**Published at : ** 18 Sep 2024

**Volume :** **IJtech**
Vol 15, No 5 (2024)

**DOI :** https://doi.org/10.14716/ijtech.v15i5.7036

Ayub, F.A., Furukawa, Y., 2024. Comparison Between Cubic and Quadratic Models of Hydrodynamic Derivatives to the Ship Course Stability Index.

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Fakhri Akbar Ayub | Naval Architecture and Marine Engineering, Faculty of Engineering, Universitas Pembangunan Nasional “Veteran” Jakarta, 12450, Jakarta, Indonesia |

Yoshitaka Furukawa | Department of Marine Systems Engineering, Faculty of Engineering, Kyushu University, Fukuoka 819-0395, Japan |

Abstract

The
mathematical equations that describe the hydrodynamic forces on a ship's hull
are important in understanding how a ship moves. These equations are based on
specific values that vary depending on the type of ship. This research will
focus on a mathematical model based on a polynomial model in order to
investigate the differences between the 3^{rd} order polynomial (cubic
model) and the 2^{nd} order polynomial (quadratic model). The course
stability index is determined by utilizing linear hydrodynamic derivatives and
examined to understand the distinctions between the characteristics of cubic
and quadratic models. In this research, the measurement data of the
hydrodynamic forces of 12 model vessels (total of 27 loading conditions) that
had been conducted for turning tests and the zigzag tests in the past at Kyushu
University were targeted. The based 2^{nd} order model and the based 3^{rd} order model are applied
for re-analysis, and the results of a comparative study on the difference and
approximation characteristics of the hydrodynamic force due to the difference
of the adopted models are shown.

Course Stability Index; Cubic Model; Hydrodynamic Derivatives; Ship Manoeuvrability; Quadratic Model

Introduction

The International Maritime
Organization (IMO) has approved Resolution A.751(18), known as the Interim
Standards for Ship Manoeuvrability, in order to improve marine safety by
removing ships with inadequate manoeuvrability. Currently, the issue of manoeuvring
subjects has become significant due to the establishment of criteria for
manoeuvring characteristics in the Standards for Ship Manoeuvrability Criteria (IMO, 2002). The IMO has identified ship
manoeuvrability as a critical factor in a ship's ability to change or maintain
its course and speed. Larger ships, in particular, often encounter greater
challenges in navigation due to their limited manoeuvrability. Any ship that is
exceeds 100 meters in length must meet the requirements outlined in the IMO's
manoeuvring standards.

Manoeuvring performance of a ship should be evaluated properly at the design phase in order to eliminate ships that, have poor manoeuvrability. Several methods to evaluate at the design stage have been developed such as direct and indirect methods (Hasanvand and Hajivand, 2019). Numerical simulations based on a mathematical model of hydrodynamic forces acting on a ship is one of the evaluation methods for indirect methods.

Moreover,
the manoeuvring standards require to prove that a newly constructed ship meets
the requirements for manoeuvring tests under fully loaded conditions in calm
weather. During this process, numerical simulation plays a key role in ensuring
the accuracy of hydrodynamic coefficients in mathematical models that depict
the forces affecting a ship's manoeuvring abilities.

The
hydrodynamic forces on a ship's hull are mathematically represented by specific
values known as hydrodynamic derivatives. These derivatives vary depending on
the type of ship and are determined by analyzing the measured forces acting on
the hull. They indicate the rate of change of the forces and moments.
Hydrodynamic derivatives can also be expressed as a function of a ship's
principal particulars, including length, beam, draught, trim, and displacement (Kijima *et al.,* 1990).

Additionally,
there are various types of errors and uncertainties present in mathematical
models of hydrodynamic forces that are utilized in predicting manoeuvring
motions. (Dash, Nagarajan, and Sha, 2015; Wang *et
al.,* 2014). These
issues are often related to the hydrodynamic derivatives that represent the
hydrodynamic forces acting on a ship hull. (Ayub, Furukawa, and Ibaragi, 2021; Shenoi, Krishnankutty, and Selvam, 2015), model tests (Woodward, 2013), facilities equipment (Gavrilin and Steen, 2016; Woodward, 2014).
Extrapolation of data may occur when applying the results of a mathematical
model to drift angles and rotation rates that have not been tested within the
model's range (ITTC Manoeuvring Committee, 2017;
2008).

