### 1. Introduction

*L*/

*B*) ratio, Froude number and beam-to-draft (

*L*/

*T*) ratio were examined. Moreover, regression analysis was performed to establish the relationship between heave and the ship length. Sayli et al. (2007) investigated the influence of the ship parameters on the heave and pitch. The functional relationships between hull form parameters and seakeeping characteristics of the fishing vessels were identified. Sayli et al. (2010) developed a nonlinear meta-model of heave, pitch and vertical acceleration of a fishing vessel. The seakeeping performance data of the fishing vessel in regular head waves were used. The influence of the hull form parameters on the heave and pitch was obtained. Tello et al. (2011) studied the seakeeping characteristics of a series of fishing vessels to establish the seakeeping criteria for irregular waves. Fishing vessels are considered to operate in sea state 5 and sea state 6 in various Froude numbers and wave directions. The roll and pitch are the most important motion responses of a fishing vessel. Hence, Tello identified the pitch and roll criteria for seakeeping performance in sea state 5 and sea state 6. Cakici and Aydin (2014) identified a relationship between ship parameters and seakeeping characteristics for the YTU Gulet series. The strip method was applied to estimate the ship motion and the statistical short-term was used to analyze the seakeeping performance. The RMS of the heave, pitch and vertical acceleration were investigated in sea state 3 at the head wave.

*L*/

*T*ratio was changed by 0.2 and 0.4 from the original value. The RMS of the roll and pitch are compared with the criteria reported by Tello et al. (2011).

*L/B*and

*B/T*ratios were changed by 10% to investigate the effects of dimension on the motion responses of a ship. The motion of the fishing trawler was estimated using a numerical method. The RMS of roll and pitch were compared with the fishing vessel criteria suggested by Tello et al. (2011). Finally, the sensitivity of the ship motions due to the influence of the ship dimensions was analyzed. These results can be used to predict the seakeeping performance of fishing vessels and ensure their safety in the design phase.

### 2. Investigation Wave Condition in the Bering Sea

### 3. Methodology

### 3.1 3D Panel Method

*ϕ*,

*ω*and

*d*denote the velocity potential, wave frequency and sea depth, respectively.

*g*,

*ζ*,

*s*and

*θ*are the gravity acceleration, wave amplitude, effective water depth and wave phase, respectively.

*ϕ*, incident potential

_{D}*ϕ*and radiation potential

_{I}*ϕ*. Each component of the velocity potential must satisfy the governing equation in Eq. (1) and the boundary conditions in Eqs. (2)–(5). Eq. (8) expresses the total velocity potential.

_{R}*X⃗*= (

*X*,

*Y*,

*Z*) is the location point on the body.

*ϕ*and

_{Rj}*x*are the potential of radiation waves caused by the ship motion and the ship motion in the

_{j}*j*direction, respectively.

*ω*is the encounter wave frequency.

_{e}*ρ*is the density of water.

*n*and

_{j}*S*are the unit normal vector and mean wetted surface of the ship hull, respectively. The total first-order hydrodynamic force can be written as Eq. (11).

*F*,

_{Ij}*F*and

_{Dj}*F*are the Froude-Krylov force, diffraction force and radiation force, respectively. These hydrodynamic forces can be calculated using Eqs. (11) – (14).

_{Rjk}### 3.2 RMS Motion

*z*denotes heave.

*ϕ*and

*θ*denote roll and pitch, respectively.

*k*is the wave number. The RMS motion value is determined as the square root of the variances. The variances and the RMS value of motion can be estimated using Eqs. (18)–(21), respectively. In order to estimate the wave spectrum

*S*(

*ω*), the ITTC spectrum was calculated using Eq. (22) (ITTC, 2014).

*H*

_{1/3}and

*T*

_{1}are the significant wave height and average wave period, respectively. In the case of the following and quartering waves, the wave frequency is considered to replace the encounter frequency to calculate the RMS motion to avoid the singularity in the encounter wave spectrum (Lewis, 1988). where,

### 3.3 Sensitivity Analysis

*S*is estimated using Eq. (24).

_{ijk}*H*

^{*}denotes the origin principal dimension of the ship.

*H*represents the deviated value of the principal dimensions of a ship.

*R*

^{*}denotes the value of the corresponding RMS motion value obtained from the original ship using

*H*

^{*}.

*R*represents the corresponding RMS motion values obtained from the modified ship using

*H*.

