Study of Applicability in Minimising Pitch Radius Gyration for Different Hull Types to Improve Seakeeping Performance

Abstract

This paper presents an optimisation study to determine the best centre of gravity (CoG) position to improve seakeeping performance. Two varied parameters used in this study were the longitudinal and vertical centre of gravity (LCG and VCG). The Radius Gyration in the y-axis ($R_y$) is introduced as a novel single-objective function to be minimised, avoiding the complexity of the conventional seakeeping optimisation process. The quality of the seakeeping performance was evaluated by response amplitude operators (RAOs) of the heave, pitch, and vertical motion. Two different hull forms are compared to investigate the applicability of the $R_y$ as the objective function in seakeeping optimisation. The patrol boat and S-60 hull form are selected as representatives of a planing hull type and a displacement hull type. The optimisation was carried out by using the Central Composite Design (CCD) and response surface methodology (RSM) to model the relationship between the CoG and $R_y$ from large and small vessels, with the objective function minimising the $R_y$. The finding shows that minimising the $R_y$ is more sensitive to the planing hull type compared to the displacement hull type in reducing the vertical motion in different Froude numbers and wave headings.

1. Introduction

The quality of seakeeping performance is essential to assess and to ensure the safety, operational efficiency, and the comfort of the passengers and crew on board, particularly in the early stage of ship design. A poor seakeeping performance can lead to excessive ship motions in waves, negatively affecting the overall performance, damaging the cargo, causing passenger seasickness, and, in the worst case, leading to capsizing.

Researchers have conducted seakeeping analyses for various purposes, including safety and mission execution, decision making, crew well-being [1], operational performance [2], and design optimisation [3,4]. Other studies have focused on operational boundaries and limits [5], operability assessments [6,7], and probabilistic evaluations of a vessel’s operability in specific sea areas [8] to ensure a safe operation within the intended operational area and environmental conditions. A seakeeping analysis can also be integrated with propulsion systems studies to enhance the overall design process and identify more efficient and effective vessel designs [9].

Before and during operation, the ship operators must determine the vessel’s loading condition. The position of the centre of gravity (CoG) has a significant influence on the hydrodynamics performance, not only in terms of the calm water resistance but also with regard to seakeeping characteristics and the added resistance in waves. Changes in the CoG affect the metacentric height (GM), which directly influences the natural periods and damping coefficients of motions, such as the heave, pitch, and roll. For certain ship types, such as fishing vessels [6] and passenger ships, the CoG location may change during the operation due to variable loading conditions.

Longitudinal changes in the CoG (LCG) affect the vessel trim. An improper LCG can cause excessive trim (by the stern or by the bow), leading to increased resistance, reduced propulsion efficiency, and higher fuel consumption. Vertical changes in the CoG (VCG) influence stability. A higher VCG lowers the GM, reducing the vessel’s ability to return to an upright position after being disturbed by waves or wind.

Many researchers have conducted trim optimisation as a practical method for improving energy efficiency and reducing fuel consumption, without altering the hull form or replacing the engine. By adjusting the trim angle to minimise the ship resistance [10,11,12], this optimisation approach is among the simplest, cheapest, and most practical to implement.

Trim optimisation has also been applied in wave conditions [13]. For example, Shivachev et al. [14] demonstrated that the trim condition affects both the seakeeping and added resistance for the KCS model. However, an optimal trim in calm water may not be optimal in waves due to the additional resistance they generate. Studies such as [15] have shown that adjusting the trim to prevailing wind and wave conditions can yield significant savings, with one example reporting a reduction of 949.3 kg of fuel on a specific route. Throughout their voyages, ships encounter varying loading conditions, including changes in the draught, trim, and CoG, which significantly influence seakeeping performance and must be considered in accurate assessments [6,7,16].

Conventional seakeeping optimisation often targets specific ship responses, such as the RMS vertical acceleration, RMS pitch, slamming probability, and green-water probability, either as single or multiple objectives to minimise. This makes the optimisation process complex. Unlike trim optimisation, which is straightforward, broader seakeeping optimisation frequently requires altering the hull form to achieve objectives such as minimising the absolute vertical motions [17,18]; vertical acceleration, heave, pitch, and slamming [19]; resistance, vertical motion, and stability [20,21]; resistance and vertical acceleration [22]; or the amplitudes of the heave and pitch [23,24]. The complexity increases further when non-conventional optimisation methods are employed.

Optimisation techniques can generally be classified as conventional or non-conventional. Conventional methods include linear and non-linear programming, gradient-based methods, and other analytical approaches relying on calculus [25,26,27]. Non-conventional or modern methods, often inspired by natural processes, include genetic algorithms, simulated annealing, and other metaheuristic and evolutionary algorithms [17,20,22,23,24,28], as well as bee colony optimisation [29] and particle swarm optimisation (PSO) [30].

Optimisation can also be categorised into gradient-based and direct search approaches. Gradient-based methods utilise first-order (gradient) or second-order (Hessian) derivative information to guide the search for optimal solutions. Common examples include the steepest descent, conjugate gradient, and response surface method. Direct search methods do not rely on gradient information, instead iterating towards an optimal solution, often requiring numerous iterations to converge. They are particularly useful for problems where derivatives are unavailable or unreliable. Examples include the simplex method, Hooke and Jeeves [19,31], and genetic algorithms.

To simplify the complexity of seakeeping optimisation, Iqbal et al. [32] proposed a new single-objective function: minimising the pitch radius of gyration ($R_y$). This parameter is correlated with the response amplitude operators (RAOs) for the heave, pitch, and vertical motion, and reducing it is expected to improve the seakeeping performance. The optimisation method employed, the response surface methodology (RSM), was chosen for its efficiency, as it reduces the number of required simulations or experiments, saving both time and resources [33,34].

