SAIL VERSUS HULL FORM PARAMETER CONFLICTS IN YACHT DESIGN

UNIVERSITY OF SOUTHAMPTON
SAIL VERSUS HULL FORM PARAMETER
CONFLICTS IN YACHT DESIGN
Written by BOYANG WANG
This project is submitted for MSc degree.
Faculty of Engineering and the Environment
SEPTEMBER 8, 2015
THE UNIVERSITY OF SOUTHAMPTON
Supervised by Grant Hearn
Second Examiner: Zhi-Min Chen

University of Southampton MSc Project report Written by Boyang Wang
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Abstract
This project aims find the influence of sail and hull form parameter to the overall
performance of yachts. The relative hull modification and section mapping will be
provided and explained. The final performance is judged by the sailing polar diagram
obtained by using engineering software. The final conclusion of how to modify the hull
form parameters with a given sail is provided, but due to the time allowance, a more
accurate conclusion could be made in the future with using the technique introduced in
this project.

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Acknowledgement
Through doing this project it needs to appreciate and show my respect to Professor
Grant Hearn who is my supervisor. Without his patient attitude and correct guidance
this project can’t be produced.
Thanks for the Andrew Petter who is a pervious student, his work in respect of the
Lewis Mapping method save a lot of time for me.
Also show my respect to my family who support me for studying in University of
Southampton.
Thanks for my girlfriend who look after me though this hard time.
Thanks all the people or organization and their works involved in this project.

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Table of Contents
This project is submitted for MSc degree......................................................................0
Faculty of Engineering and the Environment ................................................................0
Abstract..........................................................................................................................1
Acknowledgement .........................................................................................................2
1. Aims........................................................................................................................8
2. Objectives ...............................................................................................................9
3. Methodology.........................................................................................................10
4. Deliverables..........................................................................................................12
5. Literature Review .................................................................................................13
1) Delft Series....................................................................................................13
2) Method used to identify the most important hull form parameters...............17
3) Route Determination .....................................................................................17
4) Hull Form Modification: Lackenby Transformation Method.......................18
5) Lewis Section Mapping Method ...................................................................20
6) YD-40 Parameter Check. ..............................................................................20
7) Velocity Prediction Program.........................................................................22
6. Lackenby Transformation Method .......................................................................25
Nomenclatures for thi section:.................................................................................25
6.1. STEP 1 Preparation of the Hull Form Parameters ........................................26
6.2. STEP 2 Change the Hull When Block Coefficient is required to be Moved 34
6.3. NOTICE ........................................................................................................40
6.4. STEP 3 Change the Hull When LCB is required to be moved .....................40
6.5. STEP 4 Change the hull when LCF is required to be changed.....................45
6.6. Conclusion.....................................................................................................50
7. LEWIS MAPPING METHOD.............................................................................51
7.1. Lewis Mapping Method ................................................................................51

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7.2. Transverse Section Definition.......................................................................52
7.3. Improved Lewis Conformal Mapping function ............................................52
7.4. The additional equations ...............................................................................54
7.5. Matlab code for Lewis Conformal Mapping.................................................56
7.6. Example of using the Lewis Mapping...........................................................56
7.7. Accuracy Check ............................................................................................60
7.8. Conclusion.....................................................................................................61
8. Calm Water Resistance.........................................................................................62
8.1. Introduction...................................................................................................62
8.2. Hydrodynamic Forces Involved in DSYHS..................................................63
8.3. Data Analysis and Assumption .....................................................................72
8.4. The relative Matlab Code is given below: ....................................................74
8.5. Conclusion.....................................................................................................74
9. Static Stability of the Generated Yacht Hulls.......................................................75
9.1. Static Stability...................................................................................................75
9.2. GZ curve of the Hulls....................................................................................76
9.3. Conclusion of the Static Stability of the Yacht Hulls ...................................82
9.4 MatLab Code for Ploatting the 3D GZ surface..................................................82
10. The Influence of the Sail to the Overall Performance ......................................83
10.1. Introduction.....................................................................................................83
10.2. The type of the sail ....................................................................................83
10.3. Yacht hull Selection...................................................................................86
10.4. Preparation for using the software.............................................................87
10.5. Obtaining the VPP results..........................................................................88
10.6. Conclusion ......................................................................................................92
11. Conclusion of the project..................................................................................93
12. Limitation of This Project.................................................................................94

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13. Risk Assessment ...............................................................................................95
14. Nomenclature....................................................................................................96
15. Gantt Chart......................................................................................................100
16. References.......................................................................................................101
17. APPENDIX 1 PARAMETERS OF THE MODEL ........................................104
18. APPENDIX 2 DATA OF THE HULL FORM...............................................107
19. APPENDIX 3 APPENDIX 3 MATLAB CODE FOR IMPROVED LEWIS
CONFORMAL MAPPING .......................................................................................122
20. APPENDIX 4 SECTION CURVE FOR EACH STATION...........................125
21. APPENDIX 5 RESULTS OF CALM WATER RESISTANCE ....................132
22. APPENDIX 6 CODE FOR PLOTTING 3D RESPONSE SURFACE FOR
RESISTANCE ...........................................................................................................138
23. APPENDIX 7 Relative Data (GZ value and associated heel angle for each hull)
141
24. APPENDIX 8 SAILING POLAR DIAGRAMS ............................................149
Table 1 Range of Limitation with Delft Series............................................................14
Table 2 Hull Form Data of Parent Hull .......................................................................21
Table 3Simpsion's Table represents the data of parent ship ........................................27
Table 4 Actural displacement volume of the parent ship ............................................30
Table 5 Parent Hull Data..............................................................................................34
Table 6 Simpson's Table When Cb increases by 0.031 ...............................................36
Table 7 New sectional area (interpolated y) ................................................................38
Table 8 Data Error .......................................................................................................39
Table 9 Full data needed when moving LCB as -0.301...............................................43
Table 10 Simpsons table when changing LCF ............................................................46
Table 11 Fore and Aft body parameters ......................................................................48
Table 12 Data relative to LCF=-0.886.........................................................................49
Table 13 Beta and Phi with associated transver section data of station 11..................57

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Table 14 The Points to Define the Sectiona Data Through Applying the 3Parameters
Lewis Mapping Method...............................................................................................59
Table 15 DSYHS Range ..............................................................................................64
Table 16 Lackenby's & DSYHS Range.......................................................................65
Table 17 cients for the residuary resistance.................................................................69
Table 18 Data of YD40................................................................................................70
Table 19 Total Resistance of LCB and Cb ..................................................................72
Table 20 Total Resistance with Cb and LCF...............................................................72
Table 21 Total Resistance with LCB and LCF............................................................73
Table 22 Selected Hulls with Associated Hull Form Data ..........................................87
Table 23 Data of appendages.......................................................................................87
Table 24 Size of the sail...............................................................................................88
Table 25Approximate results.......................................................................................90
Table 26 Boat speed for the race course ......................................................................91
Table 27 Hull form parameter for hull number 51 ......................................................92
Table 28 GZ data of Cb and LCB Table 29 GZ data of
Cb and LCF................................................................................................................141
Table 30 GZ data of LCB and LCF ...........................................................................142
Figure 1 Unbalance pressure over a surface ................................................................15
Figure 2Wave generated by boat .................................................................................15
Figure 3 Lackenby Tranformation...............................................................................19
Figure 4 Sailing Polar Diagram ...................................................................................22
Figure 5 VPP Flow Chart.............................................................................................23
Figure 6Fractional Sectional Area Curve of Cb+0.031 ...............................................37
Figure 7 New Sectional Area Curve of Cb ..................................................................37
Figure 8 Error of changing the Cb ...............................................................................40
Figure 9 Half Beam Curve...........................................................................................46
Figure 10 Half Beam Curve LCF=-0.886....................................................................49
Figure 11 The half beam of difference LCF value while keeping sectional area
unchanged ....................................................................................................................50
Figure 12 Transverse Section with deadrise angle phi and entrance angle beta..........52
Figure 13Change of the shape of the section 11 in stage 1;2 and 3.............................60

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Figure 14Section curve of parent hull and section curve generated by using 3 parameters
methos..........................................................................................................................60
Figure 15 Accumulated errors for each station in meter..............................................61
Figure 16 Presentation Resistance Components..........................................................63
Figure 17 Response resistance surface with LCB and Cb (X:Cb and Y:LCB) ...........72
Figure 18Response resistance surface with LCF and Cb (X:Cb and Y:LCF).............73
Figure 19Response resistance surface with LCF and LCB (X:LCF and Y:LCB).......73
Figure 20 Typical GZ Curve........................................................................................75
Figure 21 GZ curve for yacht which has second peak value. ......................................76
Figure 22 Max GZ value for yacht hulls number 1 to number 25 ...............................77
Figure 23 Variation of Righting Moment with Cb and LCB.......................................77
Figure 24 Max GZ value and displaced mass for each hull.........................................78
Figure 25 GZ max and righting moment for each hull ................................................78
Figure 26 Max Gz value for hull number 26 to number 50.........................................79
Figure 27 Variation of righting momrnt with Cb and LCF .........................................79
Figure 28 Max GZ and associated displaced mass for each hull.................................80
Figure 29Max GZ and righting moment for each hull.................................................80
Figure 30 Max GZ value..............................................................................................81
Figure 31 Variation of righting moment with LCB and LCF......................................81
Figure 32 Max GZ value and associated righting moment..........................................82
Figure 33 Three types of the sail..................................................................................83
Figure 34 Sail configuration ........................................................................................84
Figure 35 Rigging system for a yacht..........................................................................85
Figure 36 Mast Definition............................................................................................85
Figure 37 Resistance and Max righting moment of the hulls ......................................86
Figure 38 Yacht Definition..........................................................................................88
Figure 39Sail Set..........................................................................................................88
Figure 40 Optimum Setting For downwind condition.................................................89
Figure 41 Optimum setting for upwind condition .......................................................89
Figure 42 Wind definition............................................................................................90
Figure 43Sailing Polar Diagram for Yacht Hull No. 51 ..............................................91

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1. Aims
Yacht performance is a function of hull form and sail arrangement and the crew. This
project is talking about the conflicts arising from impact of hull modification and sail
arrangement. The velocity prediction programmes developed by Bentley
(Bentley|SYstems, 2015) or Wolfson Unit (UNIVERSITY of SOUTHAMPTON, 2015)
is going to be applied through this project. For this project an initial yacht has to be
selected and will be used as the parent yacht for hull form modification. Though
modifying the hull forms, the improvement of performance such as stability and
resistance needs to be identified. Finally all of the hulls will be tested in an appropriate
selected course with different sail configurations, the time over course will be recorded
to judge the performance.

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2. Objectives
 Finding an initial yacht hull with suitable hull form parameters consistent with basis
of the Delft Series.
 Researching the Lackenby Hull Form Transformation method in order to
understand it and be able to use this method to modify the initially selected hull
parametrically to provide alternative yacht hulls.
 Learning how to use the Lewis section mapping method to generate the offset table
for the generated hulls according to the associate known hull form parameters.
 Learning how to use Maxsurf Integrated Software (Bentley|SYstems, 2015) to
implement the hull transformation procedure and generate the 3-D models by
replacing the position of points in the offset table.
 Understand use of the Delft series for predicting hull resistance and be able to use
it to calculate the upright resistance and the heeled resistance of different hull forms
generated in previous step.
 Finding an appropriate method that can be used to estimate the static stability of the
generated yacht hulls.
 Learn how to use the Wolfson Unit VPP program to estimate the performance of
the yacht with different wind conditions and different sail plans based on the
stability data in pervious step (Final deliverables are depends on the time allowance).
 Using the selected course to measure the performance with respect to total racing
time.

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3. Methodology
a. Select the initial hull form subject to it bearing consistent with the limitation of the
Delft series resistance regression formula.
The YD-40 (Eliasson & Larsson, 2011) is chosen as the selected candidate yacht
and hull forms of the parameters of this yacht has been proved in the range of Delft
Series hull form requirement.
The hull form parameters of the generated 3-D model could have a reasonable range
of difference with the YD-40.
b. Using Lackenby hull form parameters transform method a series of hull forms will
be generated with 3 different hull form parameter combinations (Cb and LCB; Cb
and LCF; LCB and LCB).
Where:
LCB Longitudinal Centre of Buoyancy
LCF Longitudinal Centre of Flotation
Cb Block Coeficient
c. Using Lewis mapping method to generate the offset table for each sections of the
new yacht hull forms.
d. Generating the hull through the software package.
e. Using Delft series resistance regression formula to calculate the upright resistance.
f. Applicate of the software such as Maxsurf or Wolfson unit to identify the
hydrostatics typically the GZ curve of each of the hulls.

