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ISSN : 1229-3431(Print)
ISSN : 2287-3341(Online)

Journal of the Korean Society of Marine Environment and Safety Vol.29 No.4 pp.380-388
DOI : https://doi.org/10.7837/kosomes.2023.29.4.380

Experimental Study on the Analysis and Estimation of Metacentric Height in Response to Roll Period and Moment of Inertia Variations in Ships

LeeChan Choi^*, JungHwi Kim^**, DongHyup Youn^***†

^*Researcher, Design Engineering Support Center, Research Institute of Medium & Small Shipbuilding, Busan 46757, Korea
^**Senior Engineer, Design Engineering Support Center, Research Institute of Medium & Small Shipbuilding, Busan 46757, Korea
^***Senior Researcher, Design Engineering Support Center, Research Institute of Medium & Small Shipbuilding, Busan 46757, Korea

* First Author : lcchoi@rims.re.kr, 051-974-5534

^† Corresponding Author : dhyoun@rims.re.kr, 051-974-5569

Received May 12, 2023 Review June 12, 2023 Accepted June 27, 2023

Abstract

This study estimates the metacentric height (GM) of a model ship by varying the transverse weight distribution, considering the effects of the roll period and moment of inertia, and compares it with the GM values measured by the inclining test. In the process, the relationship between the values is analyzed. Three types of ships―a 7-ton fishing vessel, 20-ton fishing vessel, and KRISO Very Large Crude-oil Carrier (KVLCC)―were used for the experiment and comparison. The roll period and moment of inertia were measured using the free roll decay and swing frame tests, and the GM was measured using inclining test. The estimated GM from the roll period and moment of inertia showed the same trend as the GM measured using the inclining test in the change of the weight distribution. However, the GM values measured using the inclining test were lower. Therefore, additional correction factors or parameters other than the roll period and moment of inertia are necessary for estimating GM. In the future, the relationship between the weight center and the estimated GM will be analyzed to derive the correction factors.

Key Words : Metacentric height , Roll period , Moment of inertia , Swing frame test , Inclining test

선박의 횡요주기와 관성모멘트 변화에 따른 GM 추정 및 분석을 위한 실험 연구

최이찬^*, 김정휘^**, 윤동협^***†

^*중소조선연구원 설계엔지니어링지원센터 연구원
^**중소조선연구원 설계엔지니어링지원센터 책임기술원
^***중소조선연구원 설계엔지니어링지원센터 선임연구원

초록

본 연구는 모형선박의 횡방향 무게분포를 변화시키면서 횡요주기 및 관성모멘트에 의한 GM을 추정하고 경사시험으로 계측되 는 GM 값과 비교한다. 그 과정에서 각 수치를 분석하여 관계성을 파악한다. 실험 및 비교에 사용된 선박의 종류는 3종류로 7톤급, 20톤급 어선과 KVLCC의 모형선을 사용하였다. 실험은 스윙프레임 테스트와 자유 횡요 감쇠테스트를 수행하여 관성모멘트와 횡요주기를 계측하 고, 경사시험으로 GM을 계측하였다. 관성모멘트와 횡요주기로 추정된 GM과 경사시험으로 구해진 GM은 선박의 무게 분포 변화에도 동 일한 경향을 보였지만, 경사시험에서 구해진 GM이 더 작은 값을 보였다. 따라서 GM 추정 시에 관성모멘트나 횡요주기 외에도 추가적인 보정 계수나 파라미터가 필요하다고 판단된다. 추후 무게중심을 변화시켜 추정 GM과의 관계를 분석하여 보정 계수를 산출할 계획이다.

키워드 : GM , 횡요 주기 , 관성모멘트 , 스윙프레임 테스트 , 경사 시험

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1. Introduction

Many ship accidents occurred due to a lack of stability, which resulted in a loss of life, environmental damage, and financial losses. Therefore, research on ship stability assessment and GM estimation methods was crucial for improving maritime safety and preventing ship accidents. Ship stability was heavily influenced by the shape and weight of the vessel, with stability mainly determined by the characteristics of the lightship to weight. Therefore, choosing the weight and center of gravity at the lightship stage was essential. During design, ship stability was calculated using the buoyancy correction method (Nowacki and Ferreiro, 2003). In actual ships, the GM was calculated through a tilt test conducted while the vessel was anchored. The tilt test was applied to new ships that were 24m or longer and to modified vessels that could affect stability. It was made mandatory by the International Maritime Organization (IMO) (International Maritime Organization, 2002). The inclining test was only valid in a static state. It was based on several assumptions, such as the assumption that the volume of the vessel remained constant regardless of the angle of inclination, that the metacentric height remained stable, and that the contour of the hull became a vertical wall (Taylan, 2020).

