1. Introduction
The accurate evaluation of roll damping ratios is crucial for predicting the roll motion of ships and ship-shaped marine structures, such as Floating Production Storage and Offloading (FPSO) units and Floating Liquefied Natural Gas (FLNG) facilities. Predicting the roll natural period (Troll) and roll motion while accounting for damping remains a significant challenge in shipbuilding and marine engineering, primarily due to the nonlinear effects caused by fluid viscosity. Ikeda et al. (1978) and Chakrabarti (2001) categorized the components contributing to roll damping, including skin friction from fluid-structure interaction, wave-making by floating structure motion, eddy-making due to relative motion between the structure and surrounding fluid, hull lift from forward motion, and bilge keel. They proposed empirical formulas to quantify the contributions from these factors. Despite these efforts, theoretical and numerical methods face limitations in evaluating roll damping ratios, particularly when addressing nonlinear effects such as eddy-making and bilge keel impacts. Consequently, free roll decay tests remain the primary experimental method for assessing roll damping ratios in floating structure models (Kim et al., 2019a).
Choi et al. (2005) utilized the quadratic damping force derived from free roll decay test results to estimate the roll motion performance of LNG FPSOs in irregular waves using the higher-order boundary element method. Jung et al. (2006) employed particle image velocimetry (PIV) to study the flow field around a rectangular floating structure under regular wave conditions, analyzing eddy making and flow pattern. They calculated roll damping ratios using the relative decrement method.
Gu et al. (2015) simulated the free roll damping of two-dimensional FPSO using Star CCM+ and determined the roll damping ratio using the relative decrement method. They researched changes in the roll natural period and damping coefficient according to the position of the bilge keel. The damping coefficient increased by more than four times when the keel was fixed at the center of the bilge compared to its lower part, and the roll natural period increased by approximately 10%. Kim et al. (2019b) conducted a motion test of a floating structure in the regular waves of a two-dimensional wave tank to identify the roll and heave motion characteristics of rectangular floating structures using numerical and experimental methods and conducted numerical analysis using Reynolds averaged Navier Stokes (RANS)-based Star CCM+. They conducted free roll decay tests and evaluated the roll damping coefficient through the relative decrement method. They also verified the numerical method by comparing the roll damping coefficients from the model test and numerical analysis results. Jiang et al. (2020) analyzed eddy generation and paths according to the position and height of the bilge keel for a Panamax class ship with a midship section coefficient (CM) of 0.97 using the free surface random vortex method (FSRVM), determining the roll damping ratio using the logarithmic decrement method. Zeraatgar et al. (2010) determined the roll damping ratio through the logarithmic decrement method based on the results of the free roll decay test of a bulk carrier model conducted in the wave tank designed for the test. They used linear damping coefficients when the initial roll angle (ϕa) was small (7° or less), and recommended using nonlinear damping coefficients when this value was larger than 7°. Kim et al. (2015) evaluated the roll damping ratio of a box-type floating structure (CM: 0.99) according to the presence/absence and installation length of the bilge keel, as well as the initial roll angle of the model using the linear-quadratic decrement method and the logarithmic decrement method, and compared the results. The results of the two methods were similar when the initial roll angle was small, but the difference between them increased as the initial roll angle increased. Park et al. (2019) calculated the roll damping ratio according to the midship section geometry and the presence/absence of the bilge keel using the logarithmic decrement method and compared the results. They evaluated the roll damping ratio by conducting free roll decay tests and excitation tests. The roll damping ratio was evaluated to be somewhat higher in the excitation test results compared to the free roll decay test results under the same initial roll angle condition.