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g. Determine sail according to the maximum righting moment of each selected hulls
with different wind condition.
h. Observing the wind flow chart and create the course with statistical data.
i. Over the generated course and wind conditions undertake a VPP analysis to
measure the total running time of each hulls.
j. Select the most suitable hull with associate sail plan.
k. Modifying sail plan such as mast height or length of the root.
l. Applying the VPP again to measure the different performance.

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4. Deliverables
Through doing this project some important results are wanted and listed as below.
 Offset tables with comparable parameters.
 Visual comparison of different hull form.
 Resistance with different hull forms.
 Comparable GZ curves with different hull forms.
 Sailing polar diagrams for different hull forms and sail plans.
 Total time of taking the generated route.
 Summary of the project and hopefully give some suggestion or indication to current
yacht industry.

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5. Literature Review
1) Delft Series
Introduction
Delft series regression resistance formula is the most comprehensive used method for
predicting the yacht hull performance. It’s based on the Delft Systematic Yacht Hull
Series carried out in the Delft University Towing Tank starting in 1974. The basic idea
is to systematically vary the hull form in order to find the impact of variation of different
hull form related parameters.
Over the last decades an extension has been undertook to the Delft Series and now the
data of it contains information about both the bare hull and appended hull resistance in
the upright and the heeled condition, the resistance increase due to the longitudinal
trimming moment of the sails, the side-force production and induced resistance due to
side-force at different combinations of forward speeds, leeway angles and heeling
angles.
In addition the new sets of formula for relative hydrodynamic forces as a function of
hull form parameters were developed to deal with larger range of yacht hull form
parameters.
Ideally this method will be valuable for us to predict the resistance while changing the
hull form parameters
Limitation of Delft Series
The range of hull form limitation with Delft series are provided in Table 1:

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Table 1 Range of Limitation with Delft Series
Based on the Delft series there are several aspect in resistance that would be helpful for
us:
1. Upright Resistance
Upright Canoe Body Resistance includes frictional resistance and residuary resistance
(viscous drag plus wave resistance).
Frictional resistance:
The frictional resistance results from energy dissipation in the viscous boundary layer
the ITTC 1957 friction line could be used to calculate this resistance and the full details
will be provided in “Calm water resistance” section.
Residuary resistance:
The residuary resistance consist two parts:
 Viscous pressure which caused by the imbalance of pressure over the surface of the
hull as illustrate in Figure 1 (Day, 2014).

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Figure 1 Unbalance pressure over a surface
As the pressure distribution over a surface showed in Figure 1, when flow passing
through the surface from right to left, the pressure is unbalance at the leading edge and
the trailing edge. This is because the flow will separate while it moving on the surface,
therefore the larger pressure in the front will “push” the surface to move afterward, thus
this pushing force is known as the viscous pressure resistance.
 Wave resistance is caused by the energy dissipated by the waves generated when
vessel travels through the water surface, see figure 2. (Day, 2014)
Figure 2Wave generated by boat
As it is showed in Figure 2, there are 2 kind of weaves generated by a moving ship in
the water, transverse wave and divergent wave which contain a great percentage of
energy generated by a fast speed ship.

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 The wave breaking resistance is also important which is generated when a ship is
breaking through waves.
The calculation of the residuary resistance is the semi-empirical regression method
which is based on the statistical analysis of experiment measurements and the full
details will be provided in the “Clam water resistance” section.
The resistance of appendages
The resistance for the keel and rudder are usually calculated separately, using same
method as mentioned in previous step. However as the aims of this project is only focus
on the bare hull and the sails, there will be no discussion or analysis about the resistance
of the appendages and it is assumed it doesn’t influence the overall performance.
2. Heeled Resistance
The introduction to yacht resistance has so far consider the yacht to be in the upright
condition. However, most of the time the yacht will experience a wind condition
generating a side force on the sail leading to a heeling moment. Therefore in order to
balance this moment the hydrostatic righting moment dependent upon to shape of the
yacht underwater body must balance the wind moment, thus both frictional and the
residuary resistance will be influenced by the yacht heel.
Change in frictional resistance of hull due to heel
It is assumed that the frictional resistance of the canoe body changes with heel as a
result of the change in wetted surface area. A regression method is used to describe the
heeled wetted surface area (Keuning & U, 2002).
Residuary resistance of the hull
As the numbers of experiments required to investigate all models at all speeds and all
heel angles are quite large, the Delft Series focus on trying to predict the change in
resistance at a single heel angle which is 20 degree which is a reasonable upwind heel
angle.
Residuary resistance of the appendages with heel
Conventionally it is assumed that the resistance of the appendages (frictional + viscous)
will not change with heel as the wetted area of the appendages is assumed to remain

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constant. However with the research done by Delft University the wave-making
resistance of the appendages does change as the keel and rudder will become closer to
the water surface, this results in an increased depression of the free surface, which leads
to changes in wave-making resistance, thus the residuary resistance will change.
2) Method used to identify the most important hull form parameters
A design chart plots the quantity to be scrutinized (for this case, total resistance and
maximum GZ) against the variation of two primary or secondary hull form parameters,
within acceptable limits, displaying the results as a three dimensional surface. (Hearn,
1999). The design chart will show how the selected quantity is influenced by the
modification of the hull form and therefore, using a series of design chart the designer
should be able to know and select the preferable advantageous changes. In order to
create the design chart the quantity (performance) must be predicted by the selected
software with all hull forms and associated hull form parameters (Petter, Optimization
of a Yacht Hull, 2012).
After the key parameters are found the next possible step (if time permits) is to identify
an appropriate optimization method such as Genetic Algorithm (GA) to identify the
optimal yacht performance and its associated hull and sail characteristics.
The engineering software which could be used to deal with the data is MatLab 2014Rb
(MathWorks, 2014) which is available from the University of Southampton Isolution.
3) Route Determination
In order to have a general concepts how a race course looks like the following materials
would be helpful.
The course of 2013 America’s Cup:
The first leg commence near the coast to a southern point turn 90 degree anticlockwise
to begin the second leg (leeward). Upon reaching the second point to the East yacht is
required turn 90 degree anticlockwise to begin the third leg. Upon arrival third point to

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the North yacht is again turn to 90 degree anticlockwise to sail back to the start point
(windward).
 After the race gun signals the start, the first leg is a short reach of around 0.5
nautical miles (0.93 km; 0.58 mi) towards the shore.
 After rounding the reach mark, the boats travel downwind to the leeward gate.
This second leg is around 2.5 nautical miles (4.6 km; 2.9 mi) in length. At the
bottom of the course, the leeward gate has two different marks. Rounding either
mark completes the leg.
 The third leg stretches around 3 nautical miles (5.6 km; 3.5 mi) from the
leeward gate to the windward gate. This upwind leg is the longest leg timewise.
 On the fourth and final downwind leg, the boats will be aiming for the leeward
mark that is closer to the shore.
 Rounding this mark puts them on a reach sprint to the finish. The fifth leg is
around 1 nautical mile (1.9 km; 1.2 mi) in length. The finish line is right in front
of America's Cup Park, at Piers 27/29.
The length of the course varies, but is around 10 nautical miles (19 km; 12 mi) and
generally takes about 25 minutes. During the 2013 Louis Vuitton Cup on the same
course, some races were raced with an extra lap around the leeward and windward gates.
This seven leg course is around 16 nautical miles (30 km; 18 mi), taking approximately
45 minutes to sail (Wikipedia, 2015).
4) Hull Form Modification: Lackenby Transformation Method
General:
A well-known derivation of the lines for a new ship from the parent ship is the ‘one
minus prismatic’ method (H.LACKENBY, 1999). However the fineness and the extent
of the parallel middle body cannot be varied independently. The Lackenby
transformation method can overcome this and permit independent variation of not only
the fineness and the LCB position, but also the extent of parallel middle in both the fore
and after body (H.LACKENBY, 1999).

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It assume that the change of the longitudinal sections is proportional to the change of
the prismatic coefficient for aft and forward body separately, see figure 3.
Normally it will be used to change the sections for the big merchant ship which has
parallel mid-body, however it can also be complied with ships which doesn’t have
parallel mid-parts
The original version only apply to change CB and LCB, however due to the fact that
LCB is the centroid of the non-dimensional sectional area, and the LCF is the centroid
of the water-plane area we can also change the LCF
Figure 3 Lackenby Tranformation
The dotted line in Figure 3 represents the new longitudinal fractional sectional area
curve and the solid line represents the initial one. By changing this curve the prismatic
coefficient of aft-body and fore-body will be changed and the hull form could be
regenerated to fit for the requirement. The full details is provided in “Lackenby
Transformation Method” section.
Parameters that are going to be changed
The parameters that are going to be modified are the length to beam ratio (L/B), beam
to draught ratio (B/T); prismatic ratio (Cp); longitudinal centre of buoyancy (LCB);
longitudinal centre of floatation (LCF) and water-plane area (Aw).
LCB is calculated by integrating every section area times its longitudinal position, and
divided by the displaced volume of the canoe body.
LCF is calculated by integrating the waterline half-beam offset of each section, times
its longitudinal position, and divided by the water-plane area.
These parameters are chosen to be modified as they are used in the Delft series related
regression formula used to predict the yacht resistance. As the waterline length and the

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displacement are to be kept fixed only the beam, draught, Cp, LCB, LCF and Aw are
going to be modified. During the modification we have to make sure all modified hull
are consistent with the parameters of the Delft series.
As yacht length is kept constant, by changing the L/B and B/T ratio, the beam and
draught could be modified to a preferred value. The modification of LCB, LCF, Cp and
Aw is more complicated and can be done by manipulation of the sectional area curve
(Cp, LCB) and the half beam curve (Aw and LCF) (Petter, Optimization of a Yacht
Hull, 2012).
5) Lewis Section Mapping Method
The Lewis mapping method used in this project is an improved one which can be
applied where the entrance angle and the dead-rise angle are not 90 degree and 0 degree
separately which is the requirement of the original version.
Through using this method the beam; draught; sectional area; entrance angle and dead
rise angle will be required and sections can be mapped into a shape generated with
coordinates.
The full details will be provided in “Lewis Mapping Method” section.
6) YD-40 Parameter Check.
The initial hull chosen is the YD-40, which is a modern cruiser-racer used as the
example in the ‘Principals of Yacht Design’ (Eliasson & Larsson, 2011). The YD-40 is
recognized as a round bilge hull which has a single fin keel; spade rudder and masthead
rig with a spinnaker. The parameters of the YD-40 shown in the Table 2.

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Table 2 Hull Form Data of Parent Hull
For the keel:
The tip chord of the keel is 1.05m and the root 1.85m, which gives a taper ratio of 0.57.
With a span of 1.50m this gives a geometric aspect ratio of 1.0 and an effective ratio of
2.0 assuming the bottom to be a flat wall.
The root of the keel is a 10.5% foil in NACA-63 form and 17.5% NACA-65 for the tip.
The section type is changed linearly between the two extremes, while the thickness ratio
has a break point 0.65m below the root, where the ratio is 14%.
For the rudder:
The taper ratio of the rudder of YD-40 is chosen as 0.46. The root chord is thus 0.688m
and the tip chord is 0.320m. The span of the rudder is 1.47m which gives a high
geometric aspect ratio of 2.9 and an area of 0.74 m2
.
The sail area/wetted area ratio is 2.4 and the sail area/(displacement)2/3
ratio is 19.7 for
the yacht in the light displacement condition, which indicate the YD-40 will have a
faster speed in light wind condition. Besides of that the high aspect ratio of fore triangle
Lbp 10.02 m
aft body
Particullars
B 3.167 m Cpa 0.639553273
T 0.616 m Lever aft body 1st 0.353922861
Volume 7.622 m^3 Lever aft body 2nd 0.178198012
Cb 0.389917249
Points involved in
general method
LCB fractional
of half length "z"
-0.06964824 Af 0.184043221
LCB from mid
ship
-0.34893768 Bf 0.492224802
Cp 0.560127045 Cf 0.071568716
Cm 0.696122874 Aa 0.186848225
Fore body
Particullars
Ba 0.59301437
Cpf 0.480700818 Ca 0.026901864
Lever fore bofy
1st
0.30856781
Lever fore bofy
2nd
0.142893426
Aw 22.308 m^2

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(7.8) and mainsail (6.4) give the YD-40 a high efficiency in upwind condition (Petter,
Optimization of a Yacht Hull, 2012).
The importance of quoting this part is all of these parameters of YD-40 are the target
value when we systematically generate the initial hull form as there is no reference
example of YD-40 drawing for us.
7) Velocity Prediction Program
Basically the VPP produce an estimate of yacht velocity of a function of changing wind
condition. Applying the VPP relationships between heel angle and hydrostatic righting
moment are needed. The basic solution requires equilibrium of the aerodynamic and
hydrodynamic forward and side forces, and the heeling and the righting moment, finally
the boat speed with all selected wind conditions will normally displayed in a polar
diagram. The important thing for anyone who concerned within the polar diagram is
the Velocity made good (VMG) as it tells what is the highest speed with associate wind
angle in a wind condition (Claughton, 2006). The Figure 4 shows a typical sailing polar
diagram (David, 2015).
Figure 4 Sailing Polar Diagram
It should be noted that the optimum boat speed is different from VMG, as the VMG is
the speed when boat heading to the wind direction, but the boat speed is the actual
velocity of the boat. It can found at the Figure 4 when boat sailing at 6.25 knots the
associate VMG is 5.3 knots at true wind angle of 38.