Measuring GM becomes challenging when the ship is underway, and the cargo has changed. Therefore, an equation using the roll period and the moment of inertia of the ship was sometimes used to estimate GM (Choi, 2015). However, the moment of inertia of the ship played an important role in determining the roll motion, and ships with a high moment of inertia acted as dependent variables because their roll motion was smaller than that of ships with a low moment inertia. Thus, the relationship between the roll motion and the moment of inertia of the ship has been extensively studied over the past few decades (Nakamura et al, 1975;Haddara and Bennett, 1989). Approximate values using the product of 0.4 times the width of the hull as the radius of gyration had been suggested, as well as simplified formulas that included the width and length of the hull and the current flow (IMO Resolution A.749, 1993).

In that study, the three model ships were divided into comparison groups. The inertia moments were determined through swing frame tests and the roll periods were measured via free roll decay tests for each model ship. The estimated GM values were calculated using the obtained inertia moments and roll periods and compared with the GM values calculated through inclining tests.

2. Method

2.1 Swing frame test

The swing frame test was a method used to determine the inertial moments and center of gravity that affected a vessel's stability, maneuverability, and actual operation (Gabl et al., 2021). It was also important for predicting the behavior of a vessel in external forces such as waves at sea. The swing frame used in the test was constructed using lightweight aluminium to minimize the potential variables introduced by its weight. The object to be measured for inertial moments was placed on the frame, as shown in Fig. 1. The frame rotated left and right around a rotation axis, and a gyro sensor was attached to the fixed part of the frame to measure the rotation angle. The frame was adjusted to maintain a horizontal position, and the wheels of the swing frame were fixed before the test to minimize shaking. During the experiment, the object of interest should have been positioned parallel to the centerline of the frame. The position of the object and the sensor should have been checked to ensure that the angle was 0º.

The measurement of the center of gravity and moment of inertia using a swing frame involved first measuring the rotation period of the frame itself. The frame was manually pushed to rotate, and the period was measured. Then, a weight was placed outside the frame, and the angle at which it stopped was determined. After completing the frame experiment, the object was placed on the frame, and a pre-experiment was performed to ensure it was level. The experiment was repeated three times to reduce human error. The moment of inertia could be calculated using the measurement data, and the formula was as follows (Delefortrie et al., 2017).

I_{f} = \frac{T_{f}^{2}}{4 π^{2}} \times W_{f} \times g \times l_{f}

(1)

I_{t} = I_{s} + I_{f}

(2)

I_{s} = I_{t} - I_{f} = g \times \frac{T_{t}^{2}}{4 π^{2}} \times (W_{s} \times l_{s} + W_{f} \times l_{f}) - g \times \frac{T_{f}^{2}}{4 π^{2}} \times W_{f} \times l_{f}

(3)

I_{G} = I_{s} - l_{s}^{2} \times W_{s}

(4)

Here, g is the gravitational acceleration [m/s²], W_f is the mass of the swing frame [kg ], W_s is the mass of the model ship [kg], l_f is the distance between the reference point and the swing frame [m], l_s is the distance between the center of gravity of the model ship and the reference point [m], T_f is the natural period of the swing frame [s ], T_t is the total natural period of the swing frame and model ship [s ], I_f is the moment of inertia of the swing frame with respect to the reference point [kg·m²], I_t is the total moment of inertia of the swing frame and model ship with respect to the reference point [kg·m²], I_s is the moment of inertia of the model ship with respect to the reference point [kg·m²], and I_G is the moment of inertia of the model ship with respect to its center of gravity [kg·m²].

The relationship between the moment of inertia and the radius of gyration is as follows.

k = \sqrt{\frac{I_{G}}{W_{s}}}

(5)

Here, k represents the radius of gyration

2.2 Free roll decay test

The ship free roll decay test was performed to measure a ship's roll period and damping coefficient (Zeraatgar et al., 2010;Igbadumhe et al., 2020). In this experiment, an initial angle was given to the ship, inducing it to roll, and the angle was measured. The roll motion was measured using a gyro sensor, considered more suitable for the tank test environment than a potentiometer, as the latter is susceptible to external environmental influences.

The experiment involved giving the ship an initial angle and allowing it to roll naturally in the water until the motion came to a stop. The experiment began in calm waters and lasted for a certain period of time, and the measured data were analyzed to derive the roll period.

The equations for deriving the roll period were as follows :

T_{m} = \frac{1}{N} \sum_{0}^{N} T_{N}

(6)

Here, N represents the number of measured periods, T_N represents the Nth period, and T_m represents the roll period of the ship.

