------
-----------------
- - -- - - - -
---- - - - -
---------------
-
----- 41 , ---------
------ J-
L -------
----- - --------
-- ---
----------------
---------- i----------------
------------------------
----------------
-----------------------
-
--
-------- ---------------- ---------------
------
-
---------- 4---------------- 4----------------
0.0
4.0
3.0
) I L 2.0
to
.0
0.0
X/L
to
4.0
3.0
2.0
P itch-90 dog. wave heading
Heave-90 dog. wave heading
2.2
1.0
----------
0.8
----------
0.7
----------
0.6
0.5
0
CCOA
cc
0.3
----- - -
0.2
0.1
0.0
- - --
-
-------------- - -------
----------------
-----------
---------
0.2
0.0
0.9
--------------- ----------------- ---------------
1.6
----------------------------
E
00.8
0
< 0.6 ----------------ir
0.4 ---------------- - -
4.0
3.0
2.0
P itch-120 dog. wave heading
Heave-120 dog. wave heading
L4
X/L
1.0
0. 0
4.0
-
X/L
1.0
00
----------- ---------- ---------
9/L=0.5
0.0
0.0
1.6
c atom aran, s /L=0.4
-catamaran.
0.0
is
catamarans/L=0.3
----e--
- ----- -
-
-
-------------------- ----------- ------
-------
--
2.0
1.8
------ ------------
-- ----- -------
-----
- - ---- -- ------- ------- --
--- ---
-------
---- ---- ----------- ----------------------- -- -- ---------------------------------------- - - -------- -- ---- ------ ----------
----------- ----------
- -------------------- ----------- ----------
---
1.4
--------------------------
------ --- -- ------- -----------
---
1.6
----- ----------
4
.2
1.0
---------- ---------- ---------------------- ---------- ---------------------------- - ------------------ -- -------- ---------- I
------------------- ---------- ----------------------- ------
- - ---------------------------- L ------ ;- ------------------------------------- - -----------
0.8
---------- T ------ ; - ------------------------ ------ ----- r ---------------------
0.6
0:4
0.2
----------- L---------- L-------------- ------
0.0
0-0
0.5
1.0
)L/L 1.5
2.0
2.5
3.0
0.0
0.5
1-0
1 / L 1-5
Figure B.5: Heave and Pitch motions response at Fn = 0.82
145
2.0
2.5
3.0
Annex B. Seakeong RemIft
Roll response - 150 dog. vvave heading - Fn = 0.92
2.5
2.0 -------------------------- t------------------------ ------------------------ ------------------------------------------------------------------------------
1.5 ------------------------------------------
------ - ------------- --
------------------
-----------------------wave spectra/2
0 1.0 ------------------------------------
----------
------- -
------ ------------------------
monohull
catamaran, s/L=0.3
0.5 --------------------------
- - - - - - - -- - - - -- - - - - - - - - - -- - - - - - - -- - - ---- - -- - - - - - - -- - - - -
-
catamaran, s/L=0.4
-
catamaran, s/L=0.5
0.0
0.0
1.0
2.0
)./L
3-0
4.0
5.0
6.0
0.82
Roll response - 120 deg. wave heading - Fn
-
4.5
4.0
-- T -------------- r -------------- I -------------- f -------------- I -------------- T -------------- I --------------
---------------- T -------------- r -----
3.5 ---------------- -------------- L---- ---- - -------------- L-------------- i-------------- It -------------- i-------------- f-------------- ----------------------------------------------- -------------- ------------------------------ T-------------- T -------------- I -------------3.0 ------------------------------r- -- -------to
2.5
------------- -------------- L-------------- I--------------
----------- ---- - ------ --------------
-------------- --------------
--------------------------- -------------- ----------------------------- ------------------------------- -------------------------------------------------- -------------------------------- --------------
-------------- ------------- I- ----------- ----------
0 2.0
1.5 ----------------------------
--------------- -------------- -------------- ------- ------------------------------------ --------------
-------------------------- --------------
1.0 ------------------------0.5 -------------- ----
-------0.0
0.0
0.5
1.0
1.5
2.0 IL/ L
2.5
3.0
3.5
4.0
4.5
5.0
-
Roll response - 90 dog. wave heading - Fn = 0.82
11.0
10.0
-
-------------------------- ------------------------
------------------------------------------------- ------------------------ T ------------------------
9.0 ------------------------- -------------- ---------- r------------------ ----------- ------------- f------------------------ ------------------------ --------------------- -----------------------8.0 -------------------------- ------------------------ ------------------------ L-------------------------------------- ------------------------ -----------------------7.0 ------------------------------------------------ ---------------------
0006.0 -------------------------- i------------------------ L----- ------------------ i------------------------------------------------- ---------------------------------- ----------------- ------ -------------------------------------------- ------------------------0 5.0 -------------------------- T------------9 4.0 --------------------------------------------------- ------------------------ ------------------------ ------------------------ -----------------------3.0 ----------------------
--i- ---------
2.0 ------------------------
1.0
-----------------
--
-
---
r ---
------------------- r------------------------ ------------------------
-------
------------------- I ------------------------ I ----------------------L
t
-------------
--------------------------- -----------------------------------------
-:
-
-
-
- -
-
-
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-------------------
-------------
--
0.0
0.0
0.5
1.0
X/L
1.5
2.0
Figure BA RoU motion response at Fn = 0.82
146
2.5
3.0
Aimex B. Seakeeping Results
P itch - 180 dog. wave heading
Heave - 180 dog. wave heading
3.0
2.5
---------------- -------------
2.0
ELS
I
0
cc to
----------------- ----------
----------------
0.5
------ ---
--------
2.0
------------- ----
1.5
0
cc
ir 1.0
-----------------
----
------- ----------------- -----------------
----------------
0. 0
-
5.0
4.0
------------
1.4 ----------------------------
-------------------------------7;
- ------------------- - ---------------------------- ----------
-- --------- -------------------------------- ----- ----1.2
E 1.0 ------------------------ -------0
,CC 0.8 ----------------------- ----------------------------- ----------------0.6 ------------------------------------------- ----------------- --------- ----------------- ----------------I
0.4 -------------------
-- ---------- ---------------
2.00
--------------- ----- -
L50
--------------- --- ----------
0
.CC too
0.50
------------------------------
----------------
--
---
-------------- -------------- ----
---------------
----------------------------------- ----------------
0.2 ------------------
0.00
-
0.0
4.0
3.0
2.0
X/L
1.0
0.0
-------- I
--------------------------------
-
1.6 ------------------ - --------
0. 0
I /L
to
I
2.0
1
1
3.0
4.0
I
P itch - 120 dog. wave heading
Heave - 120 deg. wave heading
2.0
L4
-------------------! --------------------------
1.2 ----------------------
1.8
-------------------- ------------- ---------------- -------------
1.6
-----------------------------
--------------- L ----------------
L4
- ---------------------------------------------0.8 ---------------- -E
0
-cc 0.6 ----------------- - ------------- ----------------- -----------------
ap1.2to
-
to --------------------
Ir
--------------
0.4
jf
0
---------------- ------------------------------------------------------------------------------------------ --
0.8
--------------
0,6
-------------
------- ! -------- ----------------