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A Typical VPP Calculation Sequence
In general a VPP is carried out as a iterate procedure. In order to understand how a
general VPP is undertook a flowchart showed in Figure 5 (Day, 2014).
There are 6 steps to carry out a results for a VPP.
Step 1. In put the first guess of estimated speed and heel angle.
Step 2. When holding the heel angle constant estimate the aerodynamic drive and
hydrodynamic resistance to calculate the new heel angle.
Step 3. Estimate the heel moment and righting moment while keep speed as constant
with the heel angle from previous step.
Step 4. Check if the drive equal to the resistance with applying previous results.
Step 5. If the resistance equals to the drive then the final speed and heel angle are output.
Step 6. If the resistance not equals to the drive then the iteration should restart from step
2 with applying current speed and heel angle.
Figure 5 VPP Flow Chart
Potential Problems and possible solution

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Although this simple VPP sequence works much of the time, there are numbers of
potential problems which can cause the approach to fail.
1. Failure to Converge
In some case there needs more iteration procedure to find the answer.
2. High wind speed.
This problem has big possibility to occur as in reality the high wind speed leads the
yacht has a high heel angle which exceeds the heel angle range for the coefficients.
To deal with this problem an extrapolation to extend the range of validity of the
hydrodynamic solution can be made but this will significantly increase the
inaccuracy.
3. Reef, Flat and other features.
Some VPP will introducing two variables to reflect the fact that in reality the crew
always modify the sail shape or planform to control the speed.
However as this project will only focus on the yacht itself but not the behaviour of
the crews, this problem will be ignored

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6. Lackenby Transformation Method
Nomenclatures for thi section:
A.P.0 Station 0 at After Prependicular
B Maximum Beam of the Ship
"Aa,Ba,Ca" Points used to define Aftbody Curve
"Af,Bf,Cf" Points used to define Forebody Curve
Cb Blcok Coefficient
Cm Mid-ship Coefficient
Cpa Prismatic Coefficient for the FAft-Ship
Cpf Prismatic Coefficient for the Fwd-Ship
F.P.24 Station 24 at Front Prependicular
h Distance Between Two Neighbour Station in Meter
Ka^2 2nd moment non-dimensional lever of aft body
Kf^2 2nd moment non-dimensional lever of fore body
Lbp Length Between Perpendicular
LCB Longitudinal Centre of Buoyancy
LCF Longitudinal Centre of Flotation
M.P.12 Station 12 at Mid-Ship
S.M The Simpson's Multiplier
T Draught of the Ship
V Displaced Volum of the Ship
X Distance from the end of the ship to the current station
xa 1st non-dimensional lever of Aft body
xf 1st non-dimensional lever of fore body
Y Value Fractional Sectional Area

26
z LCB forward of the mid-ship as a fraction of half length
фf Prismatic Coefficient for the Fwd-Ship
φt Prismatic Coeffcient of the Ship
φa Prismatic Coefficient for the FAft-Ship
lδфa Limitation of much Cp can be modified of aftbody
lδфf Limitation of much Cp can be modified of forebody
δфa Change of Cp of Aftbody
δфf Change of Cp of Forebody
δxa Longitudinal shift of Aftbody
δXf Longitudinal shift of Forebody
In this Section, the implementation of using Lackenby transformation (H.LACKENBY,
1999) to modify sectional area curve will be provided. With using this method the hull
form parameters could be modified to a required value.
The implementation will start from the very beginning to the final results in respect of
the fractional sectional area distribution with three group of hull form parameters
combinations, i.e Cb and LCB; LCB and LCF; CB and LCF.
From the original version, it can’t change the position of LCF but LCB, however, based
on the fact that the LCB is the longitudinal centre of the sectional area curve, and the
LCF is the longitudinal centre of the half beam curve, a same procedure can be
generated to modify the position of LCF.
All of the formulas are used according to the “Lackenby” paper (H.LACKENBY, 1999),
therefore its not necessary to give the reference for a single one.
6.1.STEP 1 Preparation of the Hull Form Parameters

27
1. Evenly divide the ship into 25 transverse stations. (There are two intermediate
station known as station 2 and 23). The distance from each of the station to the
A.P.0 (The first section, section “0”) is defined as “X”.
2. Find the fractional cross sectional area of each station, then designated them as “Y”.
The fractional area is defined as:
𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑟𝑒𝑎 =
𝑎𝑟𝑒𝑎 𝑜𝑓 𝑠𝑒𝑙𝑒𝑐𝑡𝑒𝑐𝑑 𝑠𝑡𝑎𝑡𝑖𝑜𝑛
𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑟𝑒𝑎
(The fractional area used in this paper are the result directly from the Maxsurf
(Bentley|SYstems, 2015))
3. Drawing the non-dimensional sectional area curve with point defined by point (X,
Y).
4. Determine the “Simpson’s” table with known data. For example, using the data of
the parent ship see Table 3:
Table 3Simpsion's Table represents the data of parent ship
Station X
Distance
between
the
station
Fractional
area
S.M.
Non-
dimensional
volume
Fractional
lever
First
Moment
Second
Moment
A.P.0 0.68487 0.227687 0 0.5 0 1 0 0
1 0.912557 0.227726 0.032967 2 0.065934 0.954553 0.0629375 0.060077
2 1.140283 0.455454 0.09453 1.5 0.141795 0.9090983 0.1289056 0.117188
3 1.595737 0.455453 0.238675 4 0.9547 0.8181884 0.7811245 0.639107
4 2.05119 0.455453 0.38932 2 0.77864 0.7272787 0.5662883 0.411849
5 2.506643 0.455453 0.535565 4 2.14226 0.636369 1.3632679 0.867541
6 2.962096 0.455453 0.670396 2 1.340792 0.5454593 0.7313475 0.39892
7 3.417549 0.455453 0.788473 4 3.153892 0.4545496 1.4336005 0.651643
8 3.873002 0.455454 0.882097 2 1.764194 0.36364 0.6415314 0.233286
9 4.328456 0.455453 0.950267 4 3.801068 0.2727301 1.0366655 0.28273
10 4.783909 0.455453 0.990873 2 1.981746 0.1818204 0.3603218 0.065514
11 5.239362 0.455453 1 4 4 0.0909107 0.3636428 0.033059
0.980237
0.980237
13 6.150268 0.455453 0.931444 4 3.725776 0.0909081 0.3387034 0.030791
14 6.605721 0.455453 0.85593 2 1.71186 0.1818173 0.3112458 0.05659
15 7.061174 0.455454 0.759329 4 3.037316 0.2727264 0.8283564 0.225915
16 7.516628 0.455453 0.647669 2 1.295338 0.3636358 0.4710312 0.171284
17 7.972081 0.455453 0.528492 4 2.113968 0.4545449 0.9608934 0.436769
18 8.427534 0.455453 0.410198 2 0.820396 0.5454541 0.4474883 0.244084
19 8.882987 0.455453 0.300245 4 1.20098 0.6363632 0.7642595 0.486347
20 9.33844 0.455453 0.202777 2 0.405554 0.7272724 0.2949482 0.214508
21 9.793893 0.455453 0.118533 4 0.474132 0.8181815 0.387926 0.317394
22 10.249346 0.227727 0.046008 1.5 0.069012 0.9090907 0.0627382 0.057035
23 10.477073 0.227727 0.014279 2 0.028558 0.9545453 0.0272599 0.026021
F.P.24 10.7048 0 0 0.5 0 1 0 0
66 4.8948504 2.266737
Afterbody
Non-di V
21.105258
Forebody
Non-di V
15.863127
Total Non-
di V
36.968385
7.4696333 3.760915M.P.12 5.694815 0.455453 0.980237 2 0

28
4.1.There are 25 station are generated and the first station locates at the aft-
perpendicular line denoted as section “0”.
4.2. The distance between station represents the actual distance (in “meter”) between
two stations defined as:
𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑠
= (𝑋 𝑜𝑓 𝑠𝑡𝑎𝑡𝑖𝑜𝑛 𝑁𝑜. x+1")-(X of station No.𝑥")
Example:
Distance between station 3 and station 4 is “2.05119-1.595737=0.455453”
This value is also denoted as “h” which will be used to represent the actual distance
between two neighbour stations and kept fixed for all stations.
4.3. The fractional area is defined as “2”.
4.4.“S.M” represent the Simpson’s multipliers.
4.5. The non-dimensional volume represent the product of “fractional area multiplied
by S.M”.
Example see table 3:
𝑛𝑜𝑛 − 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 𝑣𝑜𝑙𝑢𝑚 𝑜𝑓 𝑠𝑡𝑎𝑡𝑖𝑜𝑛 5 = 0.535565 ∗ 4 = 2.14226
4.6.The fractional lever represent the fractional distance from one station to the mid-
ship.
4.6.1. The fractional lever of aft-body defined as:
𝑓𝑟𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝐿𝑒𝑣𝑒𝑟 𝑎𝑓𝑡 =
"𝑋" 𝑜𝑓 𝑀. 𝑃. 12 − "𝑋" 𝑜𝑓 𝑎𝑛𝑦 𝑎𝑓𝑡 𝑠𝑡𝑎𝑡𝑖𝑜𝑛
"𝑋" 𝑜𝑓 𝑀. 𝑃. 12 − "𝑋" 𝑜𝑓 𝐴. 𝑃. 0
4.6.2. The fractional lever of Fwd-body defined as:
𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝐿𝑒𝑣𝑒𝑟 𝑓𝑤𝑑 = 1 −
("X" of F.P.24 -"X" of any fwd station)
("X" of F.P.24-"X " of M.P.12)
Examples see Table 3:
𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑙𝑒𝑣𝑒𝑟 𝑜𝑓 𝑠𝑡𝑎𝑡𝑖𝑜𝑛 3 =
5.964815 − 1.59573
5.694815 − 0.68487
= 0.8181884

29
𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑙𝑒𝑣𝑒𝑟 𝑜𝑓 𝑠𝑡𝑎𝑡𝑖𝑜𝑛 23 = 1 −
10.7048 − 10.477073
10.7048 − 5.694815
= 0.9545453
4.7. The first moment defined as:
𝑇ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑎𝑛𝑦 𝑠𝑡𝑎𝑡𝑖𝑜𝑛
= 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑙𝑒𝑣𝑒𝑟 𝑜𝑓 𝑎𝑛𝑦 𝑠𝑡𝑎𝑡𝑖𝑜𝑛 ∗ 𝑛𝑜𝑛
− 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑎𝑛𝑦 𝑠𝑡𝑎𝑡𝑖𝑜𝑛
Example see table 3:
1𝑠𝑡 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑠𝑡𝑎𝑡𝑖𝑜𝑛 10 = 1.981746 ∗ 0.1818204 = 0.3603218
4.8.The second moment defined as:
2𝑛𝑑 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑎𝑛𝑦 𝑠𝑡𝑎𝑡𝑖𝑜𝑛
= 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑙𝑒𝑣𝑒𝑟 𝑜𝑓 𝑎𝑛𝑦 𝑠𝑡𝑎𝑡𝑖𝑜𝑛
∗ 1𝑠𝑡 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑎𝑛𝑦 𝑠𝑡𝑎𝑡𝑖𝑜𝑛
Example see Table 3:
2𝑛𝑑 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑠𝑡𝑎𝑡𝑖𝑜𝑛 10 = 0.3603218 ∗ 0.1818204 = 0.065514
5. Using the data from Table 3 generated several quantities that relevant to the
procedure can be decided:
5.1. The sum of 1st
moment of aft-body and fwd-body separately. (Can be calculated as
4.89485 and 7.469633).
5.2.The sum of 2nd
moment of aft-body and fwd-body separately. (Can be calculated as
2.266737 and 3.760915).
5.3.The sum of “S.M”. (Calculated as 66).
5.4.The sum of non-dimensional volume of aft-body and fwd-body separately
(15.86127 and 21.105258), so the total amount is 36.968385.
6. Calculate the actual displacement volume with using dimensional transverse
sectional area of the parent ship, and the Simpson’s “141” rule will be used data in
this step are showed in Table 4 .