*S*is the sensitivity index for the

_{ijk}*i*RMS motion for the

^{th}*k*% change in the

*j*principal dimension. The influence of principal dimensions on the seakeeping performance, such as the RMS motion, is considered.

^{th}### 4. Target Ship and Test Condition

*L*/

*B*and

*B/T*ratios are changed by 10% to investigate the effects of dimensions on the ship’s motion response. In the case of a change in the

*L*/

*B*ratio,

*L*was changed by ± 10% and

*B*was kept constant and

*B*was changed by ± 10 % and

*T*was kept constant in the case of

*B/T*. In order to estimate the RMS motion, the numerical simulation of the motion RAO of the fishing trawler is performed at a ship speed of 6 knots. The simulation is conducted to examine the effect of regular waves on the fishing trawler in various wave directions and provide the input data of the RAO motion responses for calculating irregular wave motion responses. The range of wave directions is from 0 degrees to 180 degrees at 30 degrees intervals. The wave frequency ranges from 0.3 rad/s to 3.0 rad/s in 0.1 rad/s intervals. The wave directions are defined, as shown in Fig. 5. In the case of irregular waves, a sea state with an average significant wave height of 2.44 m and an average wave period of 6.386 s were selected to investigate the seakeeping performance of the trawler fishing vessel. Furthermore, according to the ITTC recommendation for numerical estimation of roll damping, roll damping was significantly affected by the viscous effect. In this study, the additional roll damping for the fishing vessel was divided into 5 components: wave making, hull lift, frictional, eddy making and skeg component (ITTC, 2011).

### 5. Result and Discussion

### 5.1 RAO Motion

*L*/

*B*and

*B/T*ratios are changed by 10% to investigate the effect of dimension on the ship’s motion response. The natural frequency of heave, roll and pitch are calculated due to the influence of a change in the main dimensions of the ship. The natural frequencies of heave, roll and pitch can be determined using Eqs. (25)–(27).

*m*,

*m*and

_{a}*A*denote the mass of the ship, added mass and water plane area, respectively.

_{W}*GM*and

_{T}*GM*are the transverse and longitudinal metacentric heights of the ship, respectively.

_{L}*I*and

_{xx}*I*are the roll and pitch moment of inertia of the ship, respectively.

_{yy}*J*and

_{xx}*J*are the added inertia moment in roll and pitch, respectively. Table 2 lists the natural frequencies due to the influence of the ship dimensions.

_{yy}*L*/

*B*,

*B/T*ratios and wave directions. The greatest heave RAO of the ship occurs when the wave direction approaches 180 degrees. As the wave frequency increases, the heave RAO tends to decrease. In short waves, however, the heave decreases to zero, especially when the wavelength is short compared to the ship length. The heave RAO tends to be 1 in the long waves because the heave follows the wave elevation. The heave RAO varies slightly depending on the ship dimensions. In general, the heave RAO is the largest when the

*B/T*ratio was reduced by 10% when the wave direction approaches 90 degrees. On the other hand, the heave RAO is the largest when the

*L*/

*B*ratio is reduced by 10% and the wave direction approaches 180 degrees. Hence, the heave RAO does not depend much on the ship dimension. The numerical results showed that the heave RAO is the largest when the encounter frequency is close to the heave natural frequency. The heave RAO depends on the water plane area and the ship displacement. The ship displacement and water plane area increased when the

*L*/

*B*and

*B/T*ratios were increased by 10%. The displacement and waterplane area of a ship are proportional to each other; therefore, the heave natural frequency does not change when the principal dimensions are changed. Table 2 lists the natural frequency of heave according to the

*L*/

*B*and

*B/T*ratio.

*L*/

*B*and

*B/T*ratios and wave directions. The numerical results showed that the roll RAO becomes dominant when the wave direction is close to 90 degrees and decreases as the wave direction approached 0 degrees and 180 degrees. The roll RAO with different

*B/T*ratios changes dramatically in various wave directions because of the effect of the breadth of the ship. The roll RAO decreases as the wave frequency increases. The peak roll RAO occurs at an encounter frequency close to the natural frequency of the roll. The numerical results of the roll RAO in the beam wave showed that the peak roll RAO according to the

*L*/

*B*ratio and the ship origin occurs at encounter frequency of 0.7 rad/s. This value is approximately the roll natural frequency according to the

*L*/

*B*ratio and the original ship. The peak roll RAO when the

*B/T*ratio is reduced by 10% occur at the encounter frequency of 0.9 rad/s, which is close to the roll natural frequency of

*B/T*ratio reduced by 10%. The peak roll RAO when the

*B/T*ratio is increased by 10% occurs at an encounter frequency of 0.5 rad/s, which is approximately the roll natural frequency of the

*B/T*ratio increased by 10%.