Analyzing
and evaluating errors can be a challenging and time-intensive process, but it's
necessary for validation. The accuracy of predicted results relies on the
methods used to predict manoeuvring performance. Therefore, it is essential to
make a dedicated effort to verify and validate prediction or simulation methods
in order to accurately assess their accuracy. In order to properly evaluate the
correlation between predicted outcomes and actual hydrodynamic forces, it is
important to consider uncertainties in both the predictions and the measured
data.

A brief
review of the mathematical models has been summarized by the International
Towing Tank Conference (ITTC Manoeuvring Committee,
2011). Each method has it own advantages and disadvantages. The
mathematical model for manoeuvring motions is categorized into several models
such as; cross-flow model (Yoshimura, 1988; Oltmann
and Sharma, 1984), Polynomial model (Viallon,
Sutulo, and Soares, 2012), Manoeuvring Modelling Group Model (Yasukawa and Yoshimura, 2015), Fourier expansion
model (Toxopeus, 2011a; 2011b; 2007; Kang and Hasegawa,
2007), Karasuno’s model (Karasuno *et al*.,
2003), Vectorial model (Berge and Fossen.,
2012; Fossen, 2011), RANS CFD (Liu *et al*,
2021; Islam and Soares, 2018).

The basic
principles of the equations of motion for ship manoeuvring are based on
Newton's second law of motion (Tao *et al*,
2021). The first theoretical approach focused on analyzing the ship as a
rigid body with movements in surge, sway, and yaw and explaining the hydrodynamic forces and moments on
the ship through first-order derivatives.

Nonlinear hydrodynamic forces were found to be present at high velocities
and when cross products of velocities occurred, causing forces and moments to
deviate from linear behaviour. These non-linear forces can become comparable in
magnitude to the linear component during a sharp turn with a significant rudder
angle. Moreover, the yawing moment's non-linear aspect is typically five to ten
times greater than its linear component. These non-linear terms are often
depicted using cubic or quadratic polynomial equations, with coefficients
typically established through captive model tests.

The Taylor
series expansion is employed to represent the non-linearities, leading to a
polynomial expression involving two variables (Luo *et
al*., 2016). The hydrodynamic forces and moments acting on a ship can
result in a variety of motions and orientation parameters. By utilizing the
Taylor series expansion of a function with multiple variables, these functions
can be simplified into a more manageable mathematical form. In this scenario,
the sway forces and yawing moment can be accurately represented by using only
the odd terms in the Taylor series due to the symmetry between port and
starboard.

Alternatively,
the non-linearities could be accounted for using quadratic polynomial
expressions (Fedyaevsky and Sobolev, 1963a).
Although second-order terms may not be ideal since they are even functions,
this issue can be circumvented by incorporating a modulus term and adjusting
the way the non-linearities are expressed. The quadratic functions modulus
approach can effectively demonstrate the hydrodynamic idea of cross-flow drag
at high angles of incidence, as it offers certain advantages.

Non-linear
forces and moments were computed using the quadratic form to depict the
non-linear forces on a ship hull (Fedyaevsky and Sobolev,
1963b). An accurate representation was achieved by incorporating lateral
force caused by drag from the cross-flow velocity component. Nevertheless, the
distribution of non-linear forces appeared to be more focused towards the
stern.

The linear
whole ship model gave accurate predictions for small rudder movements but
proved to be inaccurate for complete turning circles (Yang,
Chillcce, and El Moctar, 2023). In contrast, the non-linear whole ship
model accurately depicted the three degrees of freedom motion in various
manoeuvring scenarios, as validated by full-scale trials.

Inoue
(1978) suggested
using a combination of cubic and quadratic terms to incorporate non-linear
factors. While the improvement was marginal, the cubic model showed a slightly
better fit to the data compared to the quadratic model. There is a noticeable
distinction between the two models, indicating a need for further investigation
to refine mathematical representations of hydrodynamic forces.

The
hydrodynamic derivatives are also important in other manoeuvring performances
such as berthing (Zhang *et al.,* 2023),
the interaction between ships (Degrieck* et al*.,
2021), shallow water effect (Yang, and el
Moctar, 2024), and so on. (Shouji and Ohtsu, 1992) used the quadratic polynomial to express the
hydrodynamic forces and moment acting on a main hull induced by manoeuvring
motion and (Sawada *et al.,* 2021)
applied the cross-flow drag theory introduced by (Yoshimura,
Nakao, and
Ishibashi, 2009)
to describe
the hydrodynamic forces resulting from significant drift angles at low speeds.
This model, which relies on fewer hydrodynamic derivatives than the traditional
polynomial model, can effectively capture forces in both the transverse and
turning directions. Furthermore, linearized hydrodynamic derivatives, as
outlined by (Yasukawa and Sakuno, 2019). It
can be used to calculate a ship's course stability index.