----
--------------- - ---------
------------- ----------------
-- ----
--- --- ------
------ ----------------------
-----------------------
0.4
---------------- -----------------------------------
0.2 --------------
---------------------------------
0,2
- -
-------------------------------------------------
0.0
0.0
4.0
3.0
2.0
1/L
1.0
0.0
1.0
0.0
XIL
4.0
3.0
2.0
Pitch - 90 dog. wave heading
Heave - 90 dog. wave heading
2.0
1.2
-------- -----------
-----------
---------
-------------------
-----------
--- ------ --------
E
-----
- -- - -------
Ir
-
0.2
M
3.0
2.50
1.8 ------------------
-E 0.6
0
------
P itch - 150 do g. wa ve he a ding
Heave - 150 dog. wave heading
-0.8
----------------
X/L 2.0
1-0
0.0
catatnarans/L=03
---- e--catamarans/L=0A
-- acatamaran. x/L=0-5
----------------- ----------------------
0.0
2.0
to
------- ---------- --------- --------t---j
------- wave spectra/2
--------monohufl
------------
4.0
3-0
2.0
) /L
to
-------------
0.5
------------------
0.0
------------ -------------
----- ------
-----------------
---------- --------------- -------
------
--- ----- ------------ -------------
2.5 --------------- ------- --
------------- r-----------------
-----
---
----------- -----------
---- --
-
------------
----------- -----------
-
---- ------ ----------- ----------- -----------
0.5
to
---------- --------------------------------------------
L6 ----------
--------- --------------------- - --------------------------- ---------- ---------- ------------------------
L4
-------
1.2
------
1.0
-----
0
cc 0.8 ---Ir
0.6 -----0.4
---
X / L 1-5
2.0
I
2.5
I
3.0
------- ---------- ---------------------- ----------
-------------
0.2
0.0
0. 0
1.8 -----------
-------------------------- I-------- ----------------------------I---------------------
----------------------
0.0
0.0
0.5
1-0 X/L 1.5
Figure B.7: Heave and Pitch motion response at Fn = 0.74
147
------ ---------- ----------
---- - ---- - ----------------------
2.0
2.5
3.0
Annex B. Seakeeping Results
Roll response - 150 deg. wave heading - Fn = 0.74
2.3
2.0 ----------------------- ----------------- ----------------------------------------------- -----------1.8 ------------------------------------ - --
----------------------- ----------
1.5 ---------------------- - ---------- --------------------------- --------
1.3
--
---------------------------------------------
-------------- :---------------------------------------------
----------------;f ------------------------- -------------------- ---------------------
- --------
-------------------
----------- -------------- ------ -----------------------------;-------------------------------------
V
0
9
1.0 ------------------------------0.9 ----------------------- ------
--------
----------------------------------------------- ------- wave spectra/2
----- - --------------------#--k ---------------- - f---------------------- I--------------------------monohuff
0.5 ---------------------- - -- ----- --------- -------------------- -------------
catamaran, s/L=0.3
- -----------------
0.3 - -------------------- ------------------------------------------- --------------------- --------------------
-catamaran,
s/L=0.4
catamaran, s/L=0.5
0.0
0.0
1.0
2.0
3-0
X/ L
4.0
5.0
6.0
7. 0
Roll response - 120 deg. wave heading - Fn = 0.74
5.0
4.5
------------------------------ --------------- --------------
-------------- --------------- -------------- ------------------------------- --------------
4.0
--------------- --------
--------------
3.5
---------------
Vip
0
9
--------------- ---------- --------------- ------------------------------- -------------- --------------- I---------------
--------------- --------------
A 3.0
-------------- -------------- -------------- --------------- --------------
- ---------
2.5
--------------- -------------
2.0
---------------------------- - - ------------- ---------
1.5
--------------- ---------- - - --------------- --------------
1.0
--------------- -------
0.5
---------------- ----
-----------
--------- -------------- -- ------------ -------------- -------------- --------------- --------------------------- --------------------------------------------- ----------------------------------------
------------------ r------------------
------- -------------
--------------
------------- --------------- ------- --------------------- ---------------------------------------------
--------------- ------------------------------
-----------------------
------------- --------------
---- -------------- --------------- --------------
T ------------------- ----------- -------------- --------------- --------------
0.0
0. 0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Roll response - 90 deg. wave heading - Fn = 0.74
11.0
10.0 ------------------------------------------------------------------------------ ---------- ------------- - ------------------------ -------------------------------------------
9.0 -------------------------- ------------------------- ------------------------ ----------------- I------ - -----------------------8.0
7.0
A
00 6.0
0
0
------------------------ -------------------------- ------------------------ --------------------------------------------------- ------------------------------------------------ ------------------------- ------------------------ ------------------------
-------------------- ------------------------
-------------------------------------------------- ----- ------------------ ------------------------ ------------------------- -------------------------
5.0 --------------------------- ------------------------- ------------------------ -------------------------------------------------- -----------------------4.0 -------------------------- ------------------------- ------------------------ --------------------------------------------------- ------------ -----------3.0 - -------------------- --4-------------------2.0 - --------------1.0 - ------- -
-------------------------- I------------------------ -------------------------------------------------------------- ---------------------
---- ----------------------------------------
------------
---------------------------------------
0.0
0.0
0.5
1.0
1 L
1.5
2.0
Figure B.S: Roff motion response at Fn = 0.74
148
2.5
3.0
Annex B. Seakeeping Results
P itch - 180 deg. wave heading
Heave - 180 deg. wave heading
3.5
2.5
- ------- wave spectra/2
monohuU
3.0 -------------- ------- -----------------------------------
2 O ------------------------------
-catamaran,
2.5 -------------- ------
-
1.5
E
0
-------- --------------- ------------------
-------------------------
E
W.0 - ----------------
s/L=0.3
c a ta m a ra n, s /L=0.4
c a ta m a ra n. s /L=0.5
------
-cc 1.0
-
-lob
0 L5 ------------cc
------------ --------------
----------------- -----
---- -----------
------------------ - ----------- -------------------------------------------------------------------------------- -------------
1.0 - ---------------------------- -- ------------------------------- ------------------------ ------------------------------ ------------------------------ ------------------1.0
0. 0
X/L
-------------
0.0
4.0
3.0
2.0
-----------
0.5
-
0.0
-
0.5
4.0
3.0
5.0
-
3.0
-----------------------------------
1.8 ------------------ --------1.6 ------------------ j------1.4 -------------------------
-------- ----------
-----------------
--------- -----------------
0
9C 0.8 - ---------------- --- ---------- I_* ------------------ ----------------ir
----------------- ----------------0.6 - ---------------- -- ----------0.4 --------------------------------------------
-2.0
E
M.