2.3 General formula for prediction GM

In ships, GM referred to the metacentric height, a measure of stability, particularly resistance to capsizing. The metacentric height was the distance between the ship's center of gravity (G) and its metacenter (M), which was the point where the displaced water's buoyancy force (B) intersected the ship's centerline. It is shown in Fig. 2. If GM was positive, the ship was more likely to return to its original position after being tilted by waves or other external forces. However, if GM was too large, the ship could feel stiff and uncomfortable for passengers and crew, and a negative GM meant the ship was unstable and prone to capsizing.

When a ship incline, the angle of heel (ϕ) is calculated as follows (Choi, 2015):

J \ddot{ϕ} + b \dot{ϕ} + Δ \cdot g \cdot G Z (ϕ) = 0

(7)

Here, J is the moment of inertia, b is the damping coefficient in the form of a linear coefficient, and Δ is the displacement. By expressing the moment of inertia and radius of gyration in terms of each other, we can obtain the following equation:

k = \sqrt{\frac{g \cdot G M}{4 π^{2} + {(ln λ)}^{2}}} \times T_{m}

(8)

Here, λ represents the rolling decay ratio. As shown in Fig. 3, there is no significant difference when the rolling decay ratio ranges from 0.5 to 1.0. In this paper, the object is assumed to undergo undamped natural vibrations, and accordingly, λ is set to 1.

2.4 Inclining test

The ship inclining test was a procedure where a ship was tilted at various angles, and the rotation angle was observed. Fig. 4 illustrates the procedure of the inclining test. It was used to evaluate a ship's stability and seaworthiness and experimentally determine the vertical center of gravity and roll metacentric height of the ship. The test involved placing known weights at a certain distance from the ship's longitudinal centerline and measuring the inclination angle. A gyro sensor that could measure the roll angle of the heel was attached to the ship for testing. To ensure accurate measurement, it was preferred to conduct the test in an environment where external forces such as waves could affect the results were not present. The following equation calculates GM using the measured angle (Karolius and Vassalos, 2018).

G M = \frac{w \times d}{W \times tan ϕ}

(9)

Here, w is the weight of the added weight, d is the distance from the centerline to the added weight, and W is the sum of the weight of the added weight and the ship's weight.

2.5 Principal dimension of target ships

This study compares the GM estimates obtained from the swing frame and free roll decay test with the GM values measured in the inclining test for three different types of vessels. The experiments were conducted using scale model ships of two small fishing vessels with different hull shapes and tonnages, as well as a model ship of KVLCC, which is commonly utilized as a reference in various research studies. Although the primary focus of this study is on small fishing vessels, the inclusion of the KVLCC model aims to provide assistance for conducting similar research. Therefore, experiments were carried out on three ships, as shown in Fig. 5-7. The specifications of the model ship are presented in Table 1. For convenience, in this paper, we will refer to the 7 ton (G/T) fishing vessel as A, the 20 ton (G/T) fishing vessel as B, and the KVLCC as C.

Each ship was tested in three different cases by varying the weight distribution. Since the width and displacement of the ships vary by ship type, the three cases were set up by placing the weights at different distances from the centerline to the side of the ship while maintaining the weight of the weights at approximately 10% of the ship's weight. The distance of the weights was adjusted within the width so that three cases were possible, taking into account the size of the weight. The detailed numerical values are presented in Table 2.

2.6 Experiment method

Experiments were performed for three types of vessels using the free roll decay test, swing frame test, and inclining test. The experiments were conducted under identical conditions for each test and vessel type. In particular, for the free roll decay test and inclining test, which had to be conducted in a tank, a time interval was given between each test to increase the reliability of the experimental results, as the data could be affected by reflected waves or external forces. To prevent the swing frame from moving, it was fixed to the floor, and the pendulum motion was performed at a constant height to ensure a consistent external force from the operator. The position of the weights was varied for each vessel type, as shown in Table 2, to change the weight distribution of the ship. This was done to examine the relationship between the moment of inertia and GM according to weight distribution.

The swing frame test was conducted to measure the moment of inertia at the ship's center of gravity by measuring the period of the oscillatory motion of the frame on which the ship was mounted. The period refers to the time taken for the swing frame to complete one full oscillation cycle, and the difference in the period value was very small. Therefore, three experiments were conducted for each case, and the average value was used. The moment of inertia was calculated using equations (1) to (4) with the measured period and parameters.

The free roll decay test was conducted to measure the free roll motion of the vessel in still water, and data was measured using a gyro sensor over time. The initial angle was controlled to be as similar as possible for all experiments. As the experiment aimed to capture only the roll motion, Roll data was measured exclusively. To minimize human errors, an inclining test was performed for each case, and the averaged results of each case were used for final comparison.

For the inclining test, the additional weight was set to approximately 10% of the total weight, and the weight was selected, considering the narrow width of the model's hull. While the International Association of Classification Societies (IACS) recommends 4% of the lightweight displacement for the inclining test, higher ratios are permitted for small vessels. Therefore, considering that the model is tested in a full loaded displacement condition and sufficient transverse slope must be secured, we selected 10% of the total weight as the additional weight.