This
research will focus on a mathematical model based on a polynomial model in
order to investigate the differences between 3^{rd} order polynomial
(cubic model) and 2^{nd} order polynomial (quadratic model).
Additionally, when comparing the cubic model and the quadratic model, it is
commonly believed that the cubic model is more effective in accurately
estimating hydrodynamic force, especially in cases of large motion. However, it is important to note that the cubic model does not
include a term that is proportional to the square of the drift angle. This
omission may seem inconsistent with theoretical studies. However, despite this
drawback, the cubic model is still preferred due to its ability to effectively
explain physical phenomena. As a result, various research institutes may use
different models, including the selection of sway velocity

In this research, the measurement data of the hydrodynamic forces of 12 model vessels (total of 27 loading conditions) that had been conducted for turning tests and the zigzag tests in the past at Kyushu University were targeted. The based second-order model and the based third order models are applied for re-analysis, and the results of a comparative study on the difference and approximation characteristics of the hydrodynamic force due to the difference of the adopted models are shown.

Experimental Methods

The
dimensionless equations for manoeuvring motions can be described using equation
(1) by taking into account the hull, propeller, and rudder components as
follows,

The
subscripts *"H", "P",* and *"R"*

Moreover,
the hydrodynamic forces and moments experienced by a ship are dependent on its
motion and are impacted by variables such as the ship's dimensions and type of
movement. As a result, numerous parameters are needed to accurately describe
these forces. The Taylor series expansion method can be used to simplify the
complex characteristics of hydrodynamic forces into a mathematical equation
with multiple variables. It is important for the hydrodynamic forces and their
derivatives to be continuous and not approach infinity within the range of
values. This requirement is typically met when analyzing hydrodynamic bodies
like ships.

Additionally,
the Taylor expansion is structured in a specific way as expressed in equation
(2) when dealing with multiple variables.

The
combination of dimensionless sway velocity and dimensionless yaw rate or
a combination of drift angle are frequently utilized as the variables such
as

The terms
involving squared, and other terms of a higher order are
usually ignored. since their impact is considered less significant compared to
the terms outlined in Equations (3) and (4). Alternatively, the hydrodynamic
forces acting on a ship hull can be represented by equation (5) and (6) with
second order polynomials as follows,

Just like
with 3^{rd} order polynomials, the coupling termand other higher order terms are typically not taken into account. As a result,
mathematical models that rely on the Taylor series aroundare commonly presented in equation (7) and (8),

represents linear hydrodynamic derivatives,
while refer to nonlinear terms. The composition of
linear terms, as depicted by the functions of and their positions in the nonlinear terms vary among various research
institutions.

**Figure ****1** Ship manoeuvring motion in
a body fixed coordinate system

The 3^{rd}
order model, which utilizes dimensionless sway velocity (Yasukawa and
Yoshimura, 2015), is represented by equations in the ship fixed
coordinate system displayed in Figure 1.

Illustrate the dimensionless values of lateral
force and yawing moment. Kyushu University traditionally uses a second order
model based on drift angle (Kijima *et al.,*
1990), which is represented by the equations below

The 2^{nd}
order model terms ofcontain absolute symbols, signifying
hydrodynamic force changes based on motion direction. The model in Equation (9)
is referred to as the "cubic model", while the model in Equation (10)
is known as the "quadratic model”.

**Table 1** Ship models
principal dimensions

The hydrodynamic forces from captive model
tests were reanalyzed for 12 different model ships. This analysis was based on
the mathematical models in Equations (9) and (10), in order to determine the
hydrodynamic derivatives. Table 1 provides details on the ships, including the range of drift angle and block coefficient . SR108 and Esso Osaka are known for their widely disclosed hull shape and
experimental data. Ships A to J were used as test ships, with measurement data
available for fully loaded, ballast, and, in some cases, half-loaded
conditions. Table 2 and Table 3 shows the determined hydrodynamic derivatives
based on cubic model and quadratic model respectively.