In several studies on the roll damping of floating structures, the relative decrement method, which utilizes the extinction curve; the linear-quadratic decrement method, which separates linear and nonlinear terms for calculation; and the logarithmic decrement method, which employs the natural logarithm of successive peak values, have been predominantly used to determine the roll damping ratio through experimental and numerical free roll decay tests. Jung et al. (2006) assessed the roll damping ratio of a floating rectangular structure (CM: 1.00) in a two-dimensional wave tank and determined it to be 10.6% at an initial roll angle of 15° using the relative decrement method based on free roll decay test results. Jiang et al. (2020) verified through a numerical roll decay test of a Panamax-class ship that the roll damping ratio, calculated using the logarithmic decrement method, was 3.3% in the absence of a bilge keel and reached a maximum of 9.0% in the presence of a bilge keel at an initial roll angle of 20°. Zeraatgar et al. (2010) evaluated the roll damping ratio of a bulk carrier as 12.4% at an initial roll angle of 36.28° using the logarithmic decrement method. For smaller initial roll angles of 7° or less, they determined the roll damping ratio to be approximately 7%. Kim et al. (2015) investigated the roll damping ratio of a box-type floating structure (CM: 0.99) and found it to range from 0.95% to 1.45% in the absence of a bilge keel, depending on the initial roll angle (5° to 15°). In the presence of a bilge keel, the maximum roll damping ratio reached 15.91%, as determined using both the linear-quadratic decrement method and the logarithmic decrement method. The authors proposed that these two methods should be used concurrently to identify discrepancies in roll damping ratio estimates depending on the evaluation technique. Park et al. (2019) examined the roll damping ratio of a floating structure (CM: 0.99) without a bilge keel and reported values ranging from 1.5% to 1.8% for initial roll angles between 3.6° and 12.7°, based on the logarithmic decrement method. For a structure with a midship section coefficient (CM) of 1.00, they observed roll damping ratios of 3% to 6% at initial roll angles of 2.8° to 4.7° (Table 1).
Bassler et al. (2010) and Sun and Shao (2019) emphasized that the aforementioned damping ratio calculation methods are suitable for evaluating the roll damping ratio when the roll motion is small. Bassler et al. (2010) noted that the existing theoretical model may overestimate energy dissipation through damping at large roll angles, as it was originally developed for cases where the initial roll angle is small. To address this limitation, they proposed the piecewise method to evaluate the roll damping coefficient during large roll motions. This approach involves dividing the roll motion into two segments based on a critical angle and subsequently verifying the piecewise method by calculating the roll damping coefficients for each segment, denoted as δ1 and δ2, using the linear-quadratic decrement method. These results were then compared with experimental data for validation. Sun and Shao (2019) observed that while the relative decrement method is appropriate for evaluating the roll damping coefficient at small initial roll angles, nonlinear roll damping coefficients should be calculated using the linear-quadratic and linear-cubic decrement methods as the roll angle increases. This recommendation was based on the increased significance of nonlinear effects on damping coefficients at larger roll angles.
This study aimed to quantitatively compare and analyze the roll damping ratio using various evaluation methods, focusing on experimental approaches to predict the roll motion of marine structures with greater accuracy. Free roll decay tests were performed on a floating structure with a midship section coefficient of 0.99 in a two-dimensional wave tank. The roll damping ratio was evaluated and compared using the relative decrement method, the linear-quadratic decrement method, and the logarithmic decrement method. The logarithmic decrement method was further categorized into a linear regression-based approach and an arithmetic mean-based approach (referred to as the “average” of the logarithmic decrement). The results and characteristics of these two approaches were systematically compared. Additionally, the roll damping ratio was evaluated with respect to variations in the initial roll angle and the number of rolling periods observed during the free roll decay tests. Two techniques for determining the roll natural period, namely the fast Fourier transform (FFT) method and the peak-to-peak method, were also compared. The analysis of the free roll decay test results incorporated the recommendations of the International Towing Tank Conference (ITTC) (2021), which advocates the use of data from eight or more rolling periods, and the guidelines of Piehl (2017), which recommend time-series data with roll angles of ±1.0° or higher. This comprehensive evaluation aimed to ensure the robustness and accuracy of the roll damping ratio predictions.
2. Experimental Setup and Conditions
The free roll decay model test was conducted in a two-dimensional wave tank at Pusan National University using a model with a midship section coefficient of 0.99 under calm water conditions. The tank measured 30 m in length, 0.6 m in width, and 1 m in depth, with wave absorbers having a slope ratio of 1:3 installed at both ends to minimize the influence of reflected waves. Motion capture was achieved using four OptiTrack Prime 41 cameras configured in a non-contact measurement system, with five optical markers attached to the model for precise tracking . The acrylic model, positioned at the center of the tank (15 m from either end), measured 0.595 m in length, 0.3 m in width, and 0.2 m in height (Fig. 1). The key specifications of the motion measurement equipment and the model are provided in Tables 2 and 3, respectively.