30
Table 4 Actural displacement volume of the parent ship
6.1. The product defined as sectional area multiplied by S.M.
6.2.The sum of the product is 50.2397.
6.3.The displacement volume is:
𝑉 = (
1
3
) ∗ ℎ ∗ (𝑠𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑟𝑜𝑑𝑢𝑐𝑡) = (
1
3
) ∗ 0.455453 ∗ 50.2397 = 7.622
Where “h” is the distance between two neighbour stations and kept constant.
7. Input the particulars of the parent ship.
Station
Sectional
area
S.M Product
A.P.0 0 0.5 0
1 0.044801 2 0.089602
2 0.128466 1.5 0.192699
3 0.324357 4 1.297428
4 0.529082 2 1.058164
5 0.727828 4 2.911312
6 0.911063 2 1.822126
7 1.071528 4 4.286112
8 1.198763 2 2.397526
9 1.291404 4 5.165616
10 1.346587 2 2.693174
11 1.358991 4 5.435964
M.P.12 1.332133 2 2.664266
13 1.265824 4 5.063296
14 1.163202 2 2.326404
15 1.031921 4 4.127684
16 0.880177 2 1.760354
17 0.718215 4 2.87286
18 0.557455 2 1.11491
19 0.40803 4 1.63212
20 0.275572 2 0.551144
21 0.161086 4 0.644344
22 0.062525 1.5 0.093788
23 0.019405 2 0.03881
F.P.24 0 0.5 0
50.2397
Displacem
ent
Volume
7.627274

31
7.1. Length between perpendiculars; Maximum Beam and Drought of the parent ship.
Examples:
𝐿𝑏𝑝 = 10.02 𝑚
𝐵 = 3.167𝑚
𝑇 = 0.616 𝑚
8. Determine the Block coefficient of the parent ship:
𝐶𝑏 =
𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑣𝑜𝑙𝑢𝑚 V
𝐿 ∗ 𝐵 ∗ 𝑇
=
7.622
10.02 ∗ 3.167 ∗ 0.616
= 0.39
9. Determine the prismatic coefficient of the parent ship:
𝜙𝑡 =
𝑡𝑜𝑡𝑎𝑙 𝑛𝑜𝑛 − 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒 𝑓𝑜𝑟𝑚 𝑠𝑡𝑒𝑝 5.4
𝑠𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑆. 𝑀 𝑓𝑟𝑜𝑚 5.3
=
36.968385
66
= 0.56
10. Determine the mid-ship coefficient of the parent ship:
𝐶𝑚 =
𝐶𝑏
𝜙𝑡
=
0.3899
0.56
= 0.696
11. Determine the LCB forward of the mid-ship as a fraction of half length (denoted as
z):
𝑧
=
(𝑠𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 1𝑠𝑡 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑓𝑤𝑑 𝑏𝑜𝑑𝑦) − (𝑠𝑢𝑚 𝑜𝑓 1𝑠𝑡 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑎𝑓𝑡 𝑏𝑜𝑑𝑦)
𝑠𝑢𝑚 𝑜𝑓 𝑡𝑜𝑡𝑎𝑙 𝑛𝑜𝑛 − 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒
𝑧 =
4.8948504 − 7.4696333
36.968385
= −0.0696482
The negative sign means the LCB is located behind the mid-ship.
12. Determine the actual distance of the LCB from the mid-ship (in “meter”):
𝐿𝐶𝐵 = 𝑧 ∗
𝐿𝑏𝑝
2
= −0.0696482 ∗
10.02
2
= −0.3489377
13. According to the Lackenby’s method there also need to determine some particulars
of the aft-body and fwd-body separately.
13.1. Particulars of fwd-body.
13.1.1. Prismatic coefficient:

32
𝜙𝑓 =
𝑠𝑢𝑚 𝑜𝑓 𝑓𝑜𝑟𝑒 𝑏𝑜𝑑𝑦 𝑛𝑜𝑛 − 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒 𝑓𝑟𝑜𝑚 "5.4"
𝑠𝑢𝑚 𝑜𝑓
𝑆. 𝑀
2
𝑓𝑟𝑜𝑚 "5.3"
=
15.8631
33
= 0.4807
13.1.2. 1st
non-dimensional lever:
𝑥𝑓 =
𝑠𝑢𝑚 𝑜𝑓 1𝑠𝑡 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑤𝑑 − 𝑏𝑜𝑑𝑦 𝑓𝑟𝑜𝑚 "5.1"
𝑠𝑢𝑚 𝑜𝑓 𝑓𝑤𝑑𝑏𝑜𝑑𝑦 𝑛𝑜𝑛 − 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒 𝑓𝑟𝑜𝑚 "5.4"
=
4.8848504
15.863127
= 0.30856781
13.1.3. 2nd
moment non-dimensional lever:
𝑘𝑓2
=
𝑠𝑢𝑚 𝑜𝑓 2𝑛𝑑 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑓𝑤𝑑𝑏𝑜𝑑𝑦 𝑓𝑟𝑜𝑚 "5.2"
𝑠𝑢𝑚 𝑜𝑓 𝑓𝑤𝑑 𝑏𝑜𝑑𝑦 𝑛𝑜𝑛 − 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 𝑣𝑜𝑙𝑢𝑚 𝑓𝑟𝑜𝑚 "5.4"
=
2.6674
15.863127
= 0.14289343
13.2. Particulars of aft-body:
𝜙𝑎 =
𝑠𝑢𝑚 𝑜𝑓 𝑎𝑓𝑡 𝑏𝑜𝑑𝑦 𝑛𝑜𝑛 − 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒 𝑓𝑟𝑜𝑚 "5.4"
𝑆. 𝑀
2
𝑓𝑟𝑜𝑚 "5.3"
=
21.105258
33
= 0.63955327
13.2.2. 1st
𝑥𝑎 =
𝑠𝑢𝑚 𝑜𝑓 1𝑠𝑡 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑓𝑡 − 𝑏𝑜𝑑𝑦 𝑓𝑟𝑜𝑚 "5.1"
𝑠𝑢𝑚 𝑜𝑓 𝑎𝑓𝑡𝑏𝑜𝑑𝑦 𝑛𝑜𝑛 − 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒 𝑓𝑟𝑜𝑚 "5.4"
=
7.4696333
21.105258
= 0.35392286
13.2.3. 2nd
𝑘𝑎2
=
𝑠𝑢𝑚 𝑜𝑓 2𝑛𝑑 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑎𝑓𝑡𝑏𝑜𝑑𝑦 𝑓𝑟𝑜𝑚 "5.2"
𝑠𝑢𝑚 𝑜𝑓 𝑎𝑓𝑡 𝑏𝑜𝑑𝑦 𝑛𝑜𝑛 − 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 𝑣𝑜𝑙𝑢𝑚 𝑓𝑟𝑜𝑚 "5.4"
=
3.76092
21.105258
= 0.178198
14. There are certain recurrent expressions involving only the geometrical
characteristics of the parent form of the aft and fwd body separately.
14.1. The general function to define the three components are:
𝐴 = 𝜙 ∗ (1 − 2𝑥) − 𝑝(1 − 𝜙)
𝐵 =
𝜙(2𝑥 − 3𝑘2
− 𝑝(1 − 2𝑥))
𝐴

33
𝑐 =
𝐵(1 − 𝜙) − 𝜙(1 − 2𝑥)
1 − 𝑃
Where the “p” represents the length of the parallel middle body, which is zero
in our case. Therefore the definition of those components change as:
14.2. Reduced function to define the three components:
𝐴 = 𝜙(1 − 2𝑥)
𝐵 =
𝜙(2𝑥 − 3𝑘2)
𝐴
𝐶 = 𝐵(1 − 𝜙) − 𝜙(1 − 2𝑥)
14.3. Components to define the aft-body curve:
𝐴𝑎 = 𝜙𝑎(1 − 2𝑥𝑎) = 0.6396(1 − 2 ∗ 0.3539) = 0.1868
𝐵𝑎 =
𝜙𝑎(2𝑥𝑎 − 3𝑘𝑎2)
𝐴𝑎
=
0.6396(2 ∗ 0.3539 − 3 ∗ 0.1782)
0.1868
= 0.593
𝐶𝑎 = 𝐵𝑎(1 − 𝜙𝑎) − 𝜙𝑎(1 − 2𝑥𝑎)
= 0.593 ∗ (1 − 0.6396) − 0.6396 ∗ (1 − 2 ∗ 0.3539)
= 0.0269
All the value are based on the results of step 13.2
14.4. Components to define the fwd-body curve:
𝐴𝑓 = 𝜙𝑓(1 − 2𝑥𝑓) = 0.4807(1 − 2 ∗ 0.3086) = 0.184
𝐵𝑓 =
𝜙𝑓(2𝑥𝑓 − 3𝑘𝑓2)
𝐴𝑓
=
0.4807(2 ∗ 0.3089 − 3 ∗ 0.1429)
0.184
= 0.4922
𝐶𝑓 = 𝐵𝑓(1 − 𝜙𝑓) − 𝜙𝑓(1 − 2𝑥𝑓)
= 0.4922 ∗ (1 − 0.4807) − 0.4807 ∗ (1 − 2 ∗ 0.3086)
= 0.0716
15. The summary of the parent hull particulars showed below see Table 5:

34
Table 5 Parent Hull Data
6.2.STEP 2 Change the Hull When Block Coefficient is required to be Moved
16. The procedures below will illustrate how to change the block coefficient with
general method.
16.1. Determine the practical limitation of how much prismatic coefficient can be
modified for aft and fore body separately.
𝑙𝛿𝜙 𝑎 = +(−)
𝐴𝑎
2
𝑙𝛿𝜙 𝑓 = +(−)
𝐴𝑓
2
Lbp 10.02 m
aft body
Particullars
B 3.167 m Cpa 0.639553
T 0.616 m Lever aft body 1st 0.353923
Volume 7.622 m^3 Lever aft body 2nd 0.178198
Cb 0.38991725
Points involved in
general method
LCB fractional
of half length "z"
-0.06964824 Af 0.184043
LCB from mid
ship
-0.34893768 Bf 0.492225
Cp 0.56012705 Cf 0.071569
Cm 0.69612287 Aa 0.186848
Fore body
Particullars
Ba 0.593014
Cpf 0.48070082 Ca 0.026902
Lever fore bofy
1st
0.30856781
Lever fore bofy
2nd
0.14289343
Parent ship Particulars

35
16.2. Determine the change of total prismatic coefficient which is defined by:
𝛿𝜙 𝑡 =
𝛿𝐶 𝐵
𝐶 𝑚
16.3. Determine the lever for aft and fore body separately. The lever “h” in this case
is given by the constant B, therefore the lever is defined as:
ℎ𝑓 = 𝐵𝑓
ℎ𝑓 = 𝐵𝑎
16.4. Determine the change of the prismatic for fore and aft body separately.
𝛿𝜙 𝑓 =
2 ∗ 𝛿𝜙 𝑡 ∗ (ℎ 𝑎 + 𝑧)
ℎ𝑓 + ℎ 𝑎
𝛿𝜙 𝑎 =
2 ∗ 𝛿𝜙 𝑡 ∗ (ℎ𝑓 − 𝑧)
ℎ𝑓 + ℎ 𝑎
16.5. Determine the longitudinal shift (in meter) of the sections for fore and aft body
separately.
Aft body:
𝛿𝑥𝑎 =
𝛿𝜙 𝑎 ∗
𝐿𝑏𝑝
2
𝑥(1 − 𝑥)
𝐴𝑎
Fore body:
𝛿𝑥𝑓 =
𝛿𝜙 𝑓
𝐿𝑏𝑝
2
𝑥(1 − 𝑥)
𝐴𝑓
Example for STEP 2 will show the block coefficient be increased by 0.031 which is
approximately 8% of the original Cb:
𝑙𝛿𝜙 𝑎 = +(−)
0.187
2
= 0.0935
𝑙𝛿𝜙 𝑓 = +(−)
0.184
2
= 0.092
𝛿𝜙 𝑡 =
0.031
0.696
= 0.0445
ℎ𝑓 = 0.492
ℎ 𝑎 = 0.593
𝛿𝜙 𝑓 =
2 ∗ 0.0445 ∗ (0.593 + (−0.0696))
0.492 + 0.593
= 0.043