**Table 2** Cubic model
hydrodynamic derivatives

**Table 3** Quadratic model
hydrodynamic derivatives

Results and Discussion

*3**.**1**. Comparison
of approximation accuracy of hydrodynamic force*

are calculated using hydrodynamic derivatives
which were obtained based on Equations (9) and (10). The accuracy of predicted hydrodynamic force was compared
based on the coefficient of determination

Here, is the measured value of is the average value of the measured data, and is the value calculated by Equation (9) or Equation
(10). The closer the value of to 1.0, the better the approximation accuracy
results. Figure 2 compares values of the cubic model and quadratic model
for every ship and their loading conditions are shown in Table 1. Ship numbers
(1-27) in Table 1 are shown on the horizontal axis. The lines connecting each
point are added for convenience and have no physical meaning. For both values of the cubic model are generally closer
to 1, indicating that the approximation accuracy of the cubic model is
generally better than that of the quadratic model. Looking at the lateral
force, the differences in the values of Ship No. 5 to 8 (Ships A and B) are
particularly large. For these model ships and loading conditions, as shown in
Table 1, the maximum value of the drift angle at which the hydrodynamic force
was measured is 10°. Figure 3 shows the comparison between ship No. 8 and ship
No. 22, which have the same type of ship and loading condition. A large
difference appears in the calculation results of ship No. 8 based on both
models within a wide range of drift angles.

**Figure 2** Comparison of the value for cubic and quadratic models (a)
lateral force and (b) yawing moment

**Figure 3** Comparison of

On
the other hand, for the yawing moment, there is a tendency for the differences
in the values of to increase model ships and loading conditions
after Ship No. 14. The comparison between Ship No. 3 and Ship No. 14 is shown
in Figure 4. Both of these ships have the same type and loading condition. From
the figures, the nonlinearity with respect to appears to be large when is large.

**Figure 4** Comparison of by cubic and quadratic models for Esso Osaka (a)
and Ship E (b) on fully loaded condition

*3**.**2**. Effect of measurement range of drift angle ** on approximation accuracy*

To clarify the cause of the differences
in approximation accuracy shown in Figure 2, the range of drift angle used to
calculate the hydrodynamic derivatives was changed and the analysis was
performed again. The influence of the drift angle measurement range on the
hydrodynamic forces' approximation accuracy was investigated.

First, for model ships and loading conditions shown as Ships No. 1 to 4 and Ships No. 9 to 27 whose hydrodynamic forces were measured in the range of the range of the measured hydrodynamic forces data used in the analysis was limited to . After the hydrodynamic derivatives were obtained, the calculated results of lateral force and yaw moment were compared with the original results. For example, Figure 5 shows the results for the Esso Osaka (Ship No. 3) in fully loaded condition. The figures also show the measurement data in the range of that were not used in the analysis.

In Figure 5 (right), regarding the lateral forces, there
is a large difference between the calculation results of both mathematical
models in the range where the value of is large. As the value of increases, the cubic model tends to exhibit a
higher rate of increase for lateral force. Consequently, there is a widening
discrepancy between the calculated results and the measured values. Conversely,
for the yaw moment, it appears that the quadratic model yields better results
when the turning motion is minimal (when the value of is small). However, as the turning motions
develop, the cubic model has better agreement with the measured values.

**Figure 5** Comparison of by both mathematical models for Esso Osaka on
fully loaded condition in the range of

Next, Lateral force and yaw moment were determined for
model ships and loading conditions of Ships No. 1 to 4 and Ships No. 9 to 27
using the hydrodynamic derivatives obtained from the measurement data of Figure 6 shows the values
of including the measured data for . For lateral force, the
quadratic model maintains the values of

**Figure 6** Comparison of
the value for cubic and
quadratic forms (a) lateral force; and (b) yawing moment

Finally, Figures. 7 and 8 show the results of hydrodynamic derivatives for lateral force and yaw moment as functions of parameters representing hull form. The symbol indicates the hydrodynamic derivatives for Ships No. 5 to 8, and the symbol indicates the hydrodynamic derivatives for other model ships and loading conditions. The parameters on the horizontal axis in each figure have the maximum value of when polynomial approximation of the corresponding hydrodynamic derivatives is performed using the parameters.

Looking at the linear
hydrodynamic derivatives first, there is a linear relation between the
derivatives and the parameters on the horizontal axis regardless of the marks. On the other hand,
when looking at the non-linear hydrodynamic derivatives, the variation is
slightly larger than that of the linear derivatives. The symbol which shows the
hydrodynamic derivatives for Ships No. 5 to 8 shows tendency different from the
symbol which shows the
hydrodynamic derivatives for other model ships and loading conditions, as
mentioned above. It is considered that the narrow measurement range of the
drift angle affects the analysis results.