*L*/

*B*and

*B/T*ratios and wave directions. The largest pitch RAO of the ship occurs when the wave direction approaches 0 degrees due to the effect of the forward speed. The pitch RAO tended to decrease significantly as the wave frequency increases. Moreover, the pitch response is combined with the heave. Therefore, the motion responses of heave and pitch are the same phenomenon. The pitch RAO is the largest for the

*L*/

*B*ratio reduced by 10%. From the numerical results, the peak pitch RAO occurs at a wave frequency near the pitch natural frequency. The numerical results of the pitch RAO in the head wave showed that the peak pitch RAO in the original ship occurs at an encounter frequency of 1.5 rad/s. This is approximately the pitch natural pitch frequency of the original ship. The peak pitch RAO when the

*L*/

*B*ratio was increased by 10% occurs at an encounter frequency of 1.566 rad/s and a wave direction of 120 degrees. The peak pitch RAO when the

*L*/

*B*ratio is decreased by 10% occurs at an encounter frequency of 1.510 rad/s and a wave direction of 150 degrees. The peak pitch RAO when the

*B/T*ratio is increased by 10% occurs at an encounter frequency of 1.514 rad/s at a wave direction of 180 degrees. Generally, the peak heave, roll and pitch occur at an encounter frequency close to the natural frequency. Table 2 lists the natural frequency of the heave, roll and pitch due to a change in the ship dimensions.

*L*/

*B*and

*B/T*ratios because of resonance at the natural frequency. The varied ship dimensions affect the inertia moment, displacement and roll damping coefficient directly. The heave tends to increase when the displacement increases. A longer ship may have a longer waterline, which can affect the buoyancy distribution and the response to wave-induced heave. Longer ships tend to experience a reduced pitch. The longer waterline provides more resistance to pitch, and the ship is less sensitive to changes in trim caused by pitching. The increased inertia caused by the longer length contributes to a smoother response to wave-induced pitch motions. Furthermore, the roll is influenced the most by the beam of the ship. Hence, the roll period can determine the comfort of those operating on board.

### 5.2 RMS Motion

*L*/

*B*and

*B/T*. The heave is the strongest in the bow wave, head wave and beam wave. The RMS heave changes slightly as the principal dimensions of the ship changed. In the case of increased and decreased

*L*/

*B*ratio, the RMS heave varied by −4.841% and 4.312% in the head wave, respectively, compared to the original case. In the case of increased and decreased

*B/T*ratio, the RMS heave varied by −0.270% and 2.561% in the head wave, respectively, compared to the original case.

*L*/

*B*and

*B/T*ratios. The roll changes noticeably in various wave directions. The strongest roll is observed with a wave direction of 90 degrees. The breadth of a ship strongly influences the motion responses, particularly the roll. Based on the RMS roll, the effect of the ship length is not significant. In the case of increased and decreased

*L*/

*B*ratio, the RMS roll varied by 1.285% and −0.023% in the beam wave compared to the original case. In the case of increased and decreased

*B/T*ratio, the RMS roll varied by 22.514% and −41.347% in the beam wave, respectively, compared to the original case.

*L*/

*B*and

*B/T*ratios is shown in Fig. 11. The pitch changes strongly in various wave directions. The strongest pitch motion is observed at the head wave. The smallest pitch occurs at the beam wave. On the other hand, the pitch at a wave direction of 90 degrees does not become zero due to the effect of the forward speed. The pitch motion increases dramatically as the wave direction ranges nearly to quartering waves. The principal particulars of the ship strongly affect the pitch motion. Based on the RMS pitch, the effect of the ship’s length is larger than the ship’s breadth, particularly when the

*L*/

*B*ratio is decreased by 10%. In the case of increased and decreased

*L*/

*B*ratio, the RMS pitch varied by −5.910% and 6.201% in the head wave, respectively, compared to the original case. In the case of increased and decreased

*B/T*ratio, the RMS pitch varied by −4.724% and −1.150% in the head wave, respectively, compared to the original case.

*L*/

*B*and

*B/T*ratios. On the other hand, the RMS roll was slightly greater than the seakeeping criteria, except in the case of a change in

*B/T*ratio.