4p
'0 1.5
0
-cc
Ir 1.0
---------------------
0.5
--------------------------------- -----------------
0.2 ------------------
--------------------------------
---------------- ----- -
2.5
------ ------------------ -----------------
1.2 - ----------------- ----- ----------1.0 - ---------------- ---- ------------ I ;--_
-- -----
--------------- 4---
----------- ----------------------- __ -------------------------- -------
----------------
-----------
-------------
--------
-------------- ---
----
------ i--------------- j----------------
0.0
-
00
L
2.0
P itch - 150 deg. wave heading
Heave - 150 deg. wave heading
2.0
--------
1.0
0.0
------------------------
1.0
0.0
X/L
2.0
4.0
3.0
X/L
to
0.0
2.0
4.0
3.0
P itch - 120 deg. wave heading
Heave - 120 deg. wave heading
- -- - - - - - - - -- - - - - - - -- - - - - - - - -- - - - - - -- - - - - - - - -- - - - T - -- -- - - - - - - -- - -
-
2.0
1.2 ------------------- :__
-
-
14
ir
-
---------- 4-----------------
1.0 ------------------
E OS ------------------------- --------------------- ----------------'9
0 0.6 - -------------- ---- ---------------------------CC
---------------- - ------------------------------- C_ -----------------
-
------- ------ ----------------------------
-------------- -- --- ------
1.5
4p
0
------------- ----------------
Ir
0.4 --------------- - -- ----- --------- ----------------- -----------------
------------ ----------------
0.5
0.2 -------------- - - -------- -------- -------- --------- -----------------
1 /L
1.0
0.0
3.0
2.0
0.0
4.0
to
IX
1.4
to
----------- - -------- ----------- ------------------ ---------I -----------
0.8
---------
0.6
------
L2 --------to --------
--- ------ ------------------ --4---- ---- -- ------
---------- L ---------- I ---------- I---------gtpt ----------- ----------------------------------
-----------------
T
0.8
0 0.6
---------- ----------
-----
------------------
----
---- ------------ -- ---- ---------- ----------
-- --- -------------- --------------------- ----------- -----------
0.4
0.2
4.0
3.0
2.0
----
-----
---
--
----------- -----------
0.4
---
---------
0.2
--
---- ---- --
0.0
0.0
0.0
0.5
1.0
X/L 1.5
2-0
2.5
3.0
_-t ------------ -----------L----------
-
E
I/L
P itch - 90 deg. wave heading
Heave - 90 deg. wave heading
L2
0
----------------------
------
---------
0.0
-
0.0
.0
0.5
to
L 1.5
Figure B.9: Heave and Pitch motion response at Fn = 0.66
149
2.0
2.5
3.0
-
NEW
----
---
----
-
=-, "
,, ---
Annex B. Seakeeping Results
Roll response - 150 dog. wave heading - Fn = 0.66
2.5
------- wave spectra/2
monohull
2.0 ---------------------- --------------------- --------------------- -------- ------------- ----------------------
catamaran, s/L=0.3
----"-catamaran, s/L=0.4
--------------------- ---------------- .. r
L
Catamaran, s/L=O .5
A 1.5 --------------------------------- ------ - ----------------00
0
V
0
1.0 ---------------------- ------- ----------9
---------- T --
- -
------
-- - - - - - - -- - - - - - -- - - - - - - -- - - - - - - -- - - - - - - - - -- - - -
- - - - -- - - -
- - -- - --I - - - -- - - - - - - - -- - - - - - - -
- - - - - - --
-
- -- - - - - -- - - - - - - - - -- - - -
------------------------------------ --------------------------
-
0.5
------------------
0.0
1.0
0.0
2.0
3.0
X/ L
5.0
4.0
6.0
7. 0
Roll response - 120 dog. wave heading - Fn = 0.66
6.0
5.0 ---------------- -------------- 1-------------- -------------- --------------- -------------- -------------- -------------- --------------- -------------4.0 ---------------- ---------------- - -- -------------------- --------------- -------------- -------------- -------------- - ------------- -------------3.0 ---------------- -------------
-------------- --------------- -------------- -------------- -------------- --------------- --------------
--------------- ----- -
2.0 ---------------- ----------1.0 ---------------- ------
--------
-------- --------- -------------- -------------- -------------- --------------- --------------
---- -------------- -------------
------
-------- ----------
----------------- --------------- --------------
0.0
0.0
1.0
0.5
1.5
2.5
2.0
3.0
3.5
4.0
4.5
5.0
X/L
Roll response - 90 deg. wave heading - Fn
0.66
-
11.0
10.0 -------------------------- ------------------------ ------------------------ ------------------------ ------------------------- -----------------------9.0 --------------------------------------------------- ------------------------- I------------------------- I---------------------------------------------------
V
0
8.0
-------------------------------------------------
7.0
----------------------
6.0
T -------------------------I --------------------------------------------------- -------------------------
------- ----------------- ----------------------------------- ------------------ ------- : ------------------------------------------------ ------------------------ ------------------------ --------------------------:---------------------------------------------------
5.0 --------------------------------------------------- ------------------------ ------------------------- I------------------------- ------------------------------------ ----------------------------------------------------------------------------------------------- -----4.0 --------------------3.0 -------------------- ---------------- --2.0 ----------------1.0
---------------------
0.0
0.0
-----------
----------------------------------------------------------------------------- ---------------------------------------
---------------- --------
---------------------- --------------------------------------------------------- ------------------- ----------------------------------------I
---- ----------- -------0.5
1.0
X/L
1.5
2.0
Figure B.10: RoU motion response at Fn = 0.66
150
2.5
----3.0
Aimex B. Seakeeping Restdts
P itch - 180 deg. wave heading
Heave - 180 deg. wave heading
3.0
2.0
11 L
1.0
0.0
------- wave spectra/2
monchull
------------- ---- -- --- --catatn aran, s/L=0.3
catam aran, s/L=0.4
---- *------------- --- ----catam aran. s /L-0.5
------------ -- -------
3.0
2.5
E
-CA 2.0
a: 1.0
3.0
1.8 ------------------ --------- ----------------- ---------------------------------- ---------------L6 ------------------------ ----
2.5
----------
- ---------------- r---
to
-
-Cc 0.8
cc
0.6
-----------------------
---- -------
L2
----------------- ----------------
-----------
-----------
ME 1.5
0
4
Ir to
---------------
----------
---------------
------------
---------------
--------------
0.4 -----------------0.2 ----------------
-------------------------- ----------------
0.0
---------------
--
-------------
-------
0.5
---------------- L ---------------
--------
------------
0.0
X/L
1.0
0.0
4.0
3.0
2.0
2.0
4.01
3.0
P itc h - 120 de g. wave he ading
-----------------
2.0
1.8
------------- ----------------
L6
---------------- ---------- ------ --------------------------------
-------- ----------------
1.4
1.2
-------------- -
1.2 ------------------ - --- ----------- -----
to -----------------
0 0.6
1/L
2.2
1.4
0.8
1.0
0. 0
Heave - 120 deg. wave heading
- --- ----
-------------------------- -
0.4
------------
0.2
----------
0.0 t
0.0
------
-
-------- ---------------- f----------------
- ----------------- -------------------------- ----------------------- ---------------- ----------------
------------
---- T---------------- ----------- ----- ------------------------------- ---------------- --------------- ------------------------ --------- ;.r --------------------
--------------------
to
0.9
ir
0.6
------------
- --
------ - - - - - - - -I - - - - - - --
-------
-------
------------- -- -- -------------
OA
0.2
-- - - - - -
-----------------------------
--- - ------
------------
L
---------------
----------------
------
---------------
---------------
---------------
----------------
0.0
L
1.0
4.0
3.0
2.0
X/ L
1.0
0.0
4.0
3.0
2.0
P itch - 90 deg. wave heading
Heave - 90 deg. wave heading
L4
-
L2
1.0
0.8
---------------
T ------------------------------------------- :----------------- -----------------
-
---------------
2.0
I
- - - - - - - -- -
-
- - - -- - - - - - - - -- - - - - -
- ------------------
-----------------r ---------------
---------------
-------- ----------------
-------
----------
5.0
4.0
X / L 3.0
P itch - 150 deg. wave heading
Heave - 150 deg. wave heading
2.0
14
2.0
to
0. 0
------------
------------ ------------ -------------
-----------
0.0
0
------------ ---------
--------- W ---------- 1-----
-------------
0.5
4.
------ --------
------------
0 1.5
cc
-
0cc
cc
2.2
---------------- ------------------------------ -------2.0
---------------- --------------1.8 ------------------------- ------------------- ------------1.6 ---------------- ------ ----------------- --------------------------------- 4----- --------1.4
-- --------- ----------------------------- ---- ---------L2
------ -------------------- ---- ----------to
0,
--------------0.9
--------------- --- ----------- ---------------------------------------------0.6 ------------------------------------ -------------- ---------------- ---------------0.4
-------------- ---------------- --------------0.2 ----------------0.0
3.5
----------- L ---------- I--- E ------ J -----------
-- ------ --------------------
--------
E 0.6
---
L2 - ------- 4-% ---------------------- ---------- -----to - ------
-- ----
W
"ai 0.8
----------
------------------------------ --------- I-----------
-----
-----------
0
cc
0 0.6 -----cc
0.4
---------- ---------- -----------
0.2
-----------
- --------- ---------------------- - --------- -----------
r---------- ---------- -----------
0.0
--- -----------------------------4 ----
0.4
---------- --------- ----------
-- ------
---------
0.2
0.0
0. 0
0.5
1.0
X/L 1.5
2.0
2.5
3.0
0.0
0.5
to
L 1.5
Figure B.11: Heave and Pitch motion response at Fn = 0.57
151
2.0
2.5
3.0
Annex B. Seakeeping ResWts
Roll response - 150 deg. wave heading - Fn = 0.57
------------------------ ------------------------
2.5
---------------------------
2.0
------------------------ ------------------------ ------------------------- -----------------------
---------
1.5
------------------------ --------
1.0
------------------------
- -----
--------
------------------------
-------------------------------------------------
----------------
-----------------
-
------------------------
-------------------------
....... mave spectra/2
0
9
---------------------------
------------------
------------------
------------------------------------------ ---------
monohull
catamaran, s/L=0.3
0.5 --------------------------
----------- -------------------------r -------------
--------
----------
catamaran, s/L=0.4
catamaran, s/L=0.5
0.0
00
1.0
2.0
3.0
4.0
5.0
6.