**Figure 7** Hydrodynamic derivatives for lateral force in cubic form as a
function of hull form parameters

**Figure 7** Hydrodynamic derivatives for lateral force in cubic form as a
function of hull form parameters

**Figure 8** Hydrodynamic derivatives for yawing moment
in cubic form as function of hull form parameters

*3**.**3**. Course Stability Index*

The course stability index can be determined by using the linear hydrodynamic derivatives provided in Equations (9) and (10) (Yoshimura, 2001; Yukawa and Kijima, 1998).

Here, a positive (+) indicates instability, while a negative (-) indicates stability.

Then, Equation (12) can be rewritten to
equation (13)-(15),

The
course stability indices of the quadratic and cubic models are being compared
to determine the impact of using different model approaches according to equations
(12) to (15). Figure 9 illustrates the linear hydrodynamic derivatives derived
from analyzing measured hydrodynamic forces using both models, which are then
utilized in calculating the course stability indices.

Figure
10 presents a comparison of course stability indices between 2 mathematical
models for different model ships and loading conditions as detailed in Table 1.
The x-axis shows the number of ships ranging from 1 to 27. It is clear that
some ships show differences in course stability indices when using both
mathematical models. Several factors may contribute to this discrepancy, such
as differences in the mathematical properties of each model.

To achieve a better comprehension of the phenomenon, the relationship between the and the hydrodynamic derivatives of both mathematical models for each of the 27 ships are being analyzed. The ships are then categorized into three groups based on the results of calculated for easier comprehension.

I. Ships with consistent signs for cubic and quadratic models across all loading conditions.

II. Ships exhibit different signs for cubic and quadratic models under certain loading conditions.

III. Ships exhibit different signs for cubic and quadratic models across all loading conditions.

**Figure 11 **provides illustrations of the three categories, while Table 4
displays the breakdown of ships that fit into each category.

**Figure 9** Linear hydrodynamic derivatives
based on quadratic and cubic models

**Figure 10** Quadratic
model vs cubic model of

**Table 4** List of ships

Category 1 |
Category 2 |
Category 3 |

SR108 Ship A Ship B Ship H Ship J |
Esso Osaka Ship C Ship D Ship E Ship I |
Ship F Ship G |

**Figure 11** Examples of for three different categories (a) Category 1
– SR108; (b) Category 2 – Ship C; and (c)
Category 3 – Ship F

Figure 12 displays the curves of Ships No. 1 and 2 (SR108) under fully loaded and ballast conditions, fitted with cubic and quadratic models within the range The red and blue solid lines on the graph represent fitting curves based on hydrodynamic derivatives from captive model test data, with the horizontal axis showing the drift angle . The dashed lines on the graph indicate the inclines of the curves representing lateral force and yawing moment at the starting point, which are equivalent to linear hydrodynamic derivatives for and are referred to as slope lines. The figures display experimental data points marked with circles. The slope of the cubic model for lateral force is steeper than that of the quadratic model, whereas the opposite is true for yawing moment. These trends are consistent across both loading conditions.

A noticeable distinction exists between the curves produced by cubic and quadratic models over a wide range of drift angles . In general, when using the least square method to fit a curve with a cubic function, the resulting values tend to be larger beyond the input data range compared to fitting with a quadratic function. This discrepancy is attributed to the limited availability of experimental data for large drift angles.

Figure 13 displays the curves for Ship C (Ships No. 9 and 10) under ballast and fully loaded conditions in category 2. The discrepancy in signs between the two ships are primarily due to variations in the yawing moment slope line in relation to drift angle . Both ships exhibit a smaller inclination of the quadratic model slope line for lateral force compared to the cubic model. However, there is a contrasting trend in the yawing moment between Ship No. 9 and Ship No. 10, with Ship No. 9 showing a slightly larger inclination of the quadratic model slope line. This difference in tendencies is not observed in Category 1, where all signs are consistent. The varying signs are a result of differences in linear hydrodynamic derivatives between the two models, as detailed in Table 2 and Table 3.

In Category 3, Figure 14 displays the curves of Ships No. 16 and 18 (Ship F) under
fully loaded and ballast conditions. The

**Figure 12** Category 1 - The curves of
SR108 were modelled for both loading conditions (a) Fully loaded condition and
(b) Ballast condition