36
𝛿𝜙 𝑎 =
2 ∗ 0.0445 ∗ (0.492 − (−0.0696))
0.492 + 0.593
= 0.046
𝛿𝑥𝑎 =
0.046
0.187
∗ 5.01𝑥(1 − 𝑥) = 1.24𝑥(1 − 𝑥)
𝛿𝑥𝑓 =
0.043
0.184
∗ 5.01𝑥(1 − 𝑥) = 1.169𝑥(1 − 𝑥)
The Simpson’s Table (Table 6) will then be modified into the one like below:
Table 6 Simpson's Table When Cb increases by 0.031
As we can see from the Table 6 above the new longitudinal position of the stations are
given. (Notice should be made the new x of Aft body is x- Δx, where the new x of fore
body is x+ Δx).
Station X Δx new X
Distance
between
the
station
Fractional
area
S.M.
Function
of
volume
Fractional
lever
New
Lever
*First
Moment
*Second
Moment
A.P.0 0.68487 0 0.68487 0.227687 0 0.5 0 1 1 0 0
1 0.912557 0.053638 0.858919 0.227726 0.032967 2 0.065934 0.954553 0.965259 0.063643 0.061432
2 1.140283 0.102176 1.038107 0.455454 0.09453 1.5 0.141795 0.909098 0.929493 0.131797 0.122505
3 1.595737 0.183926 1.411811 0.455453 0.238675 4 0.9547 0.818188 0.8549 0.816173 0.697747
4 2.05119 0.245238 1.805952 0.455453 0.38932 2 0.77864 0.727279 0.776229 0.604403 0.469155
5 2.506643 0.286113 2.22053 0.455453 0.535565 4 2.14226 0.636369 0.693478 1.48561 1.030238
6 2.962096 0.306551 2.655545 0.455453 0.670396 2 1.340792 0.545459 0.606648 0.813388 0.49344
7 3.417549 0.306552 3.110997 0.455453 0.788473 4 3.153892 0.45455 0.515738 1.626583 0.838891
8 3.873002 0.286116 3.586886 0.455454 0.882097 2 1.764194 0.36364 0.420749 0.742284 0.312315
9 4.328456 0.245243 4.083213 0.455453 0.950267 4 3.801068 0.27273 0.321681 1.222732 0.39333
10 4.783909 0.183933 4.599976 0.455453 0.990873 2 1.981746 0.18182 0.218534 0.433079 0.094642
11 5.239362 0.102185 5.137177 0.455453 1 4 4 0.090911 0.111307 0.445229 0.049557
0.980237
0.980237
13 6.150268 0.096631 6.246899 0.455453 0.931444 4 3.725776 0.090908 0.110196 0.410565 0.045242
14 6.605721 0.173936 6.779657 0.455453 0.85593 2 1.71186 0.181817 0.216535 0.370678 0.080265
15 7.061174 0.231915 7.293089 0.455454 0.759329 4 3.037316 0.272726 0.319017 0.968956 0.309113
16 7.516628 0.270568 7.787196 0.455453 0.647669 2 1.295338 0.363636 0.417642 0.540987 0.225939
17 7.972081 0.289894 8.261975 0.455453 0.528492 4 2.113968 0.454545 0.512408 1.083215 0.555048
18 8.427534 0.289894 8.717428 0.455453 0.410198 2 0.820396 0.545454 0.603317 0.494959 0.298618
19 8.882987 0.270568 9.153555 0.455453 0.300245 4 1.20098 0.636363 0.690369 0.829119 0.572398
20 9.33844 0.231916 9.570356 0.455453 0.202777 2 0.405554 0.727272 0.773563 0.313722 0.242683
21 9.793893 0.173937 9.96783 0.455453 0.118533 4 0.474132 0.818182 0.8529 0.404387 0.344901
22 10.24935 0.096632 10.34598 0.227727 0.046008 1.5 0.069012 0.909091 0.928378 0.064069 0.059481
23 10.47707 0.050732 10.5278 0.227727 0.014279 2 0.028558 0.954545 0.964671 0.027549 0.026576
F.P.24 10.7048 0 10.7048 0 0 0.5 0 1 1 0 0
66 5.508206 2.760265
Afterbody 21.10526
Forebody 15.86313
Total 36.96839
8.384921 4.563253M.P.12 5.694815 0.455453 0.980237 2 0

37
Figure 6Fractional Sectional Area Curve of Cb+0.031
16.6. Data validation.
Although now the new fractional sectional area curve is produced (see figure 6)
according to the requirement to make the Cb increased by 0.031, it is necessary to
recheck the new Cb according to the new curve showed above. (red one).
According to the step 6 and 8 the displacement volume and the Cb of the parent hull
can be calculated with the Simpsons “141” rule. In order to make the rule still can be
used (i.e the longitudinal increment remains as 0.45545), the cubic interpolation is used
to get the interpolated sectional area based on the x of the parent hull. Then the new
displacement volume can be calculated so that the new Cb can be calculated see table
5.
The Figure 7 shows the new sectional area curve (using interpolated Y) and the old one.
Figure 7 New Sectional Area Curve of Cb

38
Table 7 New sectional area (interpolated y)
As the Table 7 shows, the new Cb is calculated as 0.421193 which is quite similar to
the value it should be as 0.421187 after the original Cb (0.390187) increased by 0.031.
If the difference is presented as percentage defined as:
𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 % =
((𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑑 𝐶𝑏 − 𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 𝐶𝑏)2)0.5
𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑑 𝐶𝑏
∗ 100%
Sectional
area
S.M Product
Sectional
area
new X X
Interplotation y
based on parent
X
S.M product
0 0.5 0 0 0.68487 0.68487 0 0.5 0
0.044801 2 0.089602 0.044801 0.858919 0.912557 0.066427965 2 0.132856
0.128466 1.5 0.192699 0.128466 1.038107 1.140283 0.181822046 1.5 0.272733
0.324357 4 1.297428 0.324357 1.411811 1.595737 0.42091556 4 1.683662
0.529082 2 1.058164 0.529082 1.805952 2.05119 0.649298276 2 1.298597
0.727828 4 2.911312 0.727828 2.22053 2.506643 0.85141501 4 3.40566
0.911063 2 1.822126 0.911063 2.655545 2.962096 1.023055953 2 2.046112
1.071528 4 4.286112 1.071528 3.110997 3.417549 1.158091979 4 4.632368
1.198763 2 2.397526 1.198763 3.586886 3.873002 1.256903105 2 2.513806
1.291404 4 5.165616 1.291404 4.083213 4.328456 1.322971812 4 5.291887
1.346587 2 2.693174 1.346587 4.599976 4.783909 1.355635297 2 2.711271
1.358991 4 5.435964 1.358991 5.137177 5.239362 1.356959402 4 5.427838
1.332133 2 2.664266 1.332133 5.694815 5.694815 1.332133 2 2.664266
1.265824 4 5.063296 1.265824 6.246899 6.150268 1.280397083 4 5.121588
1.163202 2 2.326404 1.163202 6.779657 6.605721 1.200699542 2 2.401399
1.031921 4 4.127684 1.031921 7.293089 7.061174 1.094881156 4 4.379525
0.880177 2 1.760354 0.880177 7.787196 7.516628 0.965963186 2 1.931926
0.718215 4 2.87286 0.718215 8.261975 7.972081 0.818469347 4 3.273877
0.557455 2 1.11491 0.557455 8.717428 8.427534 0.659838777 2 1.319678
0.40803 4 1.63212 0.40803 9.153555 8.882987 0.499755898 4 1.999024
0.275572 2 0.551144 0.275572 9.570356 9.33844 0.347873354 2 0.695747
0.161086 4 0.644344 0.161086 9.96783 9.793893 0.209450668 4 0.837803
0.062525 1.5 0.093788 0.062525 10.34598 10.24935 0.087779513 1.5 0.131669
0.019405 2 0.03881 0.019405 10.5278 10.47707 0.029484597 2 0.058969
0 0.5 0 0 10.7048 10.7048 0 0.5 0
50.2397 54.23226
Displace
ment
Volume
7.627274
New DV 8.233361
Cb 0.390187
New Cb 0.421193
Cb in
theory
0.421187
difference 0.001296

39
Then the difference in this case is 0.0013%. Therefore the Lackenby’s transformation
is a very accuracy method to parametrically change the hull form.
16.7. Data Expansion
In order to make a design chart later to get a full understand the change of the
performance according to the change of different parameters it is necessary to have
more different Cb.
As the limitation indicate that the variation of the block coefficient is to be +(-)16% of
the original Cb, then the variation of the Cb are from -16% to +16% with 8% percent
increment. Therefore the Cb generated are: 0.3272 (-16%); 0.3592(-8%); 0.4212(8%)
and 0.4532(16%).
16.8. Error prediction
Through using the Lackenby’s transformation method there are also some difference
happen between the result (calculated Cb) value and the target value (required Cb),
however according to the difference calculated in this project, these difference (or error)
are very small that can be ignored.
The Table 8 below shows the difference between the calculated Cb and the required Cb
in percentage against the change of the Cb.
𝑒𝑟𝑟𝑜𝑟 = 𝐴𝐵𝑆 (
𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 𝐶𝑏 − 𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑑 𝐶𝑏
𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑑 𝐶𝑏
) ∗ 100%
Table 8 Data Error
Original Cb New Cb
Change
of Cb
Difference Average
Standard
Deviation
0.390187 0.3272 -0.06299 0.001 0.00598 0.011153
0.390187 0.3592 -0.03099 0.0017 0.00598 0.011153
0.390187 0.390187 0 0 0.00598 0.011153
0.390187 0.4212 0.031013 0.0013 0.00598 0.011153
0.390187 0.4532 0.063013 0.0259 0.00598 0.011153

40
Figure 8 Error of changing the Cb
As it is showed in the Figure 8 above, the average of the error is 0.00598% and the
standard deviation is 0.011153 which are quite small that can be ignored. It is necessary
to reclaim that all of the new Cb are in the limitation of the Lackenby method to get
small errors, otherwise the error is going to be uncontrollable introduced by Lackenby
method.
6.3.NOTICE
Due to the limitation of the range parameters used in Delft Series regression formula,
the final Block Coefficient are determined as 0.36; 0.376; 0.3899 (initial); 0.4 and
0.4176.
Although the target values are changed the procedure is unchanged and still reliable.
6.4.STEP 3 Change the Hull When LCB is required to be moved
17. Normally after the Cb is changed by using procedures introduced in STEP 2, the
LCB will be changed consequently, which can be calculated with applying “11”
and “12” introduced in STEP 1.

41
This LCB value should be treated as the initial LCB value instead of the value from
parent hull parameter data.
17.1. Decide the change of “z”.
𝛿𝑧 =
𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑑 𝐿𝐶𝐵 − 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝐿𝐶𝐵
𝐿𝑏𝑝
2
17.2. Decide the change of prismatic coefficient of the ship:
𝛿𝐶𝑝𝑡 =
𝛿 𝐶𝐵
𝐶𝑚
17.3. Decide the change of prismatic coefficient of fore and aft body separately.
𝐹𝑜𝑟 𝐴𝑓𝑡 𝑏𝑜𝑑𝑦: 𝛿𝐶𝑝𝑎 =
2 ∗ (𝛿𝐶𝑝𝑡 ∗ (𝐵𝑓 − 𝑧) − 𝛿𝑧 ∗ (𝐶𝑝𝑡 + 𝛿𝐶𝑝𝑡))
𝐵𝑓 + 𝐵𝑎
𝐹𝑜𝑟 𝐹𝑜𝑟𝑒 𝑏𝑜𝑑𝑦: 𝛿𝐶𝑝𝑓 =
2 ∗ (𝛿𝐶𝑝𝑡 ∗ (𝐵𝑎 + 𝑧) + 𝛿𝑧 ∗ (𝐶𝑝𝑡 + 𝛿𝐶𝑝𝑡))
𝐵𝑓 + 𝐵𝑎
17.4. Decided the shift of the fore-body and aft-body section separately:
𝛿𝑋𝑓 =
𝛿𝐶𝑝𝑓 ∗
𝐿𝑏𝑝
2
𝐴𝑓
𝛿𝑋𝑎 =
𝛿𝐶𝑝𝑎 ∗
𝐿𝑏𝑝
2
𝐴𝑎
17.5. Applying the shift with each section to get the shift of X for fore and aft body.
𝑆ℎ𝑖𝑓𝑡 𝑜𝑓 𝑁𝑒𝑤 𝑋 𝑜𝑓 𝑓𝑜𝑟𝑒 𝑏𝑜𝑑𝑦 = 𝛿𝑋𝑓 ∗ 1 𝑠𝑡
𝐿𝑒𝑣𝑒𝑟 ∗ (1 − 1 𝑠𝑡
𝐿𝑒𝑣𝑒𝑟)
𝑆ℎ𝑖𝑓𝑡 𝑜𝑓 𝑁𝑒𝑤 𝑋 𝑜𝑓 𝑎𝑓𝑡 𝑏𝑜𝑑𝑦 = 𝛿𝑋𝑎 ∗ 1 𝑠𝑡
𝐿𝑒𝑣𝑒𝑟 ∗ (1 − 1 𝑠𝑡
𝐿𝑒𝑣𝑒𝑟)
17.6. Determine the New X.
17.7. Using the new X to get the new 1st
fractional lever.
17.7.1. The new fractional lever of aft-body defined as:
𝑁𝑒𝑤 𝑓𝑟𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝐿𝑒𝑣𝑒𝑟 𝑎𝑓𝑡
=
"𝑁𝑒𝑤 𝑋" 𝑜𝑓 𝑀. 𝑃. 12 − "𝑁𝑒𝑤 𝑋" 𝑜𝑓 𝑎𝑛𝑦 𝑎𝑓𝑡 𝑠𝑡𝑎𝑡𝑖𝑜𝑛
"𝑁𝑒𝑤 𝑋" 𝑜𝑓 𝑀. 𝑃. 12 − "𝑁𝑒𝑤 𝑋" 𝑜𝑓 𝐴. 𝑃. 0
17.7.2. The new fractional lever of Fwd-body defined as:

42
𝑁𝑒𝑤 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝐿𝑒𝑣𝑒𝑟 𝑓𝑤𝑑
= 1
−
("New X" of F.P.24 -"New X" of any fwd station)
("𝑁𝑒𝑤X" of F.P.24-"New X " of M.P.12)
17.8. Determine the new lever and the non-dimensional volume (see”4.5”) to cal
culate the new 1st
moment.
𝑁𝑒𝑤 𝑓𝑖𝑟𝑠𝑡 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑎𝑛𝑦 𝑠𝑒𝑐𝑡𝑖𝑜𝑛
= 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑎𝑛𝑦 station *"New X"
17.9. Sum the 1st
moment for fore and aft body separately.
17.10. Calculate the new LCB.
𝑁𝑒𝑤 𝐿𝐶𝐵
=
(𝑠𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 1𝑠𝑡 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑓𝑤𝑑 𝑏𝑜𝑑𝑦) − (𝑠𝑢𝑚 𝑜𝑓 1𝑠𝑡 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑎𝑓𝑡 𝑏𝑜𝑑𝑦)
𝑠𝑢𝑚 𝑜𝑓 𝑡𝑜𝑡𝑎𝑙 𝑛𝑜𝑛 − 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒
∗
𝐿𝑏𝑝
2
In this example the Cb value kept unchanged with parent hull form parameters, thus
there is no change of the initial position of LCB wich is 0.349 m after the mid-ship. The
required LCB value is -0.301, thus it required to be moved forward from it original
position.
The full data is provided in Table 9.

43
Table 9 Full data needed when moving LCB as -0.301
Increase
Cbby
0.025
StationXΔxnewX
Distance
between
the
station
Fractional
area
S.M.
Function
of
volume
Fractional
lever
New
Lever
*First
Moment
*Second
Moment
Sectional
area
S.MProduct
Sectional
area
newXX
Interplota
tiony
basedon
parentX
S.Mproduct
Fractional
area
hf0.492225A.P.00.6848700.684870.22768700.50110000.5000.684870.6848700.500
ha0.59301410.912557-0.011490.9240470.2277260.03296720.0659340.9545530.952260.0627860.0597890.04480120.0896020.0448010.9240470.9125570.04178720.0835750.032967
z-0.0696521.140283-0.021891.1621710.4554540.094531.50.1417950.9090980.9047290.1282860.1160640.1284661.50.1926990.1284661.1621711.1402830.119791.50.1796850.09453
Origional
Cb
0.38991731.595737-0.03941.6351370.4554530.23867540.95470.8181880.8103240.7736160.626880.32435741.2974280.3243571.6351371.5957370.30747841.2299110.238675
required
Cb
0.38991742.05119-0.052532.1037240.4554530.3893220.778640.7272790.7167930.5581240.4000590.52908221.0581640.5290822.1037242.051190.5060721.0121410.38932
Change
ofCb
-2.5E-0752.506643-0.061292.5679330.4554530.53556542.142260.6363690.6241351.337060.8345070.72782842.9113120.7278282.5679332.5066430.70222342.8088940.535565
changeof
Cp
-3.6E-0762.962096-0.065673.0277640.4554530.67039621.3407920.5454590.5323520.7137730.3799780.91106321.8221260.9110633.0277642.9620960.88596521.771930.670396
LCB-0.0473.417549-0.065673.4832170.4554530.78847343.1538920.454550.4414421.3922610.6146021.07152844.2861121.0715283.4832173.4175491.05018144.2007230.788473
hf0.49222583.873002-0.061293.9342930.4554540.88209721.7641940.363640.3514060.6199490.2178541.19876322.3975261.1987633.9342933.8730021.18344222.3668840.882097
ha0.59301494.328456-0.052534.3809910.4554530.95026743.8010680.272730.2622440.9968070.2614071.29140445.1656161.2914044.3809914.3284561.28230945.1292370.950267
LCBafter
Cb
changed
-0.34894104.783909-0.03944.823310.4554530.99087321.9817460.181820.1739560.3447360.0599691.34658722.6931741.3465874.823314.7839091.34335522.6867110.990873
required
newLCB
-0.301115.239362-0.021895.2612520.4554531440.0909110.0865410.3461660.0299581.35899145.4359641.3589915.2612525.2393621.35937145.4374841
changeof
LCB
0.047940.9802371.33213322.6642661.3321335.6948155.6948151.33213322.6642660.980237
changeof
Z
0.0095690.9802371.26582445.0632961.2658246.1724896.1502681.26975945.0790340.931444
Change
ofCp
-3.6E-07136.1502680.0222216.1724890.4554530.93144443.7257760.0909080.0953440.3552290.0338691.16320222.3264041.1632026.6457196.6057211.1731522.3463010.85593
Change
ofCpf
0.009877146.6057210.0399986.6457190.4554530.8559321.711860.1818170.1898010.3249130.0616691.03192144.1276841.0319217.1145057.0611741.04811744.1924690.759329
changeof
Cpa
-0.00988157.0611740.0533317.1145050.4554540.75932943.0373160.2727260.2833710.8606880.2438940.88017721.7603540.8801777.5788477.5166280.90143521.8028690.647669
Change
ofXf
0.268877167.5166280.0622197.5788470.4554530.64766921.2953380.3636360.3760550.4871180.1831830.71821542.872860.7182158.0387457.9720810.74198342.9679320.528492
changeof
Xa
-0.26486177.9720810.0666648.0387450.4554530.52849242.1139680.4545450.4678510.9890220.4627150.55745521.114910.5574558.4941988.4275340.58058121.1611610.410198
188.4275340.0666648.4941980.4554530.41019820.8203960.5454540.558760.4584050.2561380.4080341.632120.408038.9452078.8829870.42778541.7111390.300245
198.8829870.062228.9452070.4554530.30024541.200980.6363630.6487820.7791750.5055150.27557220.5511440.2755729.3917719.338440.29053820.5810770.202777
209.338440.0533319.3917710.4554530.20277720.4055540.7272720.7379170.2992650.2208330.16108640.6443440.1610869.8338919.7938930.17067340.6826910.118533
219.7938930.0399989.8338910.4554530.11853340.4741320.8181820.8261650.3917110.3236180.0625251.50.0937880.06252510.2715710.249350.0674781.50.1012170.046008
2210.249350.02222110.271570.2277270.0460081.50.0690120.9090910.9135260.0630440.0575930.01940520.038810.01940510.4887410.477070.02114520.0422890.014279
2310.477070.01166610.488740.2277270.01427920.0285580.9545450.9568740.0273260.02614800.50010.704810.704800.500
F.P.2410.7048010.7048000.50110050.239750.23962
665.0358972.375175
Displace
ment
Volume
7.627274
Afterbody21.10526NewDV7.627212
Forebody15.86313
required
LCB
-0.301Cb0.390187
Total36.96839
Calculate
dLCB
-0.30325
required
Cb
0.389917
Differenc
e
0.74796
Calculate
dCb
0.390184
Difference0.068438
7.2735643.601066M.P.125.6948150.4554530.98023720

44
The change of LCB:
𝛿𝐿𝐶𝐵 = −0.301 + 0.349 = 0.0479
Change of “z”
𝛿𝑧 =
0.0479
5.01
= 0.0096
Change of prismatic coefficient of the ship:
𝛿𝐶𝑝𝑡 =
0
0.6961
= 0
Change of prismatic coefficient of the fore-body:
𝛿𝐶𝑝𝑓 =
2 ∗ (0 ∗ (0.4922 + (0.07)) + 0.0096 ∗ (0.5601 + 0))
0.4922 + 0.593
= 0.0099
Change of prismatic coefficient of the aft-body:
𝛿𝐶𝑝𝑎 =
2 ∗ (0 ∗ (0.593 − (−0.07)) − 0.0096 ∗ (0.5601 + 0))
0.4922 + 0.593
= −0.01
Shift of the fore-body section, using station 14 see table 9:
𝛿𝑋𝑓 =
0.0099 ∗ 5.01
0.184
∗ (1 − 0.1818) ∗ 0.1818 = 0.04
Shift of the aft-body section, using station 5 see table 9:
𝛿𝑋𝑎 =
−0.01 ∗ 5.01
0.1868
∗ (1 − 0.6364) ∗ 0.6364 = −0.061
New X of station 14:
𝑁𝑒𝑤 𝑋 = 6.6057 + 0.04 = 6.6457
New X of station 5:
𝑁𝑒𝑤 𝑋 = 2.5066 − (−0.0661) = 2.5679
Applying to all stations.
The new fractional lever of aft-body defined as using station 5:
𝑁𝑒𝑤 𝑓𝑟𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝐿𝑒𝑣𝑒𝑟 𝑎𝑓𝑡 =
5.69482 − 2.5679
5.69482 − 0.68487
= 0.6241

45
The new fractional lever of Fwd-body defined as using station 14:
𝑁𝑒𝑤 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝐿𝑒𝑣𝑒𝑟 𝑓𝑤𝑑 = 1 −
(10.7048-6.6457)
(10.7048-5.69482)
= 0.1898
The new 1st
moment of any station, using station 14:
𝑁𝑒𝑤 1𝑠𝑡 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑠𝑡𝑎𝑡𝑖𝑜𝑛 14 = 1.7119 ∗ 0.1898 = 0.3249
Sum the 1st
moment of fore and aft body separately, see Table 7.
𝑆𝑢𝑚 𝑜𝑓 1𝑠𝑡 𝑚𝑜𝑚𝑒𝑛𝑡 𝑓𝑜𝑟 𝑎𝑓𝑡 𝑏𝑜𝑑𝑦 = 7.2736
𝑆𝑢𝑚 𝑜𝑓 1𝑠𝑡 𝑚𝑜𝑚𝑒𝑛𝑡 𝑓𝑜𝑟 𝑎𝑓𝑡 𝑏𝑜𝑑𝑦 = 5.0359
The new LCB:
𝑁𝑒𝑤 𝐿𝐶𝐵 =
5.0359 − 7.2736
36.968
∗ 5.01 = −0.303
6.5.STEP 4 Change the hull when LCF is required to be changed.
The procedure of changing the LCF is similar to the procedure of changing the LCB,
however, the LCF is relative to the shape of half beam curve instead of sectional area
curve.
18. Based on the STEP 1, a “Simpsons Table” associates with the halfbeam data should
be generated. “X” in this step should be still same to the step “1”, thus the distance
from any station to the station “0”.
18.1. The halfbeam value should replace the sectional area value, and associate
fractional half beam value should be generated to replace the fractional sectional
area value in Table 10.
18.2. The corresponding data table showed below:

46
Table 10 Simpsons table when changing LCF
18.3. The signs of the parameters in next steps could be the same as pervious steps,
however, it should be noticed that they are based on the water plane or calledhalf
beam curve see Figure 9 but not sectional area curve.
Figure 9 Half Beam Curve
18.4. Determine the prismatic coefficient of the half beam curve (similar to step “9”).
𝐶𝑝 =
𝑡𝑜𝑡𝑎𝑙 𝑛𝑜𝑛 − 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒
𝑠𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑆. 𝑀
=
46.472
66
= 0.7041
StationXchangeofXnewX
Distance
betweenthe
station
Halfbeamat
waterline
Fractional
halfbeam
S.M.
Non-
dimensio
nal
volume
Fractional
lever
FirstMoment
Second
Moment
newlever
Newfirst
Moment
A.P.00.6848700.684870.227687000.5010010
10.912557-0.0243745030.93693150.2277260.80.506008921.0120180.9545530.9660245810.9221220.9496880.961100843
21.140283-0.046431611.186714610.4554541.050.66413661.50.9962050.9090980.9056482060.8233230.899830.8964154
31.595737-0.0835806541.679317650.4554531.270.803289143.2131560.8181882.6289671712.150990.8015052.575361668
42.05119-0.1114424832.162632480.4554531.390.879190421.7583810.7272791.2788329090.9300680.7050341.239718564
52.506643-0.1300171892.636660190.4554531.4740.932321343.7292850.6363692.3732016291.5102320.6104172.276418537
62.962096-0.1393047713.101400770.4554531.530.967741921.9354840.5454591.0557277470.5758570.5176531.001909484
73.417549-0.139305233.556854230.4554531.5660.990512343.9620490.454551.8009481240.818620.4267431.690778252
83.873002-0.1300185654.003020570.4554541.5811220.363640.7272799130.2644680.3376870.675374454
94.328456-0.1114447264.439900730.4554531.580.999367543.997470.272731.090230250.2973390.2504851.001304826
104.783909-0.0835837944.867492790.4554531.560.986717321.9734350.181820.3588106120.0652390.1651360.325885057
115.239362-0.0464357395.285797740.4554531.5230.963314443.8532570.0909110.3503022830.0318460.0816410.314584052
M.P.125.6948155.6948150.4554531.470.929791320.929791013.535973428.390104012.95885114
0.929791
136.1502680.2373436026.38761160.4554531.40.885515543.5420620.0909080.3220022930.0292730.1382830.489807556
146.6057210.427220857.032941850.4554531.310.828589521.6571790.1818170.3013037990.0547820.2670920.442619232
157.0611740.5696288767.630802880.4554541.2040.761543343.0461730.2727260.8307719960.2265730.3864261.17712021
167.5166280.6645678398.181195840.4554531.080.68311221.3662240.3636360.4968079010.1806570.4962850.678036551
177.9720810.712037328.684118320.4554530.940.594560442.3782420.4545451.0810176660.4913710.5966691.419023324
188.4275340.7120375819.139571580.4554530.7870.497786220.9955720.5454540.5430390330.2962030.6875780.684533917
198.8829870.664568629.547555620.4554530.6280.39721741.5888680.6363631.0110970310.6434250.7690121.221859054
209.338440.569630449.908070440.4554530.4670.295382720.5907650.7272720.4296473040.3124710.8409720.496816912
219.7938930.42722303810.2211160.4554530.3070.194180940.7767240.8181820.6355008820.5199550.9034560.701735595
2210.2493460.23734641610.48669240.2277270.1510.09550921.50.1432640.9090910.1302397430.11840.9564650.13702683
2310.4770730.12460688710.60167990.2277270.0760.048070820.0961420.9545450.0917715940.08760.9794170.094162806
F.P.2410.7048010.70480000.5010010
665.8731992432.960717.542741986
Afterbody
Non-diV
29.36053
Forebody
Non-diV
17.11101
TotalNon-
diV
46.47154

47
18.5. Particulars of fwd-body.
𝜙𝑓 =
𝑠𝑢𝑚 𝑜𝑓 𝑓𝑜𝑟𝑒 𝑏𝑜𝑑𝑦 𝑛𝑜𝑛 − 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒
𝑆. 𝑀
2
=
17.11101
33
= 0.5185
18.5.2. 1st
non-dimensional lever:
𝑥𝑓 =
𝑠𝑢𝑚 𝑜𝑓 1𝑠𝑡 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑤𝑑 − 𝑏𝑜𝑑𝑦
𝑠𝑢𝑚 𝑜𝑓 𝑓𝑤𝑑𝑏𝑜𝑑𝑦 𝑛𝑜𝑛 − 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒
=
5.8732
17.11101
= 0.3432
18.5.3. 2nd
𝑘𝑓2
=
𝑠𝑢𝑚 𝑜𝑓 2𝑛𝑑 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑓𝑤𝑑𝑏𝑜𝑑𝑦
𝑠𝑢𝑚 𝑜𝑓 𝑓𝑤𝑑 𝑏𝑜𝑑𝑦 𝑛𝑜𝑛 − 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒
=
2.9607
17.11101
= 0.173
18.6. Particulars of aft-body:
𝜙𝑎 =
𝑠𝑢𝑚 𝑜𝑓 𝑎𝑓𝑡 𝑏𝑜𝑑𝑦 𝑛𝑜𝑛 − 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒
𝑆. 𝑀
2
=
29.361
33
= 08897
18.6.2. 1st
𝑥𝑎 =
𝑠𝑢𝑚 𝑜𝑓 1𝑠𝑡 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑓𝑡 − 𝑏𝑜𝑑𝑦
𝑠𝑢𝑚 𝑜𝑓 𝑎𝑓𝑡𝑏𝑜𝑑𝑦 𝑛𝑜𝑛 − 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒
=
13.535973
29.361
= 0.461
18.6.3. 2nd
𝑘𝑎2
=
𝑠𝑢𝑚 𝑜𝑓 2𝑛𝑑 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑎𝑓𝑡𝑏𝑜𝑑𝑦
𝑠𝑢𝑚 𝑜𝑓 𝑎𝑓𝑡 𝑏𝑜𝑑𝑦 𝑛𝑜𝑛 − 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒
=
8.3901
29.361
= 0.2858
18.7. Sam as in STEP 1 “14” it needs three points to define the half beam curve of
fore-body and aft-body separately, due to the actual need it’s unnecessary to
determine the point C.
18.8. Reduced function to define the three components:
𝐴 = 𝜙(1 − 2𝑥)
𝐵 =
𝜙(2𝑥 − 3𝑘2)
𝐴

48
18.9. Components to define the aft-body curve:
𝐴𝑎 = 𝜙𝑎(1 − 2𝑥𝑎) = 0.8897(1 − 2 ∗ 0.461) = 0.0694
𝐵𝑎 =
𝜙𝑎(2𝑥𝑎 − 3𝑘𝑎2)
𝐴𝑎
=
0.8897(2 ∗ 0.461 − 3 ∗ 0.2858)
0.0694
= 0.8309
18.10. Components to define the fwd-body curve:
𝐴𝑓 = 𝜙𝑓(1 − 2𝑥𝑓) = 0.4807(1 − 2 ∗ 0.3086) = 0.184
𝐵𝑓 =
𝜙𝑓(2𝑥𝑓 − 3𝑘𝑓2)
𝐴𝑓
=
0.5185(2 ∗ 0.3432 − 3 ∗ 0.173)
0.1626
= 0.5339
18.11. The summary of the fore and aft body parameters see Table 11:
Table 11 Fore and Aft body parameters
The hf equals to the Bf, and the ha equals to the Ba, see Table 11.
18.12. Following the same procedures of step “17”, the new LCF can be calculated.
The original LCF of the parent is -0.822 and the example used below changes the LCF
to -0.886, thus the LCF is moved afterward for 0.64m from the mid-ship.
The Figure 10 shows the different of the half beam curves between the parent hull and
the new hull.
CP 0.704114
Cpf 0.518515
Xf 0.343241
Kf 0.17303
Af 0.162564
Bf 0.53392
hf 0.53392
Cpa 0.889713
Xa 0.461026
Ka 0.285761
Aa 0.069351
Ba 0.830922
ha 0.830922

49
Figure 10 Half Beam Curve LCF=-0.886
Table 12 Data relative to LCF=-0.886
The data in Table 12 provides the needed data to change the LCF to -0.886. The
associate formula and calculation are the same as introduced in step “17”.
In order to draw the new half beam curve with using the original X, the new half beam
value (Y, named after “interpolated B” column in Table 12) should be interpolated.
The final calculated as -0.9036 see Table 12 and the difference between it to expected
value -0.886 is 1.987%. The difference is defined as:
Change of LCF -0.064 X new X
Half beam at
waterline
interploated B
Change of Z -0.01277 0.68487 0.68487 0 0
change of Cpf -0.02982 0.912557 0.904757159 0.8 0.818075381
change of Cpa 0.02982 1.140283 1.125424885 1.05 1.057766235
change of Xf -0.919 x(1-x) 1.595737 1.568991191 1.27 1.280210742
change of Xa 0.179796 x(1-x) 2.05119 2.015528405 1.39 1.397267252
2.506643 2.4650375 1.474 1.480383935
2.962096 2.917518473 1.53 1.534337974
3.417549 3.372971327 1.566 1.568358267
3.873002 3.831396059 1.581 1.581548715
4.328456 4.292793688 1.58 1.579168695
4.783909 4.757162186 1.56 1.55826805
5.239362 5.224502563 1.523 1.521652285
5.694815 5.694815 1.47 1.47
6.150268 6.074318047 1.4 1.383714358
6.605721 6.469010328 1.31 1.276077141
7.061174 6.87889276 1.204 1.152449661
7.516628 7.303966292 1.08 1.013637275
7.972081 7.744229058 0.94 0.86403414
8.427534 8.199681974 0.787 0.709817643
8.882987 8.670325041 0.628 0.556963636
9.33844 9.156158259 0.467 0.407998652
9.793893 9.657181628 0.307 0.2645836
10.249346 10.17339515 0.151 0.129197467
10.477073 10.4371988 0.076 0.064713751
10.7048 10.7048 0 0
New LCF Theory -0.886
New LCF
calculated
-0.90361086
difference 1.98768213

50
𝐷𝑖𝑓𝑓𝑒𝑟𝑛𝑐𝑒 =
𝐴𝐵𝑆(𝑁𝑒𝑤 𝐿𝐶𝐹 − 𝑁𝑒𝑤 𝐿𝐶𝐹 𝑖𝑛 𝑡ℎ𝑒𝑜𝑟𝑦)
𝑁𝑒𝑤 𝐿𝐶𝐹 𝑖𝑛 𝑡ℎ𝑒𝑜𝑟𝑦
∗ 100% =
0.0176
0.886
∗ 100%
= 1.987%
By using step “18” the new LCF can be obtained with a reasonable level of error.
Finally the hull with other LCF value can be generated without influencing the Cb or
LCB, as it doesn’t change the sectional area of the underwater part of the yacht hull.
Figure 11 The half beam of difference LCF value while keeping sectional area unchanged
Figure 11 above shows the half beam curve for 5 different LCF (LCF equals to -0.622;-
0.722;-0.822;-0.886 and -0.95 separately) with keeping the sectional area curve
unchanged, thus keeping the Cb and LCB unmoved (Cb=0.359 and LCB=-0.349 in
figure 11).
6.6.Conclusion
In this Section, the usage of Lackenby Sectional Transformation is introduced and the
associate algorithms of how to change the hull to have different CB, LCB and LCF
have been provided. By using these procedure a new hull with typical requirement can
be obtained by modifying the parent hull form.
Finally there are three group of parameters combination are selected as Cb and LCB;
Cb and LCF; LCB and LCF. Each group has 25 different hulls, the details of the hull
form parameters have been provided in “APPENDIX 1 PARAMETERS OF THE
MODEL”.
The associate sectional area data and half beam data have been provided in
“APPENDIX 2 DATA OF THE HULL FORM”.