0
Roll response - 120 deg. wave heading - Fn = 0.57
6.0
-
----------
------ ---------
5.0 -------------------------------- -------
---------- -------------------
--------------
4.0 --------------- ------------- ------------- -------------- -------------- f-------------------------------------------- -------------- 1-------------3.0 --------------- ----------- - - -------
----------------- -------------- -------------------------------------------- -------------- --------------
2.0 ------
-------------- --------------
-------------- -------------- -----------------
----- ---------- --------------------
1.0
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
I/L
Roll response - 90 deg. vwave heading - Fn
0.57
-
11.0
10.0 -------------------------- ------------------------ ------------------------ ------------------------ ------------------------ ------------------------
-
9.0 -------------------------- ------------------------ ------------------------ ------------------------ ------------------------ -----------------------8.0 ---------------------------- ----------------------------------------------- ------------------------ ------------------------ ---------7.0 ------------------------------------------------------------------------------------------------------ ---------------------------------------6.0
------------------------ ------------------------ ------------------------ ------------------------ -------------------------------------------------
0 5.0 -------------------------- ------------------------ ----------------------- ------------------------ ------------------------ -----------------------4.0 - ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------3.0
------------------- ----
2.0
---------------
1.0
--------- - I -
------------------------ ------------------------ ------------------------ ---------------------------------
---------- -----------
---------------------- ------------------------ --------------------------------------------------------------------------------------------
4 - - -- -- - - -- - --
-
0.0
0.0
0.5
1.0
X/L
1.5
2.0
Figure B.12: RoH motion response at Fn = 0.57
152
2.5
3.0
Annex C: Loading Results
Annex C: Loading Results
In the following annex, the forces and moments RAO graphs are presented. As elaborated
in chapter six, the exciting forces were calculated for all variable separation ratio
catamarans, at three different advancing speeds, at two wave headings. The abscissa in the
graphs is the non-dimensional value of AIL, the y-axis is the non-dimensional RAO as
explained in chapter six. The forces and moment were evaluated at five different stations
along the center-line of the catamaran.
153
Annex C: Loading Results
Shear Force
-LI2 Fn=0
@
--------------
2 .5E-0 I
------------- -----
2.0 E-01I
--------- -
2
- --
-- -- --
.5E-0 I
- ---------
---------
--2
1.0
--------
-.----------X/L 3.0
.0
4.0
------- -------- _
-6.0
5.0
Moment
Bending
/L
2.0
15
0.0
-L/4, Fn-0 .66, wave
@
6.0
5.0
4.0
3.0
heading: 150 deg
/L- .5
----------
--------
---- ---------------------- --------- ------------- -- -----
------
L.0E-01
----------
1
------ ----
5.0E-02
4-
1.0
Shear Force
3.5E-01
@
I/L
2.0
6.
3.0
4.0
3.0
0
0
0--
-------
------
2.5E-0I
-------
- --
-----
----------
midship, Fn-0.66 wave heading 180 deg
Bending Moment
.-- -s/
a---.--I/L-0 .3
3
-
1.0E-01
-------
------
-------
---
B.OE-02
2 .DE-0 2
-
----
0.0E+00
Shear Force
1/L
2.0
1.0
5.0
4.0
3.0
Bending
S
--
----
4.0E-02
O
------------------
---
---------------
Shear
@
+L/2,
F n=0 .66,
wave heading:
100
L-0
.3
/L-0.4
2/L-0.5
tO
M
aoment
@
------
-
-------------
----200
-
Bending
---------
---------
------
0.0
dog
----
------
0.0E+00
Force
deg
heading:10
------------
---
-
-------
--------
1.5E-02
6.0
5.0
4.0
I/L 3.0
2.0
1.0
6.0
-------------------------- ---- -------
-----
------------
5.0E-03
-------
0.0
5.0
4.0
3.0
Moment@ +LI4, Fe-0.66 wave
1.0E-02
------------------
----------------------------
2.0E-02
2.0E-02
--------
-----------------
- ----
/L
2.5E-02
---------
---
2.0
-=--e/
.4
s/L-0.5
.0-02----------
---
3.0 E-0 2
./L-0 .3
----------
1.0E-01 ------------------
------
- ---- ----------- ------
------- - -
-1.0
0.0
6.0
+L/4 Fn=0.66, wave heading:10 deg
@
1.2E-01
---------
----
---------
---
-
-
- - -
-
-- -
--------
------
4.0E-02
0.0
----------
----
-------
--- -----
-----------------
-------------
-
-- ----- -----
0.0E+0
L-0.4
0/L-0.5
------- --------- --------- ---------
----
------
6.0E-02
----- -- ------
-
---------- -------------------
----- --- -----
-- -----------
----
-
---------
-------
------
--- ,-
t./L1-.t
--'r2.0 E-0 I---------
1. E 05.0 E -
6.0
5.0
4.0
3.0
1.2E-0
3.OE-01
.C -0
1/L
2.0
1.0
0.0
10 deg
midship,Fn=0.66,wave heading:
. E +0
-
0.0.+00
0.0
--
- -------
-----
-------
--------- }-- -- -
-------
--------------------------------
------------
-----------
-
- 0 0 +O
.0E-02.0
-----
---------
-------
------
----
---
1.5E-0
-C
-
-
e--S/L-0.4
s/L-0.5
-
--
--
- --
---
--
2.OE-01 -- ------- ------- -------
--
- --------------- '-- ----- -------
n/L-0 .3
---
I Li-0 .3
--2.0E-01
- ------ ------ ----------- --.E- ------------
----------------
5.0
+L/2, Fn-0. 66, wave
heading,:
6.0
180 deg
7.0E-03
1.2E-01
sL-0 .3
.4
s/L-0.5
alL-S
4.0E-02
----------
2.OE-02
--
S.E+
---
0 .0
--------
0.0
-
-------
---- + --
-------- --------- +
1.0
- ---------
X/L
3.0
-----
-
----------
--------- --------
-
-
--------
----------2.0
4.0
+
5.0
----------
2.OE-03
----------------
-------- --------- ------
------
3.0E-03
L.0E-03
--------
-
---------- --------
4 .0 E-03
----
sL-0.4
-------- --- -------------- --------- y---------
- --------- --------
------- - ------ -
- - -
-
----- -------
/L-0 .3
------
.0-03s/L-0.5
.0-03-----------------
--------- -------- - ------ ----- --------- --8.E02
4.01-02
6.0E-02
--------- ---------- --------- -----------
-0.0 E+00
6 .0
-
0.0
-
------
----
---------------
1.0E-01
6 .0 E-0 3
s/L-0.4
-
-
-
--
2.5E -0 1
3.0E-01
3.0 E-0 I
2.5E-0
0
-----
- ----------
-
-
-
---------
-------
0 .0 E+00
Force @ -L/4,F-5.66, wave heading:150 deg
3.0E-01 ---------
0.0 E1+0
----
------
--- ------------------
-
- -
---------
LOE-0I
4.01-St--S=.4
6.0E-02
----------
---------
- -------
---
----------
5.0E-02
4 .5E-0 1
e
------- x /L-0 -4
-
--- --
-
------
Shear
-- -- --
--- --- --
----
2.OE-01
----
------
-~---+0.0
-0 1-------2 .5---
---------
-
5.0 E-0 2
s/L-0.3
SE-Si
-- -----------------
--- ----
--------
1.0E-0
2 --- --- --------- --------- --------- -----------
deg
10
S/L-0.5
s/L-0 .5
------- -----..----
0
0
wave heading:
/L-0.3
-9e.