Iqbal et al. [32] first applied this method to a fishing vessel, demonstrating that minimising the $R_y$ improved seakeeping performance without significantly affecting the total calm water resistance. The same fishing vessel hull was later optimised for minimal calm water resistance while also incorporating $R_y$ minimisation, yielding an optimal hull form with both reduced resistance and improved seakeeping [4].

However, the applicability of this objective function to vessels of different sizes has not yet been investigated. This research therefore aims to examine the use of the $R_y$ in optimising the CoG location for two vessels with different hull types, by varying both the LCG and VCG.

2. Methods

2.1. The Subject Vessels

The principal dimensions and 3D models of the two subject vessels are presented in

Table 1. The main dimensions of the subject ships.

Parameter	Patrol Boat	Series 60 (S-60)
Displacement (ton)	84.111	164.222
Length overall, LoA (m)	25.000	31.638
Length between water line, Lwl (m)	21.829	30.979
Breadth, B (m)	5.641	4.343
Draught, T (m)	1.500	1.737
Lpp/B (-)	3.869	7.133
B/T (-)	3.761	2.500

and Figure 1. The patrol boat and Series 60 (S-60) hull forms (available in Maxsurf Software Version 2024) were selected to represent planing hull type and displacement hull type, respectively. The longitudinal and vertical positions of the centre of gravity (LCG and VCG) were systematically varied to develop the mathematical model and assess the response, specifically, the radius of gyration about the y-axis ($R_y$). Previous work on seakeeping optimisation for the Series 60 hull [18] successfully improved vertical motion performance.

2.2. Design of Experiment

Central Composite Design (CCD) is a highly efficient and widely used approach within the design of experiments (DOE) framework for process and system optimisation. It is particularly valuable in response surface methodology (RSM) for fitting second-order polynomial models, which enables the investigation of relationships between multiple explanatory variables and one or more response variables [35,36,37]. CCD provides maximal information with a minimal number of experimental runs, making it an effective method for saving both time and resources.

A CCD involves five coded levels for each factor: −α, −1, 0, +1, and +α. These levels are selected to ensure rotatability and symmetry in the design [36]. Code 0 (centre point) represents the midpoint of the factor range. Codes −1 and +1 (factorial points) correspond to the standard low and high levels of the factors. Codes −α and +α (axial points) are located at a distance α from the centre along each axis. For two factors, α is typically set to $\sqrt{2}$ (approximately 1.414) to achieve rotatability.

In this study, the two variables are LCG ($x_1$) and VCG ($x_2$). The initial LCG position coincides with the longitudinal centre of buoyancy (LCB), while the initial VCG position is set at 75% of the transverse metacentric height (KM_t). These initial values are defined as Code 0. The LCG and VCG were then varied by ±5% from their initial positions, corresponding to Codes ±1. The choice of ±5% is based on practicality: larger variations in LCG and VCG would induce unrealistic trim changes, which are not applicable for the present vessels. The axial points (Codes ±1.414) were applied in accordance with the two-variable CCD. The actual values of LCG and VCG and their corresponding coded values are given in

Table 2. The actual value of initial LCG and VCG and the conversion to the code.

Design Variable	Code and Actual Value
	−1	0	1
Patrol Boat
LCG (m), $x_1$	8.431	8.875	9.319
VCG (m), $x_2$	2.571	2.707	2.842
Series 60
LCG (m), $x_1$	15.077	15.870	16.664
VCG (m), $x_2$	1.256	1.322	1.388

, while the complete set of variations under the CCD framework is presented in

Table 3. The design of experiment based on Central Composite Design (CCD).

Load Case	Code		Patrol Boat		Series 60
	${\mathit{x}}_1$	${\mathit{x}}_2$	LCG (m)	VCG (m)	LCG (m)	VCG (m)
Initial	0	0	8.875	2.707	15.870	1.322
LC 1	1	1	9.319	2.842	16.664	1.388
LC 2	1	−1	9.319	2.571	16.664	1.256
LC 3	−1	1	8.431	2.842	15.077	1.388
LC 4	−1	−1	8.431	2.571	15.077	1.256
LC 5	−1.414	0	8.247	2.707	14.748	1.322
LC 6	1.414	0	9.503	2.707	16.992	1.322
LC 7	0	−1.414	8.875	2.515	15.870	1.229
LC 8	0	1.414	8.875	2.898	15.870	1.416

.

The response variable in this study is the radius of gyration about the y-axis (pitch radius of gyration, $R_y$). Equation (1) is used to calculate $R_y$ for different LCG and VCG values. In this equation, $z_i$ and $x_i$ represent the vertical and longitudinal distances, respectively, of a weight component $w_i$ from the CoG. A change in CoG location alters the terms ${z_i}^2+{x_i}^2$, resulting in a different $R_y$.

The $R_y$ value affects the pitch damping ratio coefficient, as shown in Equation (2), which defines the ratio of actual damping to critical damping. This ratio is a function of the pitch moment of inertia $I_{yy}$, the added moment of inertia of pitching $I_{yy\left(a\right)}$, and the natural pitch frequency ${\omega }_n$. As the hull form remains unchanged, the values of $I_{yy\left(a\right)}$ and ${\omega }_n$ do not differ significantly from their initial values. The main variation occurs in $I_{yy}$, which depends on $R_y$ and the vessel’s displacement $\Delta$. A reduction in $R_y$ decreases $I_{yy}$, thereby increasing the pitch damping ratio, which in turn reduces the peak value of the pitch response amplitude operator (RAO).

$$R_y=\sqrt{{\displaystyle \frac{I_{yy}}{\Delta }}}=\sqrt{{\displaystyle \frac{\sum w_i\left({z_i}^2+{x_i}^2\right)}{\Delta }}}$$

(1)

$$\mathsf{\zeta }={\displaystyle \frac{C}{C_r}}={\displaystyle \frac{C}{2(I_{yy}+I_{yy\left(a\right)}){\omega }_n}}$$

(2)

