This study aims to identify the relations between seakeeping characteristics and hull form parameters for YTU Gulet series with cruiser stern. Seakeeping analyses are carried out by means of a computer software which is based on the strip theory and statistical short term response prediction method. Multiple regression analysis is used for numerical assessment through a computer software. RMS heavepitch motions and absolute vertical accelerations on passenger saloon for Sea State 3 at head waves are investigated for this purpose. It is well known that while ship weight and the ratios of main dimensions are the primary factors on ship motions, other hull form parameters (C_{P}, C_{WP}, C_{VP}, etc.) are the secondary factors. In this study, to have an idea of geometric properties on ship motions of gulets three different regression models are developed. The obtained outcomes provide practical predictions of seakeeping behavior of gulets with a high level of accuracy that would be useful during the concept design stage
.
INTRODUCTION
Existing gulets are used for pleasure trips today. Therefore it became significant to conduct a study in order to discover their hydrodynamic characteristics. A systematic series of gulet hull forms with cruiser stern is developed in order to investigate their performance (
Aydın, 2013
). Certain processes could be made to understand seakeeping characteristics of the gulets by making use of several methods. Although the strip theory is the quickest and relatively most accurate one it has restrictions because of its theoretical assumptions. It has been most preferred tool during conceptual design stage. Due to its theory is linear; solutions are more realistic for slender hulls and low Froude numbers. However, strip theory has been widely accepted and a large number of computer codes are developed.
While resistance and power outputs are sensitive to local changes of hull form, seakeeping matrix is less sensitive local changes of hull form. Seakeeping performance usually depends on main dimensions and their proportions, hydrostatic values and weight distribution. For this reason, seakeeping phenomenon must be evaluated in conceptual design stage (
Şaylı et al., 2007
).
Several studies can be found in technical literature about effects of ship geometry on seakeeping characteristics. Bales performed a criteria free rank study for 20 displacement normalized destroyers by using six form parameters. Based on the general definition of seakeeping rank, eight seakeeping responses were computed for each hull form. The responses were pitch, heave, ship to wave relative motion at Station 0 and 20, bottom slamming at Station 3, absolute vertical acceleration at Station 0, heave acceleration and absolute vertical motion at Station 20. Analyses were performed in long crested head seas for five speeds each for five modal periods (
Bales, 1980
). Kükner and Sarıöz made calculations for high speed vessels in their study. They investigated main dimensions and seconder hull form parameters effect on vertical motions (
Kükner and Sarıöz, 1995
). Brown conducted a study for gulet type boats in terms of resistance and seakeeping. Then he tried to present optimum hull parameters for gulets (
Brown, 2005
). Şaylı and others showed the effects of hull form parameters on vertical motions for fishing vessels by using multiple regression techniques (
Şaylı et al., 2007
). Şaylı and others, in their next study, performed the same configuration by using nonlinear regression techniques (
Şaylı et al., 2009
). Results were very adoptable with each other. Özüm and others observed the effect of hull parameters for high speed hull forms during conceptual design stage. They explored main dimensions and secondary form parameters make the main contribution to ship motions (
Özüm et al., 2011
).
In this study, gulet type pleasure boats were examined in terms of seakeeping properties. Although gulets were built in traditional ways previously, these crafts are built with modern technics in recent years. A study was conducted with the purpose of developing an original gulet series with cruiser stern without destroying its character. First, 21 cruiser stern gulet hull forms with different geometric design block coefficients (
C_{B}
) were designed by the author with the information gained during the technical visits. These hull forms were produced by preserving conventionalism. Detailed information can be found in related paper. (
Aydın, 2013
) Body Stations, profile and waterlines of a gulet in the series are shown in
Fig. 1
.
Body Stations, profile and waterlines of a gulet in the series.
The hull form parameters of YTU Gulets are tabulated in
Table 1
. For comparison purposes
Table 2
is constructed to show the dimension and displacement ranges of existing gulets in Turkey and YTU Gulets.
Main dimensions and some geometric properties of YTU gulets with cruiser stern.
Main dimensions and some geometric properties of YTU gulets with cruiser stern.
A comparison between existing gulets and YTU gulets with cruiser stern.
A comparison between existing gulets and YTU gulets with cruiser stern.
Effects of geometrical features of 21 different YTU Gulets on their seakeeping characteristics in terms of the following three responses are investigated:

Heave motion

Pitch motion

Absolute vertical acceleration at the passenger saloon
Location of the passenger saloon is shown in the
Table 3
:
Location of the passenger saloon.
Location of the passenger saloon.
 Design database
Displacement forces of the gulets should be brought to equal level for fair comparison of effects of geometric characteristics on specified ship motions. This value is 774.27
kN
. Thus the design database is built with the hull forms of displacement normalized YTU Gulets those are given in
Table 4
:
Hull form parameters for analyses.
Hull form parameters for analyses.
SEAKEEPING CALCULATIONS
Seakeeping performance is evaluated for head waves (
μ
=180°) and for
Fn
=0:0.5:0.3 in this paper. Hydrodynamic coefficients are calculated by using Frank Close Fit Method for each gulet section. This solution is valid for arbitrary cross sections and the velocity potential is represented by a distribution of sources on the mean submerged cross section. In this method; green function satisfying the linear free surface boundary condition is used to represent the velocity potential. The density of the sources is an unknown function to be determined from integral equations obtained by applying the body boundary condition (
Frank, 1967
). Cross sections of the gulet 11 (G11) are shown in
Fig. 2
.
Sections of the G11 and close fit points.
The assessment of seakeeping performance of a pleasure craft in a specified sea state is related to four elements:

Ship geometry

Weight distribution on ship

Transfer functions (RAO) in regular waves

Wave spectrum
As a result of these interactions, determination of ship responses can be obtained.
 Gulet response characteristics
The first step for determination of the seakeeping performance is to detect the motion response amplitudes and phase lags in the frequency domain for all six degrees of freedom. Then RAOs can be executed for each specified response such as heave motion, velocity and acceleration.
RAO graph of G11 is shown in
Fig. 3
in case of zero speed and head waves. It should be pointed out that there is a strong resonance danger in existence of restoring effect such as heave and pitch motion.
Heave transfer function for Fn =0, for G11.
 Definition of seaway
Ship motions in irregular waves should be investigated due to absence of regular waves in nature. It is important to get ship motions in random waves because of the complexity of sea surface. Modeling sea is possible by using some statistical methods. Irregular sea can be expressed by using wave spectra that is composed as regards to normal distribution. Spectral density function must be known to enter short term statistical parameters. This recommended function must fit characteristic of the sea environment where gulets will sail. It is used 2 parameter ITTC (Bretschnider) wave spectrum in analyses which is proposed in STANAG 4194 documents by reason of gulet type boats mostly operate in East Mediterranean Sea. Analyses are performed at sea state 3. Characteristic wave height and modal period of sea state 3 are shown in
Table 5
:
Characteristics of east mediterranean sea at SS 3.
Characteristics of east mediterranean sea at SS 3.
 Prediction of motions
It is very common to use 2D and 3D analytical methods on prediction of ship responses in operational sea environment. At the shortterm analyses, average, observed and most frequent motion amplitudes are obtained by the help of response function curve which is plotted with superposition RAO and wave spectrum curve (
Figs. 4