----
--- -------- -------
3.0 E-01I
0 .0 E +0 0
-L/2, Fa-.6 6,
@
3.0E-01
1.5E-01
cc
oment
Bending M
wave heading: 10 deg
.66
3.5E-01
1.0
2.0
X/L
3.0
4.0
5.0
Figure C.1: Shear force and bending moment RAO for 1800 wave heading,
at Fn = 0.66
154
6.0
-7
Am" C: Loading Results
1AE41
2 1"1
LS"I
6OZ42
-------
4AE-02
-
-----
LOZ-01
-----
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14AZ-02
---------------------
...... .....
",02
-- -
---OA
-------r------ i--------------
OAIW
-----IA I/L 3A
10
6D
SA
4.0
6.9
SA
4A
2A 7JL 310
la
OA
3B
2A
LO
OA
6A
3.0
4A
nO 4%
414,
Bamft Momme
hadW& UO 49
MwWm Kamm* @ V4. Ps-0.66,w
de
LOEM
LSFIOI
------
IA&01
-----
SID63
-NILP-OA
SILPO.5
--
IA"I ------ - --
- ------ -------
SBE-02
---------- ------- ------ -------------I
C18941 ----- ---------L
4C
-----------5AZ412
----------
Wh43
r
------ r ------ r-
7AS42
------
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-----
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r ------ i-------
DFA2
1
9.0942
-SAPOA
I
r ------ ------- r
6A"2 --------
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LOM
TSM02
2AE-01
-----------
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j ------ j ------ I ------
2AF02
O4z'O
---------------------
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SDE-02 -----
------------ 4-------
---- 4 ------
-------
L2"I
-
---- ------
----
-&W3
91DOA
5ArOj
--------------
1&M43
-SarO3
1AN41 -------------------------IA&01 ----------stbos
------------E
j -------------------LM81 ------ - -- ---------
LAM
Dooft Momw @ -V2,ft-OA6. mwebeaft: 120 d#g
Tmi=Momo@4J2,F&466,wwebeodbWU064
Sbw Form 0 -VZ, PWOA6. WSW MWiW UO 49
SAZ.O2
------
7AWA2
------
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r-----------
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------
4AVO2
--------------------- ------ r------
--------------
L------ ------
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-------------
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2MR-02
------ ------ ------
------
-----
IFM-02
3MA2
6JOE42
snGA
-------------------- -
2AWO2
r
101-02
2AE-02
-----5.0
4A
0.09100
2IL3.0
2D
LO
0.0
6.0
OOK400 0
OA
2A
ILO
6.0
SA
4.0
)/L 3B
-&M,03
----------
ISM
6AN42 ------
onro3Ai
a
------------------ ------
4fiF,02 ---------- ------
----
SAM
z
0 2-9m
-------------------
ZB,02
------------
L5E42
--- -------- -------------
IAFI02
5.O,03
------ ------
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3AM
3.0"2
4AM
------- ------------2AF'02 --- -----
-
--------------
WIPO.5
--------
------ ----
3MA2
------ ------ -------
6AEO2
----
--- ------- r-------------------------
----
--- ------
-------------
------ T ------
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r------------2AM
-------------------
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---- ---- --
0-0
IA
OA
UL 3JO
2.0
2A M 3.0
TwwionMowW#-V4.FirOAd,
3AM
XhmF*m@4V4,F*OA6,vmbe*dkFIZOdeg
-------L-------L-------L-
74AFM
LSE-02
0
13PE-02
------ ------ ------ ------
2AZ42
------- I-----I ------ L------ I
IDE42
--------------------
1DE-02
--- --- ------------ --- ------
-- --
--
Sam
GAR400
0.0
LO
4.0
SJO
6A
OAZHOO
------ ------ ------5A
4A
2A u0A
GA
6A
-----------------
I
UZ-82
-------- ----
lol92
-----
SAM
----
6AZ4)3
-
42M
- -
-
------
------ ------
-----
------ ------
------------------------- -------
2AM
6A
LD
OA
-------------4D
2D IfL 3,0
D4MftMGMIt@W2'F&4A6'MM'
MowF*m@4V2,Fm-OA6,mwbosdIW120dog
SA
6A
-@nod%
32Z-03
2-,03
6AS42
0
5D
-SaPOA
500.5
----------- I
-
ODIB-100
SA
4A
---I
:
0
I------- ------
4A
j/L 3A
T--
Bamft Me=& @ -1114.ft-OAd. MW b"M 120 dog
MI&O2
bt&ft. 120 dog
-------- -------------
------ ----2A I/L 3A
LO
2A
to
OA
I
-IM,03
2-%W ---------------------------- -NIDGA
5/IrO5
----- ------------- ------ -----2AM ----
6AZ42 -------------- ------ -----3AFA2
1A
O'A
0
SA
4A
6A
1b03
-.
4-002 ------ - ----
------- - -----
9AS42
5.0
40
5BM
7.O,02
L2E-Ot
3A
Bewft Momat @ midm6p.F".66. inn bad1w 120 dog
TamcsMommt @wWdipft-OA6,vmbeadw 120 ft
SbwFom@sddddp.F&466,wmbadkF]20des
"
2A
ILO
OB
-SILP03
"F102
------ ------
-------------
-
-----------
250,63 ------
SILPOA
SID6.5
2.OE-O3 ------- -----------------
4AN42
-------------------- ------
7
0 3AW02
--------------- -------
--- ---
----------
S/bO.5
------ -------------------
------
----------- ------ ------- ------19FIO3
-------------
2.010-02
L8942
0109-100
OA
------ -
----- -------
- --- -----------CO
2.0 2,/L 3fi
SA
------ I ------
SAE-04
- - - - - -4
6.0
OAR-100
-----LO
OA
I------ ------
---
SAFAW
OAEADO
2A
X/L 310
42
64
OB
LO
------2.0 UL 3A
4A
510
C.2: Shear force, torsion moment and bending moment RAO for 1800 wave
Figure
CP
heading, at Fn = 0.66
155
6A
Annex C: Loading Results
4 .0 E-01
5.0 E-0 1
4.5E-O I
s/L-0.3
/L-O .4
s1L-O .5
----------------------------------------------------4.0 E-0 1 ---------- ----------------3 .5E-O I ------------------ -- - -
-
- T- - - - - - - -
2.5E-O I ---------------- ------------ - --------- --------- ---------------2 .0 E-0 I ---------- -- ----- I-------- ---------- -------I.SE-0 I
------ --------- :--------- ----------I --------1:
-1.0 E-0 I ::::-::
--------- --------- ------------------ -----------------------------------------------5.0 E -0 1
- T----------- -------------------------------------------------------------------------------- ----------
-IJ4
@
ShearForce
--------- ---------- ---------- --- --- ----- --------
I.5E-O I
------------
LOE-O I
-------- - ------- --------- ----------------- --------------------
----------------------------
6.0
5.0
4.0
)L/L 3 .0
2.0
1.0
-------
----3.0 E-0 I - --------- ---
--------- ----
1.5 E-0 1
----
CIC
--------- ---------
1.0 E-0 I ------------- ------ --------- -
-------- ---------
2.OE-01
--------- -- ------ - --------- -- ---------------------------------------I -------- 11----
------ --------- ---------
-----
----------
5.OE-02
-
I.OE-01
L-------- ----------1---------
2.OE-01 ------------
-------- ---------
- ---------
--. w-s/L=0.4
s/L-0.5
--------- ---------
2.5E-01
-------------------------------
--------
-----
4 .0 E-0 1 ----------
g/L-0.3
-s/L-0.3
0 ".3
aI
/L__0
--0.4
s/L
.'LO
s/L-.0,55
----- ------------
---
5.0 E-0 I --------------------
0.0 E+O 0
0 OE+OO
-s/L-O.3
4.0 E-0 I ----------
--------
--
-- -------------
a/L-O .4
---------
s/L-O .5
3.5E-0 I -------------------- -----3.0 E-0 I ------------------------------ -------------------2.5E-0 I
2.0 E-O I
I.5E-O I
----
1.0 E-0 I
----
-----------------------------r ------------------------ r
1:1 9.0 E-0 2
---------
6.OE-02
------ I ---------
4.0 E -0 2
---------
---------
--- -----
--------
------ I---------- T----------------
-------
----
------
-------- -
--------- -------- ---------
---- ------ -
-------- ---
-----------
------ ------------------
------- --------- --------
--------------------
--------
----
6.0
5.0
4.0
I/L 3.0
B ending M o meat @ midship, Fn-O.S 2 , wave hooding 15 0 d eg
1.4 E-0 I
9/L-0.3
---------0.4
S/L
------------------1.2 E-0 I
s/L-0.5
1.0 E-0 I --------------- ---1 ---- - -- --------- --------------------
Fn-O .8 2 , w ave head ing: 18 0 d eg
@ mid ship,
F orce
Shear
I
2.0
1.0
0.0
6,0
5.0
4.0
3 .0
X/L
2.0
1.0
0.0
------- -------- ---------
2.0 E -0 2
-------------------------------------------------------
5.OE-02
0.0 E+O 0
0.0 E+O
@
Force
Shear
111,
2.0
1.0
0.0
I L--O .4
--
1.4 E-0 I ----------------1.2 E-0 I ----------
I.OE-02
s/L-0.3
-
--
----
2.OE-02
----
-------- ---------
--
----------------------------
Shear
@
Force
VL
2-0
--------
--
...........