2.3. Optimal Solution

Once the $R_y$ values for each load case in Table 3 have been determined, a mathematical model can be generated from the sample data using a regression method, resulting in Equation (3). After the coefficients from $a$ to $f$ are obtained, the response surface can be created by plotting Equation (3) with inputs $x_1$ and $x_2$. The ranges of $x_1$ and $x_2$ must be constrained to ensure realistic longitudinal and vertical centres of gravity (LCG and VCG), thereby avoiding extreme shifts that could result in excessive trim and poor stability.

$$R_y\left(x_1,x_2\right)=a+b{x}_1+c{x}_2+d{x_1}^2+e{x_2}^2+f{x}_1{x}_2$$

(3)

The optimal condition from the generated mathematical model is determined using Equations (4) and (5). These equations identify the stationary point, where the first derivative of the function equals zero. This indicates that the function $R_y\left(x_1,x_2\right)$ is neither increasing nor decreasing, representing either a minimum or maximum. In this study, the response surface is expected to exhibit a minimum.

$$R_{\mathrm{y}min}\left(x_1,x_2\right)={\displaystyle \frac{d{R}_y\left(x_1,x_2\right)}{d{x}_1}}=0$$

(4)

$$R_{\mathrm{y}min}\left(x_1,x_2\right)={\displaystyle \frac{d{R}_y\left(x_1,x_2\right)}{d{x}_2}}=0$$

(5)

2.4. Seakeeping Calculation

Once the optimal centre of gravity (CoG) is identified, the seakeeping simulation is conducted using Maxsurf Motion Software Version 2024, based on the two-dimensional (2D) strip theory method. The hull is divided into a series of strips—each treated as a 2D section. The original section is transformed into a simpler geometry (e.g., a circle or ellipse) using a conformal mapping function [38]. Potential flow equations are solved for the transformed geometry, which is mathematically more tractable, and the results are then transformed back to the original geometry using the inverse mapping. The hydrodynamic responses of all strips are superimposed to obtain the overall vessel response in waves.

The sections obtained through conformal mapping are geometrically equivalent to the originals (Figure 2). As each transformed section closely matches its original shape, the hydrodynamic coefficients accurately represent the actual hull geometry. These coefficients are then integrated along the hull to derive the ship’s overall hydrodynamic coefficients, from which the seakeeping responses are determined.

The higher-fidelity methods, such as CFD, have already been carried out in our earlier works [4,32], where the RAOs based on CFD result for both the initial and optimised CoG (obtained by minimising $R_y$) of fishing vessel were compared. Those studies demonstrated that minimising $R_y$ reduces the RAOs in heave, pitch, and added resistance. In the present paper, our intention is to show that the same trend can also be captured using 2D strip theory (Maxsurf Motion), where a lower $R_y$ leads to reductions in pitch, heave, and vertical motions.

The experimental seakeeping test of planing hulls in irregular waves [39] was employed as a validation study by comparing the vertical acceleration results at the LCG and the bow, obtained from Maxsurf Motion (2D Strip Theory), with the experimental data. The 3D hull form in IGES format is available and was used for simulation. In this validation study, Model C was selected. The 3D model and the conformal mapping section are shown in Figure 3. The principal dimensions and test conditions are presented in

Table 4. Main dimensions of test conditions for Model C [39].

Main Dimensions		Test Conditions
Parameters	Value	Parameters	Value
L (m)	2.00	Scale factor	5.435
B (m)	0.43	Wave Spectrum	Jonswap
T (m)	0.09	H1/3 (m)	0.092
Δ (N)	243.40	Tp (s)	2.57
L/$\nabla$1/3	6.86	Heading	180°
Β (°)	22.5	FrB, $V/\sqrt{gB}$	5.67
LCG (%L)	0.33	$V$ (m/s)	11.659

. The results, summarised in

Table 5. The vertical acceleration comparison results obtained from Maxsurf Motion and experimental test from [39].

Position	Maxsurf Motion	Experiment	Error
LCG	8.88 m/s2	8.60 m/s2	3.2%
Bow (FP)	15.78 m/s2	15.42 m/s2	2.3%

, show that the values from Maxsurf Motion are marginally overestimated compared to the experimental measurements, by approximately 3.2% at the LCG and 2.3% at the bow. The small discrepancies indicate that the numerical method provides sufficient predictive accuracy of vertical accelerations, thereby validating its applicability for further seakeeping analyses.

To strengthen the analysis and support the use of the 2D Strip Method, we have also employed ANSYS Aqwa (hydrodynamic diffraction) software, Version 2024 R2, based on the 3D Panel Method. Accordingly, the RAO results obtained from Maxsurf Motion were supported with those from ANSYS Aqwa. It is not expected that the RAO values will be identical; rather, the objective is to examine the consistency of the RAO trends between the initial and the optimised CoG achieved by minimising $R_y$. The generated mesh of patrol boat and Series 60 is shown in Figure 4.

2.5. Simulation Conditions for Seakeeping Simulation

Seakeeping simulations for both the initial and optimal CoG positions were performed at various speeds and at head seas, as shown in

Table 6. Simulation conditions.

Parameter	Patrol Boat	Series 60
Froude Number	0.2, 0.3, 0.4	0.2, 0.25, 0.3
Wave Heading	180°, 150°, 120°	180°, 150°, 120°
${\omega }_e$ Range (rad/s)	0–4	0–4
Remote Location for Vertical Motion ($x,y$)	FP, T	FP, T

. Changing the CoG position alters the radius of gyration about the x-, y-, and z-axes and thus the moments of inertia for roll, pitch, and yaw. In this study, only pitch motion is analysed at wave headings of μ = 180°, 150°, and 120°. Shifting the CoG from the even-keel condition produces a trim by bow or stern, depending on whether the shift is forward or aft.

3. Results and Discussion

3.1. Results of Response Surface

The $R_y$ values for each load case and vessel type are shown in

Table 7. The response of $R_y$ for every load case.