3. Result

3.1 GM calculation from swing frame and free roll decay test

The time series roll results of the free roll decay test are illustrated in Fig. 8 to 10. Regardless of the ship type, the roll period increases as the weight distribution shifts to the side. In addition, despite changes in transverse weight distribution, the roll angle remains relatively stable and exhibits a consistent pattern. Tables 3 to 5 show the moment of inertia and the radius of gyration according to the ship types. It can be observed that the radius of gyration increases as the weight distribution moves to the side.

The GM value was estimated using the roll period and moment of inertia obtained from the free decay and swing frame test. The results are presented in Table 6-8.

Tests were conducted to increase the moment of inertia for each ship type. Still, it was observed that as the moment of inertia increased, the free roll decay test period also increased, resulting in similar GM values.

3.2 GM calculation from inclining test

The results of the inclining test are presented in Table 9-11. The experimental results show that the transverse inclination angle is within the recommended range of 2-4° by IACS. The experiment was conducted by varying the weight distribution of the ship for each case, but the GM values were found to be the same since the weights were moved horizontally rather than vertically.

3.3 Compare predicted GM and inclining test GM

GM estimated from the moment of inertia and roll period, and GM measured from the inclining test were graphed in Fig. 11-13. While there was no significant change in GM due to weight distribution in all ship types, GM estimated from the roll period and moment of inertia showed differences from the GM value obtained from the inclining test. The observed discrepancies in the results are believed to be due to uncontrollable variables or unforeseen factors in the experiments. Therefore, in order to address this issue, it is necessary to enhance variable control and revise the estimation equations from multiple perspectives.

4. Conclusion

This will be an important point for vessels such as fishing boats where the weight distribution changes quickly or stability evaluation is important. In this study, the estimated GM values obtained through the moment of inertia and roll period were compared with the GM values measured through the inclining test to compare the static and dynamic GM values. The following conclusions were drawn from the experiments.

The impact of weight distribution on the moment of inertia, roll period, and GM of ships was examined through various tests. It has been observed that as the transverse displacement of weight distribution increases, both the roll period and moment of inertia also increase. This phenomenon is applicable to all types of vessels.

The GM values obtained through the inclining test remained unchanged even with the horizontal movement of weights, and the transverse inclination angle was within the recommended range of 2-4° by IACS.

The GM values estimated from the moment of inertia and roll period were compared with the GM values obtained from the inclining test, and the results showed differences between the two values, suggesting that additional parameters or new estimation equations are necessary to estimate the dynamic GM accurately. However, as the dynamic GM tends to be higher than that from the inclining test, a conservative approach is required.

For small ships, in particular, the variation in GM due to weight changes is significant, making it essential to estimate the GM to ensure stability. However, it is not feasible regarding cost and time to perform tests such as the inclining test while the ship is docked. Therefore, if the GM value of a ship can be estimated only from the rolling period, the stability can be checked in real-time. As observed in the previous results section, in order to reduce the discrepancies in values, it appears necessary to consider various modifications and variables. In the future, we plan to analyze the relationship between the estimated GM and the change in the center of gravity and derive correction factors.

Acknowledgment

This research was supported by a grant (20015029) of Regional Customized Disaster-Safety R&D Program, funded by Ministry of Interior and Safety (MOIS, Korea).

Figure

Fig. 1.

The appearance of the swing frame test.

Fig. 2.

The schematic diagram for GM.

Fig. 3.

The basis for assuming λ of 1.

Fig. 4.

The appearance of the inclining test.

Fig. 5.

Body plan of 7 ton (G/T) fishing vessel.

Fig. 6.

Body plan of 20 ton (G/T) fishing vessel.

Fig. 7.

Body plan of KVLCC.

Fig. 8.

Experimental data obtained from the free roll decay test of A.

Fig. 9.

Experimental data obtained from the free roll decay test of B.

Fig. 10.

Experimental data obtained from the free roll decay test of C.

Fig. 11.

Estimated GM and inclining test GM of A.

Fig. 12.

Estimated GM and inclining test GM of B.

Fig. 13.

Estimated GM and inclining test GM of C.

Table

Table 1.

Principal particulars of model ships.

Table 2.

Case description.

Table 3.

Experimental data obtained from the swing frame test of A

Table 4.

Experimental data obtained from the swing frame test of B

Table 5.

Experimental data obtained from the swing frame test of C

Table 6.

GM of A obtained by free decay & swing frame test

Table 7.

GM of B obtained by free decay & swing frame test

Table 8.

GM of C obtained by free decay & swing frame test

Table 9.

Inclining test result of A

Table 10.

Inclining test result of B

Table 11.

Inclining test result of C

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