6
). This procedure is applied with Eq. (1). Response function curve must be plotted in the case of head waves and vertical responses. Maximum
Fn
is taken as 0.3 because the gulets are displacement type boats and
Fn
is a limiting factor for the strip theory.
Typical RAO curve.
Typical wave spectrum curve.
Typical response curve.
MULTIPLE REGRESSION ANALYSES
The multiple regression equation is derived from the Least Squared Method and it is alike two variable regression analyses. Instead of a single independent variable, two or more independent variables are used to find a dependent variable values. The multiple regression equation is given Eq. (2):
where
P
is and estimated dependent variable which represents RMS values of specified responses in case of head waves. Independent variables must represent dependent variables very well. Otherwise obtained results might be not adoptable. Used regression models are based on main dimensions and hydrostatic values since ship motions are generally are function of them. Therefore
X
_{1}
,
X
_{2}
⋯,
X_{n}
independent variables represent main dimensions and their proportions, form coefficients etc. On the other hand
A
_{1}
,
A
_{2}
⋯,
A_{n}
coefficients are regression coefficients which shows how independent variables affects dependent variables. These all variables should be written in matrix format in Eq. (3) to calculate regression coefficients.
Eq. (4) must be solved to find matrix of the regression coefficients.
while
m
shows number of independent variables
n
shows number of equation.
X^{T}
represents transpose of
X
matrix. Number of independent variables,
m
, is determined from selected regression model. In this respect, multiple regression coefficients are computed by using regression solver software with a very high
R
^{2}
. 21 equations are solved for each response and regression model. Recommended regression models are given next chapter.
 Recommended regression models
Three different regression models are presented for the purpose of determining effects of independent variables on dependent variables. These regression models are given in
Table 6
. Dependent variables which are used in regression analyses are RMS responses of specified ship motions at SS3 for displacement normalized gulets. Computed RMS values represent most frequent motion amplitudes at SS3. Besides, computations are repeated for each model for seven different Froude number (
Fn
=0:0.05:0.3).
Used models for regression analyses.
Used models for regression analyses.
while Model 1 consists of only main dimension proportions, Model 2 additively consist of
C_{WP}
,
C_{VP}
and
C_{P}
hydrostatic form coefficients. Model 3 contains main dimension proportions, hydrostatic form coefficients and
L_{CB }

L_{CF}
locations as addition. Adequate consideration of models is extremely important to evaluate multiple form parameters at this kind of analyses. Therefore number of models is selected as three.
Multiple regression equations for Model 1;
Multiple regression equations for Model 2;
Multiple regression equations for Model 3;
Computed regression coefficients can be found
Tables 7

15
. While
Tables 7

9
show regression coefficients for Model 1,
Tables 10

12
show regression coefficients for Model 2. Finally
Tables 13

15
present regression coefficients for Model 3.
Regression coefficients for heave motion for Model 1.
Regression coefficients for heave motion for Model 1.
Regression coefficients for pitch motion for Model 1.
Regression coefficients for pitch motion for Model 1.
Regression coefficients for vertical acceleration at saloon for Model 1.
Regression coefficients for vertical acceleration at saloon for Model 1.
Regression coefficients for heave motion for Model 2.
Regression coefficients for heave motion for Model 2.
Regression coefficients for pitch motion for Model 2.
Regression coefficients for pitch motion for Model 2.
Regression coefficients for vertical acceleration at saloon for Model 2.
Regression coefficients for vertical acceleration at saloon for Model 2.
Regression coefficients for heave motion for Model 3.
Regression coefficients for heave motion for Model 3.
Regression coefficients for pitch motion for Model 3.
Regression coefficients for pitch motion for Model 3.
Regression coefficients for vertical acceleration at saloon for Model 3.
Regression coefficients for vertical acceleration at saloon for Model 3.
DISCUSSION
After researching recommended regression models it is now possible to evaluate the effects of hull geometry on seakeeping characteristics. Rsquared values which specify goodness of fit are around 0.95. It is a very good prediction on determining dependent variables. It means that independent variables represent dependent variables very well. However, when
Table 8
is examined; one can see a slight decreasing
R
squared value. It is determined around 0.88 and corresponds to
Fn
=0.15 case. This situation leads to a slight difference between strip theory and recommended models that can be seen in
Figs. 6(b)
and
Fig. 8(b)
. There is no specific reason for this state due to calculation of motions is dependent of so many parameters. Hull form requirements for good seakeeping for the gulets are given in
Table 16
:
Requirement for good seakeeping.
Requirement for good seakeeping.
When regression coefficients of Model 1 are investigated the effects of main dimension proportions on ship motions are seen clearly. Increasing of
L_{WL}
/
B_{WL}
and
B_{WL}
/
T
values, decreasing of
L_{WL}
/ ∇
^{1/3}
values became useful to reduce heavepitch motions and absolute vertical acceleration at saloon. In particular, being positive or negative value of regression coefficient is a solid symptom that shows how it affects. The role of the parameters given with question marks is not clear. When Model 2 is examined it is understood that higher
C_{WP}
lower
C_{VP}
and
C_{P}
values are better for heave motion. It is also seen from tables that lower
C_{VP}
and
C_{P}
values are better for pitch motion and vertical acceleration. Model 3 has to be checked to obtain the influence of
L_{CB}
and
L_{CF}
. These points are should be closer to bow for pitch motion. In other respects, while
L_{CB}
should be closer stern
L_{CF}
should be closer bow for the sake of heave motion and vertical acceleration. If the regression coefficient tables are observed in detail the rate of influence of parameters also could be understood. Comparing rate of influence is simply possible when each parameter is grouped each other such as separation of comparing form coefficients and
L_{CB}
and
L_{CF}
.
Figs. 7