Load Case	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{R}}_{\mathit{y}}$ Patrol Boat (m)	${\mathit{R}}_{\mathit{y}}$ Series 60 (m)
Initial	0	0	6.191	7.172
LC 1	1	1	6.144	7.253
LC 2	1	−1	6.133	7.254
LC 3	−1	1	6.284	7.176
LC 4	−1	−1	6.272	7.177
LC 5	−1.414	0	6.321	7.203
LC 6	1.414	0	6.124	7.316
LC 7	0	−1.414	6.186	7.174
LC 8	0	1.414	6.202	7.172

. Regression analyses, yielding coefficients for Equation (6a) for the patrol boat and Equation (6b) for the Series 60 hull, were conducted. Statistical results, including the regression statistics (

Table 8. The regression statistics.

Variables	Patrol Boat	Series 60
Multiple R	1.000000	0.999826
R Square	0.999999	0.999653
Adjusted R Square	0.999997	0.999075
Standard Error	0.000112	0.001569
Observations	9	9

), analysis of variance (

Table 9. Analysis of variance (ANOVA).

Component	df	SS	MS	F	Significance F
Patrol Boat
Regression	5	0.040148913	0.008029783	639174.769	3.088 × 10−9
Residual	3	3.76882 × 10−8	1.25627 × 10−8
Total	8	0.040148951
Series 60
Regression	5	0.021274	0.004255	1728.454	2.19 × 10−5
Residual	3	7.38 × 10−6	2.46 × 10−6
Total	8	0.021282

), and coefficient significance (

Table 10. The coefficient model and its significance.

Variable	Patrol Boat		Series 60
	Coefficients	p-Value	Coefficients	p-Value
Intercept	6.191307	1.31 × 10−14	7.17249	2.31 × 10−11
$x_1$	−0.069686	4.06 × 10−10	0.03898	6.35 × 10−6
$x_2$	0.005729	7.30 × 10−7	−0.00060	0.360305
${x_1}^2$	0.015499	1.68 × 10−7	0.04316	2.13 × 10−5
${x_2}^2$	0.001467	1.97 × 10−4	−5.44 × 10−5	0.956545
$x_1{x}_2$	6.432 × 10−5	0.334312	3.18 × 10−6	0.997025

), were evaluated.

$$\begin{gathered} R_y\left(x_1,x_2\right)=6.191307-0.069686+0.005729x_2+0.015499{x_1}^2 \\ -0.001467{x_2}^2+6.43·{10}^{-5}{x}_1{x}_2 \end{gathered}$$

(6a)

$$\begin{gathered} R_y\left(x_1,x_2\right)=7.17249+0.03898x_1-0.00060x_2+0.04316{x_1}^2-5.44 \\ ·{10}^{-5}{x_2}^2+3.18·{10}^{-6}{x}_1{x}_2 \end{gathered}$$

(6b)

For the patrol boat and the Series 60 hull, the multiple $R$ values are extremely high (≈1), indicating a strong correlation between the predicted and actual $R_y$. The $R^2$ values are 0.999999 and 0.99653, respectively, meaning that approximately 99.999% and 99.653% of the variation in the $R_y$ is explained by the model variables $x_1$ (LCG) and $x_2$ (VCG). Adjusted $R^2$ values (0.999997 and 0.999075) confirm the robustness of the models even after accounting for the number of variables. The low standard errors indicate minimal deviation between predicted and observed values, despite the relatively small dataset (nine observations).

As shown in Table 9, the F-statistics are very high (639,174.769 and 1728.454), with significance levels far below 0.05, confirming that the regression relationships are statistically significant. Table 10 shows the coefficient significance, where for the patrol boat, both $x_1$ (LCG) and $x_2$ (VCG) are significant predictors of the $R_y$, except for their interaction term ($x_1{x}_2$). The linear and quadratic terms for both variables exert a strong influence, indicating that small changes in the LCG and VCG can meaningfully alter the $R_y$. This information highlights the importance of careful load distributions of small vessels to ensure optimal seakeeping.

For the Series 60 hull, $x_1$ (LCG) exerts a more significant effect than $x_2$ (VCG), both linearly and quadratically. The VCG terms, $x_2$ (VCG), have p-values above 0.05, suggesting a minimal influence. Interaction effects are also negligible. This implies that for large, high L/B ratio merchant vessels, LCG shifts affect the $R_y$ far more than VCG shifts.

3.2. Optimal Solutions

The optimal solution, obtained by solving Equations (4) and (5) from Equation (6a,b), is compared to the initial condition in

Table 11. The comparison between initial and optimum design variables ($x_1$ and $x_2$).

Load Case	${\mathit{x}}_1$	${\mathit{x}}_2$	LCG (m)	VGG (m)	${\mathit{R}}_{\mathit{y}}$ Cal (m)	${\mathit{R}}_{\mathit{y}}$ Equation (6a,b) (m)
Patrol Boat
Initial	0.00	0.00	8.875	2.707	6.191	6.191
LC 9 (Optimum $R_y$)	2.252	−2.003	9.874	2.436	6.108	6.107
Difference (%)					−1.339	−1.360
Series 60
Initial	0.00	0.00	15.870	1.322	7.172	7.172
LC 9 (Optimum $R_y$)	−0.452	2.50	15.512	1.488	7.164	7.162
Difference (%)					−0.122	−0.148

and Figure 5. The regression model predicts an optimal $R_y$ of 7.162 m, which is a reduction of 0.148% from the initial value. The actual calculated $R_y$ is 7.164 m, which is a 0.122% reduction. The prediction error is −0.028%, indicating high model accuracy.