---------------
6 .0 E-0 2
- -------- --
4 .0 E-0 2
2 .0 E -0 2
- --------
- ------
----
- E
---
--------- ---------- ---------
- -
------------
-------
0
2.5E.03
M
2.OE-03
----------
2.0
1/1,
4.0
5.0
/L-0.3
-
/L-0.4
s/L-0.5
T
---
F ---------!-- - -- - - - - -
- - -- - - - -
--------------------------------
-------------- - !--------- ----------------------1.0 E -0 3 - ------- ------------------- -- ----- -------------- ------------------------------------ --------- - -----------------5.0 E -0 4
-------
3.0
=
- -------L -------- I---------- t----------------- --------- ----------
-------------
- -------
0 O E +0 0
1.0
wave heading:150 deg
U E-03
--------- ----------
O.OE+00
0.0
6.0
5.0
4.0
3.OE-03 - --------- -- ------ -- - ----
- ------I----------
X/L 3.0
- ---------------
3 AE-03 - --------- --- -----
---------------- ----------------
----- ------
2.0
1.0
- ---------- -------:
:
-------4.011-03 - --------- ---- ----
s/L=O .4
s/L-O .5
9 .0 E-0 2
--------
---
------
Bending Moment @ +IJ2, Fn-0.92
5.OE-03
4 SE-03
1.0 E-0 I I --------- ------
--- ------ ------------------
--------------------------
j--1 ------- -------------------------------- ---------
0.0
6.0
1.2 E-0 I
-
----------------- --------- ---------
------------ -
0 OE+OO
190 deg
, wave heading:
Fn-0.92
+JJ2,
--------- -------
--------
5.OE-03
5.0
4.0
3.0
2.OE-02
--------- --
/L-0.4
:IL-0.3
s/L-0.5
-- 4o-
-----------
--------
--------- ----------
O.OE+OO
1.0
----------I
LOE-02
---------------
---0.0
--------- -------
2.5E-O 2
1.511-0 2 -----------
--------------------------------
-------------- 2-
------------
'r
------
-
-----
----
--------
3.OE-02
-------------
.: -----
---------
--------- ---------
--------
6.OE-02
-02
4 OE
- ----------
--
------- --------
-------
o/L-0.5
-------------------
-
---------
---------------------
1.6 E-0 I ----------
1.0 E-0 1
6.0
5.0
4.0
3.SE-02
-
1.8 E-0 I
3.0
VL
BendingMoment@ +IJ4, Fu=0.82 wave heading: 180 deg
deg
wave heading:180
Fn-0.82
+L/4,
2.0
1.0
0.0
6.0
5.0
4.0
3.0
2.OF-OI
CC
----------- -------------------
----------- ---------
3.OE-OI
6.OE-01
4.5E-0
---------
s/L-0.4
s/L-0.5
B end ing M o went @ -IJ4 , Fn-O.S 2 , w ave heeding: 15 0 d og
ISO deg
wave heading:
Fn=0.82
--------------- ---------
0 .0 E +0 0 -1%-0.0
6.0
5.0
4.0
VL 3.0
2.0
1.0
2 .5E-O 1
2 .0 E-0 I
5.OE-O 2
- ------
O.OE+00
0.0
------------------ --------- -------------------------------
3 .5E-01
3 OE-01
-
Pr 3.0 E-0 I ----------
<
cc
wave heading: 180 deg
Bending Moment@ -V2, Fn=0.82
Sheaf Force @ -IJ2, Fn=o.$ 2 . wave heading: 180 d eg
0.0
6.0
1.0
...... 4-2.0 1/1.
3 .0
4O
5.0
Figure C.3: Shear force and bending moment RAO for 180' wave heading,
at Fn = 0.82
156
6.0
Annex C: lAading Residts
Bsmft Momd @-IL/2,ft4J2.WWbMMlW 120 &1
Te"im Mom" 0 -1/2,ft-0.82, MW W&AW no 4"
UO 6*
3AE-01
3AZ41
4.0"1
2j"I
------------- 2--------- ------ -
am*A
-- - - - - -- - -- - - -- - -- - - -
2_4"l
2A"l -------
0
24WO1
------ LLi
209-01
-------
-WIA3
41WA
-
3
-s/L-0.3
_
l/EOA
3M41 --------------- ------ ------
------ ------
-------------
- --------L ------
.L
15E.01
-------
------ -- -- ------------- ------ ------
Mwel
------
-
--
--------
------------
-------
cc
---------
---
------
IDE-01
------
------
0AE400 V-1
-----------------------------------------
------
-----
5=
--------------4A
2A WL 3A
ID
SAF102
------
OAX400 r
CA
6.0
SA
---
--_---------- ----------
SA
410
2A I/L 3A
LO
LON41
91DOA
SILF0.5
SAE42
---- -
4AZ42
-
OALMO
OD
__ ------- ------ ------ ------
- ------..........
2.0 VOA
W
6.0
JA
4A
OAR400
OA
------ r --- --r__
SAV,02
------2A
LO
OAR-100
0A
------------------
--
-----
------ r
----
4A&02
-
------I ------
------
3A
------
-
18941
------
XAF,02
------
-------------
" 3.0
4D
_j
WOOF 120
1
70FA12
6OZ-02
SAE-02
x
0 4AFAY2
LO
A
2A YL 3.8
3.0
4.0
6.0
-
----- -
----- --
-------
3AZ-02
----
20Z-02
- -
-------------------- r------ r -----
IOZ-02
r----- i------ i------ r------ r ----- I
d0
0
4,b
2)p - - 0
lb
OAZ400
-IOZ-02
8ftftMG=W@41/4,Fv0J2,WMW8MW UO dog
25542
231M
Z.OW02
-------
UY02
------
LSIM
-----
----------------------- r _
------------- ------
------------
TaMmK*=W@41L/4,FvOA2.wwbn&WVOft
3AR42
---
-------------
------
21WA
2AF,02 ----- -- ----
------ ------
0
- - ------------------- - ------
LqVM&2
<
cc
----
------
------ ----------------
- -
---- -------2A
la
CA
34OZ-02
IALPO.5
------------------- ....... ------ ------------
OAE400
I/IPG.5
I
------- __
-
9A,02
r
r ------------ T__
--------------------
------6A
510
4A
-----
------- T ------ r -----
Delft Mo=W 0 MkIft, F"32,
2AR*2
_811r03
a 6SE-02
6p
- - -- ------------------- ------
8bwFw=@4V4,F".$2,w&nbe&dhW UO
------
SA
1@120dog
lll%,YvOA2,ww'
--
------------ ------------
2BF",02
OBE-W
0 S.OZ.02
it
4.09"42
L4"1
Uy"Al
-----------------4.0
2jL3.0
------- -------------
0
6AM
20
la
LOF101
-- --------
--
621"42
MIN
-Map.03
4E
---
----
Te=b=M*mW@
IA941
SAIW2
LOR41 ------
5A"2 ------ - -----
------ ------ ------
--
I.OE"42
-------
I" des
LOE41
E
0
SOL-0-1
----- ------ -----
-
L2141 -------
------ ------ -------I------
4AF,02
------
------
------
UZ41 ---------- ---------- j -------T
-----
------- ------
5A,42
-------
0 6AZ.02
<
--------------
----
Mom" 0 -Vq. ft*.U. UM hadim no
B=ft
saOA
-
18"1 ------------------------
-----
6A
SA
4A
3fi
2.0 "
LO
IA"l ------
31"l ------- ------ ------ r ------ r- -
LOFAI
------ ------- ------- L --------------
0"1
1.2Z41
0 22FAI
-----
0AE40
CA
610
-------
------
SAM
TordORMOMW @ -1/4.F&O.S2. UM hMdhW UO d*S
8bwyoc=@-1/4.FvOA2,=9beaft:1Z0 do$
4A"
------ -------
-------------------------- ------ ------ ------
4AF,02
-------
21OR42
0.
