Comparison Between Cubic and Quadratic Models of Hydrodynamic Derivatives to the Ship Course Stability Index

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

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.

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" represent the hydrodynamic forces that have been non-dimensionalized and are acting on a hull, propeller, and rudder. In this study, particular focus is placed on investigating the lateral force  and yawing moment  as they have a significant impact on the accuracy of predicting manoeuvring.

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

Table 3 Quadratic model hydrodynamic derivatives (Cont.)

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  defined by equation (11).


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  and yawing moment by cubic and quadratic models for ship B (a) and Ship H (b) on fully loaded condition

    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  close to 1.0 in many model ships and loading conditions, whereas the cubic model has a lower value of  compared to Figure 2. Although it was shown in Figure 2 that  values of the cubic model for Ships No. 5 to 8 have good approximation accuracy in the range of  it may be necessary to be careful when applying the conditions of motion outside the measurement range. On the other hand, the influence of the measurement range of on approximation accuracy of the hydrodynamic forces using the quadratic model is considered relatively small. There is no significant difference in the yaw moment due to the difference between them.

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),

where,

 represents the location where the yaw damping force is applied, while  represent the locations where the sway damping force is applied. The positions of these points where forces are applied relative to each other can be used to assess the course stability of a ship. The stability of a ship depends on the positioning of the yaw damping force application point relative to the sway damping force application point. The ship is considered to be stable when the point at which the yaw damping force is applied is located in front of the point where the sway damping force is applied. On the other hand, if this is not the case, the ship is considered unstable.

        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   curves show no significant difference between fully loaded and ballast conditions. Yet, the slope line of the quadratic model is less steep compared to the cubic model when it comes to lateral force and yawing moment in relation to drift angle . Conversely, the slope line inclination of the quadratic model for  in relation to the  is greater than that of the cubic model. This trend is also evident in Figure 9.

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

Figure 13 Category 2 - The  curves of Ship C were modeled for both loading conditions (a) Fully loaded condition and (b) Ballast condition

Figure 14 Category 3 - The   curves of Ship F were modelled for both loading conditions (a) Fully loaded condition and (b) Ballast condition

     Previously, the focus was on comparing loading conditions to analyze the variation in hydrodynamic derivatives and   between both mathematical models of the same ship types. Many ships display a difference in   between the two models. Understanding the reasons behind these results is crucial. Consequently, ships within each category are being compared once more.
    The discrepancy between the cubic and quadratic models results can be attributed to differences in the values of linear hydrodynamic derivatives. By examining the course stability index values in Equation (12), it is evident that ships with different signs of   have varying linear derivatives between the two models. For instance, Ship No. 17 exhibits a different sign in the  . Figure 15 shows that both models have different absolute values for the 1^st and 2^nd terms of the   for all ships. In the quadratic model,  has a smaller absolute value compared to  whereas the cubic model shows the opposite. This discrepancy is due to the distinct characteristics of the two functions. In general, when the nonlinearity of the data is low, a quadratic function tends to have smaller linear derivatives compared to a cubic model. This is supported by the data in Figure 15, where the absolute values of the   terms in the quadratic model are consistently smaller than those in the cubic model.

Figure 15 Comparison of the linear derivatives absolute value (a) cubic model and (b) quadratic model

        Moreover, most ships with fully loaded and ballast conditions, including Esso Osaka Ship C, Ship, Ship E, and Ship I, show pretty much all the different signs of   in both the cubic and quadratic models. However, there are exceptions, such as Ship F and Ship G, where both ships show distinct signs of   between the two models in all loading conditions.

        Both models in Category 1 consistently yield the same results regardless of whether  is greater than  or vice versa. However, there is a difference in results in categories 2 and 3. The value of  is greater than  only for the quadratic model. This indicates the importance of considering the discrepancies in linear hydrodynamic derivatives between both mathematical models when assessing the course stability of a ship in ballast conditions.

      Next, we compare ships from Category 1 with ships from Category 2 or 3 for all loading conditions. Ship No. 1 and Ship No. 2 from Category 1 (SR108) are compared with Ship No. 16 and Ship No. 18 from Category 3 (Ship F). These ships were chosen because they are both container carriers, but they have different   tendencies that need to be investigated.

        Figure 16 evaluates the differences in values between the first and second terms without taking into account the negative signs in Equation (12) between Ship No. 1 and Ship No. 16. In Ship No. 1, both mathematic models show that the absolute value of   is smaller than . However, the cubic model for Ship No. 16 shows a different outcome. This discrepancy in absolute values is attributed to the limited experimental data and mathematical differences between the two models, as previously discussed.

Figure 16  Comparison (a) ship no. 1 and (b) ship no 16

        It has been verified that there are variations in the results obtained from the cubic and quadratic models when measuring hydrodynamic forces at different drift angles. To further explore these differences in calculating , the locations where yaw and sway damping force are applied to all ships are analyzed. It is observed that the quadratic model generally yields a lower value compared to the cubic model.

Manoeuvring motion and two mathematical models for lateral force and yawing moment using cubic and quadratic polynomials have been discussed. The cubic model is more accurate for hydrodynamic forces measured over a large range of motion, while the quadratic model may be more accurate for smaller ranges. It is important to consider these differences when using the cubic model for turning tests and CMT on a small range of  and  On the other hand, the comparison of  results between the cubic and quadratic models are examined. The discrepancy in  is attributed to the varying linear hydrodynamic derivatives in the two models. The presence of measured hydrodynamic forces across a wide range of drift angles leads to divergent outcomes between both mathematical models. This highlights the importance of carefully considering the differences in linear hydrodynamic derivatives from the cubic and quadratic models when assessing the course stability index under ballast conditions.

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