### 5.3 Sensitivity Analysis

*L*/

*B*and

*B/T*ratios is done by deviating each

*L*/

*B*and

*B/T*ratio by 10%. Figs. 12,–14 show the effects of the

*L*/

*B*and

*B/T*ratios on the RMS heave, roll and pitch, respectively. The sensitivity of the RMS heave is the highest in the head wave and the following wave. Moreover, the

*L*/

*B*ratio has the greatest influence on the RMS heave. Similar to the same trend of the RMS heave, the

*L*/

*B*ratio also has the greatest impact on the RMS pitch. The reason for such relationships is the coupling of the heave and pitch motions. The sensitivity of the RMS pitch increases noticeably with the following wave and stern waves. In contrast to the sensitivity of the heave and RMS pitch, the RMS roll sensitivity becomes zero in the following wave and the heave wave because of the direction of the incident wave. The RMS roll sensitivity is strongly affected by the

*B/T*, ratio, especially at a wave direction of 120 degrees. The

*L*/

*B*ratio has a negligible influence on the RMS roll sensitivity.

*L*/

*B*and

*B/T*ratios are important geometric parameters of a ship that significantly influence its RMS motion. The

*L*/

*B*and

*B/T*ratios influence the ship motions through their effects on stability, buoyancy distribution, displacement and the response of the ship to wave-induced forces. The sensitivity index of the RMS heave changes slightly in various wave directions. In other words, the sensitivity of RMS heave is negligible under the influence of the

*L*/

*B*and

*B/T*ratio. Nevertheless, the sensitivity index of the RMS roll varies significantly in various wave directions because of the influence of the B/T ratio and the roll is influenced the most by the breadth of the ship. In the case of pitch, the sensitivity index of the RMS pitch changes drastically due to the influence of the

*L*/

*B*ratio.

### 6. Conclusion

First, the sea condition in the Bering Sea was investigated based on the available data. The statistics of significant wave height and average wave period were analyzed. A sea state was then chosen using the average value of the significant wave height and wave period.

Second, the

*L*/*B*and*B/T*ratios were changed by 10% to determine the effect of the dimension on the motion responses of a ship. As shown in the RAO and RMS motion results, the*L*/*B*ratio affected the heave and pitch. In contrast, the roll was influenced substantially by the*B/T*ratio. Moreover, the influence of the*L*/*B*and*B/T*ratios on the RMS motion was also investigated. The maximum RMS roll and pitch were compared with the seakeeping criteria suggested by Tello et al. (2011).Third, the sensitivity index of the

*L*/*B*and*B/T*ratios on the RMS heave, roll and pitch was analyzed. The sensitivity of the RMS heave was highest in the head wave and the following wave. The heave and the pitch were coupled. Therefore, the sensitivity trends of the heave and the pitch are similar and the*L*/*B*ratio has the largest impact on the RMS heave and pitch. In contrast to the sensitivity of the RMS heave and RMS, the sensitivity of the RMS roll becomes zero in the following wave and heave wave because of the direction of the incident wave. The sensitivity of the RMS roll is strongly affected by the*B/T*ratio, especially at a wave direction of 120 degrees. The*L*/*B*ratio has a negligible influence on the sensitivity of the RMS roll.

*B/T*ratio has a good seakeeping performance based on a comparison of the roll and pitch with the seakeeping criteria. On the other hand, the increased beam can cause ship stability problems due to a change in the righting arm (GZ) curve because the inflection point of the GZ curve occurs at a small inclination angle. The reduced beam shortens the natural period of a ship. Its acceleration and comfort are affected. In addition, the

*B/T*ratio influences the wave-making resistance. Generally, lower

*B/T*ratios are associated with lower wave resistance, leading to better propulsion efficiency. Nevertheless, extremely low

*B/T*ratios may increase the susceptibility to parametric rolling. On the other hand, the change in

*L*/

*B*has less seakeeping performance, particularly in roll based on a comparison of roll seakeeping criteria. A higher

*L*/

*B*ratio generally contributes to better stability in waves. Longer ships tend to have smoother motions and are less prone to rolling, which is particularly important for passenger comfort and safety. Longer ships experience less wave resistance, leading to improved fuel efficiency, but extremely long and slender ships may face challenges related to structural strength. Although these ratios provide insights into the seakeeping performance of a ship, they are just one set of parameters that naval architects should consider. The actual impact of these ratios depends on various factors, including the specific design, the purpose of the ship, operational conditions, and intended trade routes.