I/L 3A
2D
LO
OA
SAE-83
------ ------ I
OADIOG
OAE400
------------SA
4A
6A
19
CA
I
Tanim MewW
3.0
240
4A
SA
dh
------ ------
3SE42
----- r- ---
2AFoG2
----
------ -----......
= A
----- r------7--------
T ------
------
ISE-02
------
--------- - --
:t
W7,03
snAA
0
LO
2D
YL 3A
4A
SA
6.0
_
91DOA
SIDGJ
----------------------- -------
------------------------ ----------- L------
IOE4)3
----
------
----- ------
------ ------
------
SAFAW
------
__4
OAFADO
OAR-400
CA
6A
------ A------ I ----- - ------
-----
2,09413
------ ------
LO"3
5Ar,04
OBE400
------ ------
25Z-03 -------
------------. ...........
--
................
------------------------....
------------
5.0
4.0
3.0
SnAj
2-1743 -------- ----- ------ -----4SE42
2J) m
1.0
3.019-03
3AW03 -------------- ------ -------- -
WLOA
--------------- -
------ ---
SAR42
CA
qaWftMemu9@-HV2,FvQJ2,Mftbl&ft: 120 dog
4V2. PW-0,82, um bndhV IZO dog
35E.03
6DE-02
--- -----
------
SAW03
"09,
-
------- ------
- -
------------------------------
-
------
-
Ls"I
CA
LO
2.0
UL 3A
4.0
SA
640
Do
LO
--- _
2.0 WL 310
4A
3D
Figure CA Shear force, torsion moment and bending moment RAO for 1W wave
heading, at Fn = 0.82
157
610
......
......
. . .. ..
Annex C: Loading Results
Shear
Force @ -L/2,
wave
F=0.99,
Bending M
180 dog
heading:
7.0E-01
0.0
----------------
Z
-------------- - a----------------4.00-014
3.0E-01
-----------
2.0E-01
--------------------------
--------
1.0E-01
-------
------------------
---------
a/L-0.3
-
C
- -
---------- -
-------
--------
---
0.0,-01
LOE-01
.0 E+0 1
--
---------
1/L
2.0
1.0
3.0
5.0
4.0
Shear Force @ midship, Fn=0 .99 wave heading:
7.0E-01
Bending M
2.0
---------
j-
--
3
+L4, Fn-0.99,
@
.
--
2.3E-02
-----
10 0 d og
wave heading:
-
-
- -
--
-------
Bending Moment
4.5E-02
s/L-0.3
e-
s/L=-0.4
9/L=-0.5
-- - - - - - - - ----------
-------- ------------------- --
- ------
:- ----------
4.OE-02
----------- ------- -
3.5E-0 2
----------- -
3.0E-02
----------
----
--------
----
--
-
-- - -
--------
-
3.0
4.0
50
wave
+12,Fn-0.99,
- -.
1.2E-01
------------
1.0 E-0I
--------
0.0E-02
6.0E-02
-
-----
--
--------
- ---- -----------
- --.
----
----------------
-
--------
--------
0
0 .0
1.0
----------------------------------------------4 .0
2.0 X/L 3 .0
-
@+L/2,
-
--
Fn-0.99, wave heading: 190
3.5E -03
----------
.. 3.0 E-03
----------
deg
-
o--/L-0.3
.-
s/L-0.4
s/L-0.5
------ -------- - - - - - - - ---- ---- --- ---- - ------ j- - - - - - - - - -
--
2
-
--
---
-
-
- - - - - - - - - - -
-----
------------------
2 .0 E -0 3
6.0
5.0
4.0
4.OE-03
2 .5E -0 3 --------- -
1.5E -03
---------
5.0
X/L 3.0
2.0
---------- -------- --------- --------I- ----------- ------ - + - - --
4.5E-03
---------
---------------
--------
2 .0 E1-0 2-----
IL-=0 .3
/L .4
- s/L-0
.4
-----
------
-
-
-
Moment
--- -
5.OE-03
1.5E-032
IL-0.
--------------
4.0E-02
0 .0E+0 0
-
--------
--
-
---
------
1
s/L 0.3
- -----
-----
-----
1.0
/L-0.3
-
-
----
Bending
deg
heading:100
1.6E-01
1.4 E-01I
0.0
6.0
- -/
+
Shear Force@
I/L
6.0
-- - - - -t- ----
----
-----------1.0E-02
-1.0E-02 -------------- --------- ---- 3.31-034
5.0E-03-
--
-
-
---- - -- ------ --------
----- ---------
0..E+00
-
2.0
1.0
-
5.0
4.0
3.0
---- L---------- L--
-.
-
---
- - - - - - - --
2 .0E -0
3.0E+00
0.0
-
-
+IJ4, Fn-0.99, wave headung:180 deg
@
---
-
-------
-------
1L
2.0
1.0
0.0
2.5E-02
5.OE-02
-
--
-
--
--
-------
-------
-
----
--- -
----
4.0E-02
6.0
5.0
0
---------- ----- -- --------
1.0E-01
4---
-
F orce
X/L
.0
o.
1.5 E-01
- - -
-
2
1.0
----------- - -
1
----
.0-E+00
3.0E-01
2.5E-01
- ------- 40------------- -- - -------+
-
Shear
----------------------------------- - -- - - - - - - --------
-- -
-----1 - -- I. -0
0.0 1+00 L
0.0
-
- ----- --
----------
6.OE-02
--
-----------
-
L-0.4
0/L-0 .5
.
..
--
-- -
1.0 E-0 3 5.0 E-0 4 0 X0 E+0 00.0
--
6.0
-
------------
- - - -
1.0
-
-------- ------------------ - - --- - - - - -
----------------- ----------------- -------2.0
I/L 3.0
-
1.0E-01
-----------------
-
2.0E-01
oment @ midship. Fn-0.99, wave heading 180 deg
1.4E-0
1-------. --------- -----1.0 E-0 I ------ -- - -------8.0E-02
-------
---------
------
--------
-------
---------
3.0 E-01
-------
6.0
5.0
4.0
3.-0
/L
1.8 -0 ----- --- ------- --------- ---------r ---s/L-0.
1.6 E-0 1
-
----------------
------ --------- - --
1
E-0
e/L-0 .3
--
--------
.0-01---------
2.0
1.0
0.0
s/L-0.5
4.0E-01
---------- ------- -------- -----
6.0
./-0.4
----
---
5.0 E-02
0.0 E+ 00
180 dog
--------------
---------
--------
-
1.0E-01
-
6.0E-01
--
-- ---
---------
.