51
7. LEWIS MAPPING METHOD
7.1.Lewis Mapping Method
Normally the geometrical shape of a ship hull is defined by the points specified in each
longitudinal section or water-plane section. One method for generating ship-like section
from knowledge of transverse sectional area and associated beam and draught is the
two parameters Lewis Mapping (Lewis.F.M, 1929). The Lewis mapping method is a
conformal mapping technique. An inbuilt assumption of the 2 parameters method is
that the hull transverse section is wall-sided at the free-surface with entrance angle is
90 degrees and dead-rise angle is 0 degrees. The figure 11 provides the definition of the
transverse section and associated angle. This method is applicable for generating the
transverse sections of larger form merchant ships, but is unacceptable in this project as
the chosen parent yacht hull (form YD-40) has different entrance angles and dead-rise
angles for each sections.
Accommodate the entrance and dead-rise angle a 3 parameter method is utilised. The
improved Lewis mapping method requires two extra stages in its implementation. The
theory is based on the conformal mapping is underpinning preserved at regular points
and it is changed at singular points (where the derivative of the mapping function is
zero). This provide the original Lewis mapping produced a chance to change the
entrance and the dead-rise angle. The first stage is modify the entrance angle and the
second stage is modify the dead-rise angle. The transverse sections of the ship can be
mapped as an offset table which can be used as input will commercial software to
generate a 3-D model. The procedure of this new mapping method will be provided in
section “7.2”.
The errors will happen when using this new mapping method to generate the hull. The
main source of the error is the angle determination, especially when doing the validation
with the parent hull. This will be demonstrated in later pages.
It should be noted that although there are errors (i.e the generated hull is slightly
different with the parent hull), but they are small enough to give high confidence for

52
the method validation. It should also be noted that both of the improved method and the
original Lewis mapping method are aimed to generate the underwater part of the hull.
7.2.Transverse Section Definition
Figure 12 Transverse Section with deadrise angle phi and entrance angle beta
Figure 12 represents the shape of a typical transverse section subject to the following
parameter definitions.
 “A” represents the sectional area.
 “t” represents the draught of this section.
 “b” represents the half beam of this section.
 “Beta” represents the entrance angle of this section.
 “Phi” represents the deadrise angle of this section.
7.3. Improved Lewis Conformal Mapping function
7.3.1. The nomenclatures for next steps.
A Sectional Area Input
a1,a1,
a3
Coefficients 7.4.1
Alpha α Coefficient 7.4.5

53
b Beam Input
b lewis Coefficient equal to Za 7.4.3
Beta β Entrance angle Input
F Coefficient 7.4.4
G Coefficient 7.4.3
Gama γ Coefficient 7.4.5
Omega
ω
Coefficient 7.4.5
Phi φ Dead-rise angle Input
t Draught Input
T Coefficient 7.4.4
Z
Point used to define the
transverse section with changing
dead-rise angle and the entrance
angle
Output
Za
transverse section with using 2
parameter Lewis Mapping
Method
7.4.3
Zb
transverse section with changing
dead-rise angle
7.4.3
There are three stages in this function.
7.3.2. The first one uses the three parameter Lewis mapping (Lewis.F.M, 1929) to map
the unit circle defined in complex plane by ζ=eiθ
into a ship-like section with a
defined beam, draught and area.
𝑍 𝑎 = 𝑎1 𝜁 +
𝑎2
𝜁
+
𝑎3
𝜁3
7.3.3. The second stage change the deadrise angle of the section and is developed by
(C & F, 1983)

54
(
𝑍 𝑎 −
𝑇
𝛾
𝑍 𝑎 +
𝑇
𝛾
) = (
𝑍 𝑏 − 𝑇
𝑍 𝑏 + 𝑇
)
1
𝛾
7.3.4. The final stage will change the entrance angle of the section.
(
𝑍 𝑎 −
𝑏
𝛼
𝑍 𝑎 +
𝑏
𝛼
) = (
𝑍 − 𝑏
𝑍 + 𝑏
)
1
𝛼
7.4.The additional equations
7.4.1. The equations for the coefficients contented in the Lewis Mapping are:
𝑎1 = 0.5 ∗ (𝑏𝑙𝑒𝑤𝑖𝑠 + 𝑡𝑙𝑒𝑤𝑖𝑠) − 𝑎3
𝑎2 = 0.5 ∗ (𝑏𝑙𝑒𝑤𝑖𝑠 − 𝑡𝑙𝑒𝑤𝑖𝑠)
𝑎3 =
1
4
(−(𝑏𝑙𝑒𝑤𝑖𝑠 + 𝑡𝑙𝑒𝑤𝑖𝑠) + √|(𝑏𝑙𝑒𝑤𝑖𝑠 + 𝑡𝑙𝑒𝑤𝑖𝑠)2 + 8 (𝑏𝑙𝑒𝑤𝑖𝑠 𝑡𝑙𝑒𝑤𝑖𝑠 −
4𝐴𝑙𝑒𝑤𝑖𝑠
𝜋
)|)
7.4.2. As an additional two mapping functions are used the beam, draught and area of
the section generated by the Lewis mapping will differ from the beam, draught and area
of the final section. The area of the Lewis section (Alewis) is found by multiplying the
desired final sectional area by a section shape factor.
𝐴𝑙𝑒𝑤𝑖𝑠 = 𝐴𝑆𝑓
7.4.3. The section shape factor will depend on the entrance and dead-rise angles.
Therefore it is important to determine those two angle s as accurate as possible. The
shape factor for each sections in this project will be determined with iterative method
until the difference of the curve that between the lewis section and the parent section
are small enough. The equation below is used to define the section shape factor and a
value of 2 for AF has proven a good starting points in test.

55
The points za=blewis and za=-itlewis correspond to the points z=b and z=-it respectively.
Therefore, it is necessary to work through the final two stages in reverse for the points
z=b and z=-it to find blewis and tlewis. Substituting z=b into the third stage the right hand
side becomes zero, so therefore
𝑧 𝑏 −
𝑏
𝛼
= 0
And
𝑧 𝑏 =
𝑏
𝛼
Substituting this into the second stage the right hand side (G) becomes as:
𝐺 = (
𝑏
𝛼 − 𝑇
𝑏
𝛼
+ 𝑇
)
1
𝛾
By rearranging,
𝑍 𝑎 = 𝑏𝑙𝑒𝑤𝑖𝑠 =
𝑇
𝛾
(
1 + 𝐺
1 − 𝐺
)
T is defined as the value of zb corresponding to the point z=-it, ie T=zb (z=-it). Therefore
the third stage o fthe mapping can be skipped and zb=T substituted into the second stage.
This results in the right hand side becoming zero so therefore:
𝑧 𝑎 = −𝑖𝑡𝑙𝑒𝑤𝑖𝑠 =
𝑇
𝛾
And
𝑡𝑙𝑒𝑤𝑖𝑠 = 𝑖
𝑇
𝛾
7.4.4.T can then be defined by substituting z=-it into equation three thr right hand side
(F) becomes as:
𝐹 = (
−𝑖𝑡 − 𝑏
−𝑖𝑡 + 𝑏
)
1
𝛼
And

56
𝑧 𝑏 =
𝑏
𝛼
(
1 + 𝐹
1 − 𝐹
)
So
𝑇 =
𝑏
𝛼
(
1 + 𝐹
1 − 𝐹
)
7.4.5. Finally, in equations 7.3.2 and 7.3.3,
𝛾 = 2(1 −
𝜔
𝜋
)
Where
𝜔 =
𝜋
2
− 𝜙
And
𝛼 = 2 (1 −
𝛽
𝜋
)
Theses equations require all angles to be in radians.
7.5.Matlab code for Lewis Conformal Mapping
The corresponded Matlab Code are provided in “APPENDIX 3 MATLAB CODE FOR
IMPROVED LEWIS CONFORMAL MAPPING”
7.6.Example of using the Lewis Mapping
In this example the data of section 11 of the YD 40 will be used.
The inputs data provided in Table 13:

57
Table 13 Beta and Phi with associated transver section data of station 11
We can then obtain the function in pervious to calculate the coefficients of a1, a2 and
a3.
7.6.1. Step 1
𝛼 = 2 (1 −
1
3.1415926
) = 1.3634
7.6.2. Step 2
𝜔 =
3.1415926
2
− 0.1266 = 1.4442
7.6.3. Step 3
𝛾 = 2 ∗ (1 − (
1.4442
3.1415926
)) = 1.0806
𝑆𝑓 =
1.1
𝑠𝑖𝑛1 + 𝑐𝑜𝑠0.1266
= 0.6
7.6.4. Step 4
𝐹 = (
−0.604𝑖 − 1.48
−0.604𝑖 + 1.48
)
1
1.3634
= −0.1643 − 0.9864𝑖
7.6.5. Step 5
𝑇 = (
1.48
1.3634
) ∗ (
1 + (−0.1634 − 0.9864𝑖
1 − (−0.1634 − 0.9864𝑖
) = −0 − 0.9197𝑖
𝐴𝑙𝑒𝑤𝑖𝑠 = 1.3321 ∗ 0.6 = 0.7992
tan Phi 0.127272727
Phi radians 0.126592127
Phi degree 7.253194736
tan Beta 1.345454545
Beta radians 1
Beta degree 57.29578049
A 1.3321
b 1.48
t 0.604

58
7.6.6. Step 6
𝐺 = (
1.48
1.3634
− (−0 − 0.9197𝑖)
1.48
1.3634
+ (−0 − 0.9197𝑖)
)
1
1.0806
= 0.2666 + 0.9638𝑖
7.6.7. Step 7
𝑏𝑙𝑒𝑤𝑖𝑠 = (
0.604
1.0806
) ∗ (
1 + 0.2666 + 0.9638𝐼
1 − 0.2666 − 0.9638𝐼
) = 1.1185 − 0.0000𝑖
7.6.8. Step 8
𝑡𝑙𝑒𝑤𝑖𝑠 = 𝑖 ∗ (−
0.9197𝑖
1.0806
) = 0.8511 − 0.0000𝑖
7.6.9. Step 9
𝑎3
= 0.25(−1 ∗ 1.1185 + 0.8511)
+ √|((1.1185 + 0.8511)2) + (8 ∗ (1.1185 ∗ 0.8511 −
4 ∗ 0.7992
3.1415926
))|
= −0.0345 + 0𝑖
7.6.10. Step 10
𝑎1 = 0.5 ∗ (1.1182 + 0.8511) − (−0.0345) = 1.0193 − 0.000𝑖
𝑎2 = 0.5 ∗ (1.1185 − 0.8511) = 0.1337 + 0𝑖
Then the next stage is use the mapping function in “7.3” to map the unit circle into a
ship like section and by using the extra two stages to change the section shape to fit for
different entrance angle and dead-rise angle.
With selecting a range of angle between π and 2π, the section below the waterline is
defined, the associated point to define the section with stage 1; stage2 and stage3 are
provided.

59
Table 14 The Points to Define the Sectiona Data Through Applying the 3Parameters Lewis Mapping Method
Then we plot the shape of the section by using section data (See Tabel 14) to see how
the shape will change in stage 1, 2 and 3.
Figure 13 shows how will the section curve change through 3 stages. After the stage 2
the deadrise angle is modified and finally in stage 3the entrance angle is modified and
the curve z represent the section shape generated by Lewis mapping method.
Real Part Imagined Part Real Part Imagined Part Real Part Imagined Part
xa ya xb yb x y
-0.0001 -0.8511 0 -0.9197 0 -0.604
0.0758 -0.85 0.0629 -0.9116 0.055 -0.597
0.1511 -0.8469 0.1321 -0.902 0.11 -0.591
0.2256 -0.8416 0.2027 -0.891 0.164 -0.584
0.2988 -0.834 0.2733 -0.8785 0.219 -0.577
0.3701 -0.8242 0.3429 -0.8644 0.274 -0.571
0.4394 -0.812 0.411 -0.8484 0.329 -0.564
0.5061 -0.7973 0.4769 -0.8304 0.384 -0.557
0.5701 -0.78 0.5403 -0.8101 0.438 -0.55
0.6309 -0.76 0.6008 -0.7875 0.493 -0.542
0.6885 -0.7371 0.6581 -0.7622 0.548 -0.535
0.7425 -0.7114 0.712 -0.7343 0.603 -0.526
0.793 -0.6826 0.7622 -0.7036 0.658 -0.517
0.8397 -0.6507 0.8088 -0.6699 0.713 -0.507
0.8826 -0.6158 0.8516 -0.6333 0.767 -0.496
0.9218 -0.5777 0.8905 -0.5937 0.822 -0.484
0.9571 -0.5367 0.9257 -0.5512 0.877 -0.471
0.9888 -0.4927 0.9572 -0.5057 0.932 -0.455
1.0168 -0.4459 0.9849 -0.4575 0.987 -0.437
1.0411 -0.3965 1.0091 -0.4067 1.041 -0.417
1.062 -0.3447 1.0298 -0.3535 1.096 -0.393
1.0795 -0.2908 1.0471 -0.2981 1.151 -0.364
1.0937 -0.235 1.061 -0.2408 1.206 -0.331
1.1046 -0.1776 1.0718 -0.182 1.261 -0.289
1.1123 -0.1191 1.0795 -0.1221 1.315 -0.239
1.117 -0.0598 1.084 -0.0613 1.37 -0.175
1.1185 -0.0001 1.0855 -0.0001 1.425 -0.093
Za Zb Z
Stage 1 Stage 2 Stage 3

60
Figure 13Change of the shape of the section 11 in stage 1;2 and 3
We can also plot the lewis section curve with the parent hull section curve together for
each underwater sections see Figure 14.
Figure 14Section curve of parent hull and section curve generated by using 3 parameters methos
The details of each station for parent hull and associated section curve generated by
using 3 parameters method are provided in the “APPENDIX 4 SECTION CURVE FOR
EACH STATION”.
7.7.Accuracy Check
The Lewis Mapping method used in this project is a very accurate method but requires
to estimate (measure) the entrance angle and the deadrise angle of the parent ship as
accurate as possible. Within this way all of the curves that generated by the Mapping
function are highly similar as the parent hull section curve.

SAIL VERSUS HULL FORM PARAMETER CONFLICTS IN YACHT DESIGN

SAIL VERSUS HULL FORM PARAMETER CONFLICTS IN YACHT DESIGN

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