The response surface analysis, shown in Figure 5, shows that the patrol boat exhibits a minimum in both the LCG and VCG directions, which is consistent with earlier findings [4,32]. In contrast, the Series 60 hull’s response surface is nearly flat with respect to the VCG but exhibits a quadratic minimum with respect to the LCG at −0.452. The disparity is likely to be due to differences in the hull form, with the Series 60′s higher L/B ratio (7.133) compared to the patrol boat’s 3.869, making it less sensitive to VCG changes.

3.2.1. Equilibrium Condition for Patrol Boat

Changing the CoG alters the vessel’s equilibrium condition. For the patrol boat (

Table 12. The comparison of equilibrium condition between initial and optimal CoG for patrol boat.

Parameter	Initial	Optimal	Difference (%)
Draft amidships (m)	1.5	1.543	2.87
Displacement (ton)	84.112	84.111	0.00
Prismatic coeff. (Cp)	0.782	0.778	−0.51
Block coeff. (Cb)	0.444	0.397	−10.59
LCB from zero pt. (+ fwd) (m)	8.875	9.913	11.70
VCB (m)	0.988	1.002	1.42
LCG from zero pt. (+ fwd) (m)	8.875	9.874	11.26
VCG (m)	2.707	2.436	−10.01
GMt (m)	0.9	1.26	40.00
GML (m)	35.607	39.701	11.50
KMt (m)	3.607	3.696	2.47
KML (m)	38.314	42.123	9.94
Trim angle (+by stern) (deg)	0	−1.5181	-
Radius gyration in x-axis (m)	1.8031	1.7839	−1.07
Radius gyration in y-axis (m)	6.1913	6.1084	−1.34
Radius gyration in z-axis (m)	6.2715	6.1953	−1.22

and Figure 6), the optimal solution shifts the LCG forward, producing a trim by bow of 1.5181°. The longitudinal and transverse KM values change by 2.47% and 9.94%, respectively. The GM increases when the VCG is reduced. The radius of gyration decreases in all axes.

3.2.2. Equilibrium Condition for Series 60 Hull Form

For the Series 60 hull (

Table 13. The comparison of equilibrium condition between initial and optimal CoG for Series 60 hull form.

Parameter	Initial	Optimal	Difference (%)
Draft amidships (m)	1.737	1.733	−0.23
Displacement (ton)	164.220	164.210	−0.01
Prismatic coeff. (Cp)	0.699	0.693	−0.86
Block coeff. (Cb)	0.685	0.631	−7.88
LCB from zero pt. (+fwd) (m)	15.87	15.508	−2.28
VCB (m)	0.917	0.918	0.11
LCG from zero pt. (+fwd) (m)	15.87	15.512	−2.26
VCG (m)	1.322	1.488	12.56
GMt (m)	0.441	0.286	−35.15
GML (m)	34.406	35.304	2.61
KMt (m)	1.763	1.774	0.62
KML (m)	35.728	36.791	2.98
Trim angle (+by stern) (deg)	0	0.5878	-
Radius gyration in x-axis (m)	1.3451	1.3472	0.16
Radius gyration in y-axis (m)	7.1725	7.1637	−0.12
Radius gyration in z-axis (m)	7.2216	7.2125	−0.13

and Figure 7), the optimal LCG shifts to the stern direction, producing a trim by stern of 0.58°. The KM remains largely unchanged, but the optimal VCG reduces the transverse GM by 35.15%, as the optimal VCG is higher than the initial one (12.56%). The minimal $R_y$ reduces $R_z$ but increases $R_x$ slightly (0.16%).

3.3. Seakeeping Performance for Patrol Boat

3.3.1. Wave Heading 180°

Figure 8 compares the RAOs of the heave, pitch, and vertical motion of the patrol boat in head seas, based on the Strip and Panel Methods. At a Froude number of 0.2, the optimal centre of gravity (CoG) position, defined by both the optimal longitudinal and vertical CoG, produces only a negligible reduction in the heave RAO curve compared with the initial condition. When the Froude number is increased to 0.3 and 0.4, the optimal CoG slightly increases the RAO at encounter frequencies (${\omega }_e$) lower than the frequency corresponding to the peak RAO and, conversely, reduces the RAO at a higher ${\omega }_e$.

The RAO obtained from the Panel Method exhibits similar behaviour, with the initial and optimised RAO results lying in close agreement. This consistency indicates that the optimisation of the CoG through the minimisation of the $R_y$ does not significantly alter the overall heave RAO characteristics.

In contrast, the optimal CoG consistently yields lower pitch RAOs than the initial configuration across all Froude numbers and encounter frequencies. This result suggests that a small vessel, such as a patrol boat, is more sensitive to the pitch motion than to the heave when the $R_y$ value is minimised. As shown in Table 11, the optimal solution can reduce the $R_y$ by up to 1.339%. The influence of the CoG position on pitch RAOs becomes more pronounced at higher Froude numbers, with the optimal CoG offering a superior pitch performance compared to the initial CoG position.

The pitch RAO obtained from the Panel Method also demonstrates a clear difference between the initial and the optimised CoG. Specifically, the pitch RAO of the optimised CoG is lower than that of the initial configuration, which is in agreement with and further supports the findings from the Strip Method. This consistent trend across both methods highlights the effectiveness of the CoG optimisation in reducing the pitch motion, thereby strengthening confidence in the robustness of the numerical predictions.

Considering the combined effects of the heave and pitch, the peak RAO of the vertical motion at the forward perpendicular (FP) is also reduced under the optimal CoG. This demonstrates an improvement in seakeeping performance. At a lower ${\omega }_e$, the RAO values of the optimal CoG are slightly higher than those of the initial condition, whereas at a higher ${\omega }_e$, the optimal CoG produces a notably lower RAO. Additionally, the optimal CoG not only lowers the peak RAO but also shifts the peak ${\omega }_e$ towards a lower frequency, mirroring the behaviour observed in the pitch.