9
shows the comparison between strip theory and multiple regression calculation for gulet 11. While
Fig. 7
shows respectively heave pitch motions and vertical acceleration for Model 1,
Fig. 8
shows respectively heavepitch motions and vertical acceleration for Model 2,
Fig. 9
shows respectively heavepitch motions and vertical acceleration for Model 3. It could be easily seen from
Figs. 7

9
that RMS strip theory values for heavepitch motions and vertical acceleration are predicted very close to multiple regression computations.
(a) Model 1 heave comparison between strip theory and regression. (b) Model 1 pitch comparison between strip theory and regression. (c) Model 1 vert. acce. comparison between strip theory and regression.
(a) Model 2 heave comparison between strip theory and regression. (b) Model 2 pitch comparison between strip theory and regression. (c) Model 2 vert. acce. comparison between strip theory and regression.
(a) Model 3 heave comparison between strip theory and regression. (b) Model 3 pitch comparison between strip theory and regression. (c) Model 3 vert. acce. comparison between strip theory and regression.
CONCLUSION
The development of the effect of hull form parameters of YTU Gulets on ship motions is presented in this paper with several processes. The process started with prediction of transfer functions for 21 different gulet forms for different Froude numbers. Then these transfer functions are combined with specified spectral curve. Finally, the effects of hull form parameters are determined by the help of multiple regression method. At the end of the study, hull form requirements for good seakeeping for the gulets are determined. The comparison between strip theory and multiple regression calculations for gulet 11 is shown in figures. The obtained results ensure practical predictions of form parameter contribution to motions with a high level of accuracy that would be useful during the concept design stage. As a future work, the habitability indices of gulet type pleasure hulls will be investigated in terms of comfort on board.
NOMENCLATURE
ABBREVIATION LIST
Aydın M.
2013
Development of a systematic series of gulet hull forms with cruiser stern
Ocean Engineering
58
180 
190
Bales N.K.
1980
Optimizing the seakeeping performance of destroyertype hulls
Proceedings of the 13th Symposium on Naval Hydrodynamics
Tokyo, Japan
October 1980
479 
502
Brown D.
2005
A forward analysis of a gulet type hull form, Mar 398 Project and Report
University of Newcastle
Newcastle
1967
Oscillation of cylinders in or below the free surface of deep fluids, Technical Report 2375
Naval Ship Research and Development Centre
Washington D.C.
Kükner A.
,
Sarıöz K.
1995
High speed hull form optimization for seakeeping
Advances in Engineering Software
22
179 
189
Özüm S.
,
Şener B.
,
Yılmaz H.
2011
A parametric study on seakeeping assessment of fast ships in conceptual design stage
Ocean Engineering
38
1439 
1447
Şaylı A.
,
Alkan A.D.
,
Nabergoj R.
,
Uysal A.O.
2007
Seakeeping assessment of fishing vessels in conceptual design stage
Ocean Engineering
34
724 
738
Şaylı A.
,
Alkan A.D.
,
Ganiler O.
2009
Nonlinear meta models for conceptual seakeeping design of fishing vessels
Ocean Engineering
37
730 
741