### 1. Introduction

*k*) and VCG on roll damping. Collectively, these references contribute to understanding roll-damping phenomena and the associated factors.

_{xx}### 2. Roll Decay Test

### 2.1 Model Test and Set-Up

*GM*in the inclining test. These preliminary steps play a significant role in ensuring data accuracy for the subsequent stages of the study. Table 2 lists the VCG conditions corresponding to GM

_{T}_{T}, where VCG1 is the design value for the KVLCC2.

*GM*). This test involved determining the inclination angle when the mass within the model ship was displaced horizontally while adjusting the height of the weight attached to the stern. The estimation of

_{T}*GM*can be determined by Eq. (1), which involves calculating the heel angle resulting from the change in weight distribution. Fig. 3 defines parameters in the inclining test, where

_{T}*w*,

*W*,

*l*, and

_{y}*ϕ*are the moving weight, displacement of the model ship, lateral distance of the moving weight, and heel angle, respectively. Fig. 4 presents the inclining test performance with the moving weight moved to port (Fig. 4(b)) and starboard (Fig. 4(c)). The heel angle (

*ϕ*) was measured using a digital protractor attached to the model ship.

### 2.2 Experimental Roll Decay

### 2.3 Roll Decay Results

*w*were confirmed to the target periods initially calculated from Eq. (2) for the natural frequency, where

_{n}*m*,

*g*, and

*GM*are the mass, gravity acceleration, and transversal metacentric height, respectively.

_{T}*I*is the roll mass moment of inertia.

_{44}*a*is the roll-added mass moment of inertia that was determined to be the simplified 0.25

_{44}*I*for conventional ships by the United States Naval Academy (USNA). Kianejad et al. (2019) predicted that the non-dimensional roll-added mass moment of inertia was approximately 0.25 for various frequencies at roll angles smaller than 15°. Table 3 lists the measurement roll periods of different VCGs with discrepancies to the target period lower than 1%. Therefore, time histories of the measurement roll decay are used to determine roll damping.

_{44}### 2.4 Roll Damping Estimation

*I*is the mass moment of inertia. a

_{44}*and c*

_{44}*denote the added mass moment of the inertia and hydrostatic restoring coefficient, respectively. b*

_{44}*is the damping coefficient, which depends on the roll amplitude. The term*

_{44}*ϕ(t)*is the instantaneous roll motion, where

*t*represents time.

*ϕ*,

*ϕ̇*, and

*ϕ̈*represent the roll angle, angular velocity, and angular acceleration, respectively. Dividing the inertia term (

*I*) from Eq. (3), the equation of motion can be rewritten as where

_{44}+a_{44}*p(ϕ)*is the damping coefficient calculated using Eq. (5), and

*ω*is the natural frequency calculated using Eq. (2).

_{n}*p*,

*p*

_{1}, and

*p*

_{2}can be established as Eq. (8):

*ϕ*and

_{k}*ϕ*

_{k}_{+1}refer to two successive peaks in the roll decay motion;

*T*denotes the roll period. The roll damping coefficients

_{k}*p*,

*p*

_{1}, and

*p*

_{2}are obtained from the roll decay time records using various methods. Fig. 7 shows the time series history of the roll decay test for VCG2.

*δϕ*against

*ϕ*, where

_{a}*p*

_{1}and

*p*

_{2}can be obtained using a regression procedure. In this approach,

*k*refers to a positive peak, while the successive

*k+*1 peak refers to a negative one. The non-dimensionalized roll-damping coefficient is expressed as

### 3. Motion response

### 3.1 3D Panel Method

*ϕ*), diffraction potential (

_{I}*ϕ*), and radiation potential (

_{D}*ϕ*), as expressed in Eq. (21).

_{R}*x, y, z*) in the fluid domain and incident wave potential is estimated in Eq. (24). where

*n*is the generalized normal vector, and χ is the wave direction. Radiation for diffraction velocity potential located inside the fluid domain can be expressed in Eq. (25)

_{j}*P*is located in the source on surface S, and

*G(x, y, z)*is the Green function, which describes the flow at

*(x, y, z)*. The Green function satisfies the Laplace equation and the boundary condition everywhere except the body surface, and the boundary condition on the surface as

*r*is the distance between source point

_{PQ}*P*and field point

*Q*. The solution of three-dimensional velocity potential for ship motion with specified motion can be obtained using the panel method,