Based on Figure 6, although the values of the RAOs are not identical between the two approaches, both methods exhibit a consistent trend. For all considered Froude numbers (Fr = 0.2, 0.3, 0.4), the optimised hull form corresponding to the minimization of the $R_y$ demonstrates a reduction in the peak vertical motion RAO compared to the initial hull form. This trend is clearly visible in both the Strip and Panel Method results, confirming that the optimisation strategy effectively contributes to reducing vertical responses in head seas.

The discrepancy in the absolute RAO values is attributed to the limitations inherent in each method. The Strip Theory, being based on a two-dimensional sectional approach, tends to underestimate hydrodynamic interactions, particularly in the vicinity of resonance, where three-dimensional effects become significant. On the other hand, the Panel Method provides a higher-fidelity representation of the fluid–structure interaction, capturing the three-dimensional flow effects and therefore predicting larger motion amplitudes. Despite this difference in magnitude, the consistency of the trend across both methods strengthens the conclusion that reducing the $R_y$ through hull form optimisation has a beneficial effect on the seakeeping performance in terms of vertical motion.

3.3.2. Wave Heading 150°

Figure 9 presents the RAOs for the heave, pitch, and vertical motion at a wave heading of 150°. Relative to the 180° heading, the RAO peaks decrease, and the peak frequencies shift to the right (higher frequency) for all motions and Froude numbers. Higher Froude numbers also result in increased peak RAOs. The influence of the CoG position on RAOs follows the same pattern observed for the 180° heading.

At Fr = 0.2, the optimal CoG for the heave motion again produces only a marginal reduction in the RAO curve compared to the initial one, with a slight increase at the lower ${\omega }_e$ and a reduction at the higher ${\omega }_e$ as the Froude number increases to 0.3 and 0.4. The optimal CoG also reduces pitch RAOs at all Froude numbers and encounter frequencies. For the vertical motion at the FP, the peak RAO decreases, and although RAOs are slightly higher at the lower ${\omega }_e$, they are reduced at the higher ${\omega }_e$.

The vertical motion RAO results at an oblique wave heading of μ = 150° show a similar trend to those at head waves (μ = 180°). The findings confirm that the proposed objective function within the seakeeping optimisation framework effectively reduces vertical motion amplitudes for planing hulls, not only in head seas but also in oblique wave conditions. This demonstrates the robustness of $R_y$-based optimisation as a practical approach for enhancing the seakeeping performance of high-speed vessels.

3.3.3. Wave Heading 120°

Figure 10 compares the RAOs for the same motions at a wave heading of 120°. The trends are similar: RAO peaks decrease, and peak frequencies shift to higher values for all motions and Froude numbers. Increasing Froude numbers further elevates the peak RAOs. The influence of the CoG position on RAOs is consistent with the results for 180° and 150° headings.

The optimal CoG that minimises the $R_y$ significantly affects pitch RAOs but has a more limited effect on heave RAOs. Consequently, the vertical motion RAOs are also reduced, further improving the seakeeping performance. These findings demonstrate that optimising the CoG alters the RAO characteristics of the patrol boat regardless of the wave heading or speed/Froude number. While variations in the heading and speed influence the magnitude of RAOs for both the initial and optimal CoG configurations, the relative trends between them remain unchanged.

Based on Figure 10, it can be observed that minimising the $R_y$ consistently reduces the vertical motion response at a wave heading of μ = 120°. A similar trend is also evident at other wave headings, namely μ = 150° and μ = 180°. Furthermore, this reduction is not limited to a specific Froude number; rather, the same tendency is observed across different operating speeds. These findings provide an important insight that once the $R_y$ is minimised, the vertical motion amplitudes are effectively reduced regardless of the wave heading and Froude number. This consistency highlights the robustness of the $R_y$-based optimisation approach in improving the seakeeping performance of planing-hull-type vessels under various operating conditions.

3.4. Seakeeping Performance for Series 60

3.4.1. Wave Heading 180°

Figure 11 compares the RAOs of the heave, pitch, and vertical motion for the Series 60 hull form (S-60) in head seas. In contrast to the patrol boat results, the optimal CoG for the S-60 has a more pronounced effect on the heave RAO than on the pitch RAO. The peak heave RAO for the optimal CoG is reduced compared to the initial CoG, without any significant shift in the natural frequency (denoted as the ${\omega }_e$ corresponding to the peak RAO). According to Table 11, the optimal solution for the S-60 reduces the $R_y$ value by up to 0.122%. This relatively small reduction may account for the negligible change observed in the pitch RAO, particularly at Fr = 0.3. On the other hand, the Panel Method reveals a clearer distinction between the initial and optimised CoG in the pitch RAO, demonstrating that minimising the $R_y$ can also reduce the pitch motion for the displacement hull type.

Increasing the Froude number produces a similar trend to that observed for the patrol boat, whereby the peak RAO increases and the ${\omega }_e$ shifts to a higher frequency (i.e., to the right). This behaviour is typical in the seakeeping performance across vessel types.

The vertical motion RAO, which is influenced by both heave and pitch RAOs, shows negligible differences between the initial and optimal CoG. As seen with the patrol boat, the vertical motion is more strongly influenced by the pitch than by the heave. Consequently, when pitch reductions are significant (as in the patrol boat), the vertical motion RAO decreases accordingly. In the case of the S-60, since the optimal CoG has an insignificant effect on the pitch RAO, the resulting changes in the vertical motion RAO are also negligible.

Both heave RAOs obtained from the Strip and Panel Methods exhibit a clear distinction, with the minimisation of the $R_y$ for the S-60 hull resulting in a reduction in the heave RAO. In contrast, the difference between the initial and optimised CoG in the pitch RAO is more pronounced in the Panel Method than in the Strip Method. Consequently, the vertical motion RAO derived from the Panel Method demonstrates a reduction in vertical motion for the S-60 hull, a trend that is not as clearly reflected in the results obtained from the Strip Method.