0.0
-
--------
----
-------------- ---- --
-
------------------- -------- ----------
-
---
1.5E-0 I
- ----- ---------
----
------- +----------+--
-
-------
-----------
---------------
2.0E-01
---
-
1.0 E-01I--
---
--
--
-------- r--------- ---------
--
-
--------- --------
-------
--------
- -------
-------
--
x/L=0 .5
- ------- j-
-------- ---- --- -----
-
2.0 E-0I
1
2.51-01
-
3.0 E-0
0
E-0 I
s/L-0.3
----
------------------
-
s/L-0.5
---------
Fu=0.9, wave heading:180 deg
1
3.5E-01
-o---.O/L-0.4
- ----
--------- -
-L/4,
/L-0.3
-.---
--
---- -- -
5 .0 E-0 I ------------
@
Bending Moment
6.0
5.0
4.0
3.0
-
- -------------1 - --------5.0E
----- --- ---0-------
1L
2.0
1.0
0.0
E-0
.-
f
r
-
6.0
5.0
4.0
---------- ------
--- ----
- ---------
-
- - - -
--------
----- ---------
---j-------- L-
-
-- - - - - -
4.0
6.0 E-0 1
---------
---------
2.OE-01
-.-.-----
--
-
-
-
---------------
-
7.0E-01 ----------
1
3.0E-
L-.
--------
-----
.3
4
-
0
--
8.0E-01
0.0E+00
---------
wave heading: 10 dog
Fn-0.99
L/4,
4.0E-0I
/ -
.-
-
--------3.
/L
2.0
Force
--------------------
-
Shear
5. 0E - 01
---------
---------
LO
00
0
wave heading:150 dog
-
0 .0 E +O 0
C)
.99,
-
2
2.0 E-0
F-0
-/2,
@
- - L-0
L1-0.1
3.0 E-01
4.0E-0
4.0
oment
6.0E-01
4.0
5.0
Figure C.5: Shear force and bending moment RAO for 1800 wave heading,
at Fn = 0.99
158
6
.0
Annex C: lmding Results
u
b0*dkw nod%
TbCwMM6=&t@-v2.ft-Qqq'V
3-MO1
4
------
------- -----
4A,01
------ -------------------
3SE41
------
LWA::
- ------
--------
LOZ4
----
-----
--- -----4---------- ------ ------------
18
-----
-----
-----
LOZ-01
-----
-----
L------ ------ j------
SAM
--------------
10342
------
4.0
5jO
OAR"
---------1.0
2,0 I/L 3A
6101
40
3A
0.0
6AI
WD-03
-
4-VAI -------------- ------ ----------- ------
------ ------ ------ ------ ---------- ------ T-------------------
---------- --
1-4"1
-----
-----
------------ I--------- I --------------I------
LO,01
0.0
LODOI
------
6.01
-BILP03
-94POA
SIDej
6.OZ.02
-----
....... .......
Z.OF'01 ....... ..
LJWOI ------
------ -------
-
------
------- ------
----
- ---------- - ------------
- -
------------------ ------
-----
-
------------- ------
------L -----
5APA2
------
------
- --- ------
---
----
O.OMOO
L
-----LO
3.0
------
2A"z
---0.02 Weo
-SfWA
4AZ.02
------
4B
2-VO1
L27AI -------
3 4"1 ------- -3.OF,01 ------- -- -
2.0 JJL 3A
wbt.3
UE-01 ------
WDOA
4A"I -------
10
sawft mama@4j4,ft4",=whndw UO des
h"MW 120 dIS
4/4,ft-O".
16"1
"FAI
-----
----------------------------- ------ -------------
-
-
OA
2A VL 3A
TwWomMosat
0 2-1"1
ir 2AZ-01
--- ------------- ------
---
--------
---- -
0.01400
OA
LM41
-- ------------ - ----- ------
-------- --
181-01
----- ------ ------ ------
------ ------ ------ ------
!------- ------- ------ T
--------
--
SAZ-02
LV41
------ T
------
1.0&01
-------------2A"I ------- - --
------------ ------
2.0m
-
----2-411-01 -------
2-mel
------
- ------ -------------------- I---
I
wba.3
-
: -----1.
------
------
3"Al
------- ------
------ ------
-
-WA1 ------
3AW101 -------
S UO do@
-@mo=a@4j2,P&4",wm'
31&01
4A)
"3A
2A
5A
TonimMema
F"99,w&wbwdkWnOdtg
MwwFom@
14 Pkv-O-",
6A
sh
4.0
jjL3A
2.0
1.0
0.0
6.0
is a]
5A
.0
OA
I
NsIft memo @ MMdjp'p"-"'ww bew1w no
bwdhwnO dog
1AF,01
2AFAK
MZ-01 ------- ----
---
L2FAn -------
----- ------ ------ ----------- ------------ ------
LOZ.01
&GIA2
-------
6AZ412
------
4.OM
----
2.OZ4)2
--- --
LI&OI --------------------
SA"2 ------ 6.OFA2
4AE42
--
----- I------- ------ -------
-- --
-
----
------
2AFI*2
----------10
4A
6.0
------ L -- -
L"A I
------
0 IOFAI
8AE4)2
-----------
-
2A 1& 30
4A
L
0 2AM
---------------------
LSWAOU
-------------- ------
r ------ I ----------- ------ ------ ----- - ----------- -------I ------ L - ---- I ------
2AE-02
3.0
4A
5.0
5AM
3.0
4A
2'0
LOZ42
-----
SDE43
- -
6A
-----
4.07,02 ----------
--
--
------ -----
-----
j
r -- ------------ --------
4AR-03
AAM-403
------------- - -----
--- ---
la
OA
----
------2A WL 3'0
------ ------
SJO
4.0
6.0
-- ----
2.01&02
------ ------
--- ------
----- - ----- ------
------
------ ------------------------- ---------------
2.Oz-O3
L4WA*
----------
W
--------------- T ------
----------------- - ------------
----
IBM
-
LOE43
-------
SaOA
A
-
--- ------
-----
2AM
------------
---
2AR-03
-------------
-WL-03
sAPOA
Muej
5.OE." -------------- ----- - -----
----- ------ ------
------ r
------ -----------------
3GE4)3
'r---------------------- r ------
----
---
*
------ ---
Wbej
------ ------------ ------
B=ft MOM& @-IV2,ft-O".WMW6dkW nO don
6.OeO3
---- ------ .......
------sIbOA
X.OZ42 -------- - --- ------------- IWD6.5
j--------------------------7A&02 -----------
3,OE-02
-----
7em@4V2,ft-O-".mwbndhW 90 dog
9.09-02
IDE-02
L"M
-SID63
SILPOA
---------
---------
2A,02
O.OZ*O
1A
OA
6.0
18"1
-6A"2
--------
5,OZ*4
-
M
"
2A
1A
25"2
-- --- -- - --------I----- ------
O'0E-w
OAEV4"
OA
6A
des
08
------ :-------------
IOZ-02
4.0*02
5.0
4A
-------
1
6.0502
------
6.01
.ID03
------ j -Lf"A
3AFAa
Wbej
2M,02 ------ - - --- -------------------- ------
- -------
-- -------
-- ------
--- -------2D )JL 3.0
to
OA
JA
------
3,OVA2
3M.02
----- ------
-
----- ------------ ------
......
------------
TmjonMo=@d@41/4,7v*",mwhesdhW 120 dog
-54,03
sAEPOA
------ L-------- -
- ----
-
IA
COM
LSZ-01
L4,41
-----
4- OA
be&dLW UO dog
WwwFww@-W4,FW-O",
-----
4AM
2.AZA42
-----------------
GAZ400
yL 3.0
2.0
Ift
62Z.02
----
-----
------ k------- ------
GAR"
OA
SAZ42 ------
------------
--
0
-------------
-- -
------
...................
10141 ------- --
VD05
r------ r
------ ------
-------
q,0.3
#11POA
--
sm03
&qrOA
al"i
ISFAII ------- ------ ------ ------- L"." --------------
--- ------
---------------- I
---------------
IOFAr2
OA
------LO
---------------2.0 ;jL 3.0
Oav"
CD
5.0
6A
----4.0
VL 3.0
8.0*100
-
OBE400
OA
Lo
2A
5.0
6A
0
a
LO
2.0 YL 3A
4A
3.0
Figure CA Shear force, torsion moment and bending moment 040 for 1800 wave
heading, at Fn = 0.99
159
6.0
(/1
-
jilt