3.4.2. Wave Heading 150°

Figure 12 presents a comparison of RAOs for the Series 60 hull form between the initial and optimal CoG at a wave heading of 150°. As the wave heading changes from 180° to 150°, both the initial and optimal CoG cases show the peak shifting to the right (towards higher frequency). This trend is consistent with that of the patrol boat and is common in the seakeeping analysis when altering the wave heading.

However, examining the influence of the CoG reveals a key difference: based on the Strip and Panel Method, the optimal CoG of the S-60 shifts the vertical motion RAO towards a higher frequency, whereas the optimal CoG of the patrol boat shifts it towards a lower frequency. As noted earlier, the frequency corresponding to the RAO peak is close to the natural frequency. This implies that when the optimal CoG for the patrol boat is shifted forward towards the bow, the natural frequency becomes lower than in the initial condition. Conversely, in the S-60 case, shifting the optimal CoG aft towards the stern raises the natural frequency.

Another difference between the 180° and 150° wave headings lies in the trend of the RAO curves. For the heave RAO at 150°, there is both a reduction in magnitude and a shift towards a higher frequency, with both methods showing a good agreement in capturing this behaviour. This phenomenon is less distinct for the pitch RAO, where the change in the heading to 150° primarily produces a frequency shift, while the reduction in the RAO magnitude varies depending on the method employed. Consequently, the vertical motion RAO exhibits a trend similar to that of the pitch RAO.

These findings, based on the Strip Method, indicate that at a wave heading of 150°, the optimised CoG shifts the vertical motion RAO towards a higher frequency without altering the overall curve shape. The Panel Method shows that at the same heading, the optimised CoG also shifts the RAO curve and lowers it. At a wave heading of 180°, the Strip Method shows neither a change in the curve shape of the vertical motion RAO nor a frequency shift, whereas the Panel Method shows both a frequency shift and a reduction in the RAO.

3.4.3. Wave Heading 120°

Figure 13 compares the RAOs of the heave, pitch, and vertical motion for the Series 60 hull form at a wave heading of 120°. A similar trend to that seen at 150° is observed: both methods agree that the heave RAO for the optimal CoG decreases and shifts towards a higher frequency, while the pitch and vertical motion RAOs merely shift to a higher frequency with or without significant changes in the curve shape according to the method. The typical effects of the wave heading and Froude number are also present here: higher Froude numbers result in increased RAO magnitudes, and moving the vessel’s heading to 120°, away from head seas, reduces both heave and pitch motions, as indicated by lower RAO curves compared to 150° and 180°.

3.5. Comparison Between Two Vessel Types

In general, the Strip and Panel Methods show that the results from both the patrol boat and the Series 60 hull form indicate that a minimal $R_y$ produces a notable effect on the pitch RAO, while the effect on the heave RAO is comparatively minor. When the reduction in the $R_y$ is significant, the pitch RAO and, consequently, vertical motions are also reduced. The influence of the heave RAO on the vertical motion is minimal, which can be explained by the definition of the vertical motion as the sum of the heave at the CoG and the additional vertical displacement arising from the pitch angle and the longitudinal distance from the CoG to the FP. When the pitch is zero, the vertical motion at the FP is identical to the heave motion at the CoG.

Based on Table 11, of the two CoG parameters, longitudinal (LCG) and vertical (VCG), the LCG has the greater influence on the $R_y$. The vessel’s hull type also affects the influence of the VCG: for the planing hull type, such as the patrol boat in this study, the VCG has a significant impact on the $R_y$, both linearly and quadratically, similar to the LCG. In contrast, for the displacement hull type, such as the Series 60, the influence of the VCG on the $R_y$ is far smaller. This can be explained by Table 13, which shows that the transverse metacentric height (KM), calculated as KB + BM, for the Series 60 is relatively low (1.736 m), almost identical to the draught (1.737 m). This low KM limits the range over which the VCG can shift, leading to negligible changes in the VGG. In comparison, the patrol vessel’s transverse KM (3.607 m) is more than twice its draught (1.50 m), shown in Table 12, allowing a greater height for the VCG adjustment. This difference stems from the hull design: the patrol boat has a higher BM, which is the ratio of the second moment of the waterplane area ($I$) about the rotation axis to the displacement volume ($V$).

The influence of the speed (Froude number) and wave heading follows the same general trend for both the patrol boat and the Series 60 hull form. Higher speeds result in higher RAO magnitudes and a shift towards higher frequencies. The wave heading affects the RAO peak regardless of speed: both heave and pitch responses diminish when the heading moves away from head seas (180°) towards oblique seas (150° and 120°). Both the initial and optimal CoG configurations for the patrol boat and the Series 60 exhibit this same behaviour under varying speeds and wave headings.

The findings from this study suggest that when the optimal LCG and VCG produce a significant reduction in the $R_y$, as in the case of the patrol boat, pitch responses decrease, leading to reduced vertical motion at the FP and an improved seakeeping performance. Conversely, when the reduction in the $R_y$ is minimal, as with the Series 60, where LCG changes are small, the effect on pitch responses is negligible, and the vertical motion at the FP remains largely unchanged compared to the initial condition.

Table 14. A comparative summary between two hull types.

Phenomena	Planing Hull Type (Patrol Boat)	Displacement Hull Type (Series 60)
The effect of LCG on $R_y$	- Significant for linear and quadratic.	- Significant for linear and quadratic.
The effect of VCG on $R_y$	- Significant for linear and quadratic.	- Insignificant for linear and quadratic.
The effect of minimal $R_y$ on heave RAO	- Insignificant. - The RAO curve is almost similar to the initial condition.	- Significant. - The RAO curve is lower than initial condition.
The effect of minimal $R_y$ to pitch RAO	- Significant. - The RAO curve is lower than initial condition.	- Insignificant. - The RAO curve is almost similar with the initial condition.
The effect of minimal $R_y$ on vertical motion RAO	- Same trend with pitch.	- Same trend with pitch.
The effect of wave heading on the sensitivity of minimal $R_y$	- The RAO curves in wave headings of 150° and 120° are lower than in wave heading of 180°. - The RAO peak shifts to lower frequency in wave headings of 150° and 120°.	- The RAO curves in wave headings of 150° and 120° are lower than in wave heading of 180°. - The RAO peak shifts to higher frequency in wave heading of 150° and 120°.
The effect of Froude number on the sensitivity of minimal $R_y$	- Insignificant. - The RAO curve has similar trend at each Froude number.	- Insignificant. - The RAO curve has similar trend at each Froude number.

summarises the comparison between the patrol boat and the Series 60 hull form in relation to the phenomena observed, providing insights into the applicability of minimising the radius of gyration about the y-axis ($R_y$) to enhance the seakeeping performance.

3.6. The Implication and Applicability of Minimising R_y

Minimising the $R_y$ for a small vessel can reduce the pitch RAO regardless of the vessel speed or wave heading. When the $R_y$ is significantly reduced, as in the case of the patrol boat in this study, the vessel’s mass moment of inertia about the y-axis ($I_{yy}$) is altered. A minimal $R_y$, as shown in the pitch RAO curve, slightly reduces the natural frequency ${\omega }_n$. The pitch damping ratio, as expressed in Equation (2), can therefore be increased since both the $I_{yy}$ and ${\omega }_n$ are lower than their initial values. The pitch damping coefficient $C$, due to the small influence of the viscous damping on the pitch, and the added moment of the inertia of the pitch $I_{yy\left(a\right)}$ (a frequency-dependent term) are assumed to insignificantly change. This is because the trim difference between initial and optimal conditions is slight and does not involve substantial changes in the hull form. Consequently, the damping ratio coefficient $\mathsf{\zeta }$ for pitching can be increased, leading to a reduction in the peak of the pitch RAO.

Minimising the $R_y$ by identifying the optimal positions of the longitudinal centre of gravity (LCG) and the vertical centre of gravity (VCG) can be implemented either in the early stages of vessel design or for existing ships. Once the optimal LCG and VCG are established, operators of existing vessels can adjust loading conditions to achieve these positions. For vessels still in the design stage, designers can not only plan for the appropriate loading arrangements but may also alter the vessel structurally, for example, by adjusting the main deck arrangement, as illustrated for the patrol boat in Figure 14. In the case of the Series 60 hull form, since the optimal $R_y$ is insignificantly different from the initial condition, such structural adjustments are unnecessary. Moreover, altering a merchant-type hull form to avoid a flat bottom is impractical.

The findings from this study indicate that the applicability of the $R_y$ minimisation is far more pronounced for the planing hull type than for the displacement hull type. Optimal LCG and VCG adjustments in the planing hull type can lead to a significant reduction in the $R_y$, with a corresponding clear reduction in the pitch motion. Unlike the displacement hull type, which typically only carries crew, many planing hull types also transport passengers. For such vessels, reducing the pitch leads to lower vertical accelerations, which in turn minimises slamming and deck wetness, thereby improving comfort and safety. For the displacement hull type, optimising the LCG and VCG with minimal $R_y$ variation can still yield improvements, but these are more evident in the heave performance, as observed for the Series 60 results in this study, rather than in the pitch motion.

4. Conclusions

This investigation into the applicability of $R_y$ minimisation through optimal LCG and VCG adjustments was carried out using two contrasting hull form types: (1) a planing hull type, represented by the patrol boat, and (2) a displacement type, represented by the Series 60 hull form. The LCG variable demonstrated a significant influence on the $R_y$ for both vessel types, whereas the VCG variation influenced the $R_y$ only for the patrol boat and not for the Series 60. The seakeeping performance in this study was analysed with the Maxsurf Motion software, based on the 2D Strip Method, and the results from Maxsurf motion were supported by the results from ANSYS Aqwa (Hydrodynamic Diffraction), based on the 3D Panel Method.

The optimal LCG and VCG for the patrol boat achieved a much greater reduction in the $R_y$ than the Series 60 hull form. Since the optimal $R_y$ for the Series 60 was not markedly different from the initial condition, no significant differences were observed in the pitch RAO or vertical motion between initial and optimal cases. In contrast, the patrol boat displayed a clear improvement: the minimal $R_y$ reduced both pitch and vertical motion RAOs more than in the Series 60, thereby enhancing the seakeeping performance. However, the Series 60 did show a favourable reduction in the heave RAO when the $R_y$ was minimised, performing better in this regard than the patrol boat.

The influence of the wave heading on the sensitivity of the $R_y$ showed that RAO curves for both the patrol boat and the Series 60 hull form were lower at wave headings of 150° and 120° than at 180°. For the patrol boat, the RAO peak shifted to a lower frequency at 150° and 120°, whereas for the Series 60, the RAO peak shifted to a higher frequency. The effect of the Froude number on the sensitivity of the minimal $R_y$ was found to be negligible: the RAO curves for both vessel types displayed similar trends across different Froude numbers. Therefore, when the optimal LCG and VCG can significantly minimise the $R_y$, the pitch and vertical motion RAOs can be reduced regardless of changes in the Froude number or wave heading.

It should be noted, however, that the $R_y$ provides only an indirect assessment of the seakeeping performance and cannot replace a full seakeeping analysis. Consequently, the reduction in RAO peaks achieved through this approach is imperfect. Nevertheless, the method remains useful, particularly for certain hull types, such as planing hulls, where minimising the $R_y$ can contribute meaningfully to improved seakeeping behaviour.

Future work will focus on an operability analysis. Minimal RAO curves do not guarantee reduced vessel responses in irregular wave conditions. Vessels operating in different sea states can exhibit varied motion responses even when their RAO curves are identical. An operability analysis can therefore provide recommendations on whether a ship is capable of performing effectively in a specific operational area, by identifying the vessel’s operability index.