During the free roll decay test, the model was allowed to float freely with its bow and stern aligned along the width of the wave tank. One side of the model was inclined to generate a specific heeling angle, after which it was released from a stationary position. The water depth-to-draft ratio (h/T) was maintained at 5.00, as recommended by Jung et al. (2022), and the initial roll angle (ϕa) was set to 5°, 10°, 15°, and 20°. Each test was conducted at least three times to ensure reproducibility. Calibration of the motion capture system confirmed a maximum measurement error of 0.05°. Data were collected at a sampling rate of 100 Hz over a duration of approximately 60 s. The experimental conditions are summarized in Table 4.
3. Roll Natural Period and Damping Ratio Estimation Methods
3.1 Roll Natural Period
The roll natural period (Troll ) was determined and compared using the Fast Fourier Transform (FFT) method and the peak-to-peak method. For this analysis, measurement data comprising at least the minimum number of rolling periods recommended by ITTC (2021), specifically eight, were utilized. For the FFT method, the roll natural period was computed while varying the frequency resolution (dω) through the zero-padding technique (Lyons, 1997). This technique enhances frequency resolution by maintaining the spectral shape and increasing the number of data by appending zeros to the existing data. Table 5 provides the frequency resolutions and the number of data used in the roll natural period analysis. The peak-to-peak method calculated the roll natural period by averaging the periods (Troll(i)) between successive double peaks, as illustrated in Fig. 2. The period was directly determined by measuring the amplitude peaks in the time domain.
3.2 Roll Damping Ratio Calculation Methods
Roll motions smaller than ±1.0° are generally insignificant for evaluating the roll damping ratio; however, they can introduce inaccuracies in the calculations. Therefore, only peaks exceeding this threshold were considered during the roll damping ratio calculations, with ±1.0° selected as the threshold value, as suggested by Piehl (2017).
where ϕ̈, ϕ̇, and ϕ are the roll acceleration, velocity, and displacement, respectively. a is the virtual mass moment of inertia, which is the sum of the rolling moment of inertia and the added mass moment of inertia, b is the damping moment coefficient, and c is the restoring moment. The virtual mass moment of inertia was determined as the sum of the added mass moment of inertia that used HydroStar v8.2.1(Bureau Veritas, hereafter “BV”) and the moment of inertia of the model. Various methods can be used to calculate the damping coefficient b from the free roll decay test. In this study, the roll damping ratio calculation methods were divided into three categories—relative decrement method, linear-quadratic decrement method, and logarithmic decrement method—and the logarithmic decrement method was again divided into two methods—linear regression method and the average of logarithmic decrement. Thus, the roll damping ratio (ζEX, ζEQ, ζL, and ζMean) was calculated through these four methods. The specific calculation methodologies for each approach are detailed below.
3.2.1 Relative decrement method
The relative decrement method, which utilizes the extinction curve, is a technique for analyzing the relative decrement of each peak (Bhattacharyya, 1978). This method illustrates the relationship between the average roll amplitude and the decrement during each period as the extinction curve, and the roll damping ratio is calculated from the slope of this curve. The average amplitude of each peak for a given section (ϕm ) and the decrement of the roll amplitude per period (∆ϕi ) are determined using Eqs. (2) and (3).
ϕi is the amplitude of the i-th roll motion. The roll decrement according to the average roll amplitude is calculated through Eqs. (4) and (5) (Fig. 3), and the extinction curve is obtained using the linear regression technique. The slope of the curve is defined as K1 (Eq. (4)). Based on this, the roll damping coefficient is obtained (Eq. (5)).
3.2.2 Linear-quadratic decrement method
The linear-quadratic decrement method (Kim et al., 2015) describes the relationship between the value obtained by dividing the decrement of the roll amplitude per period by the average amplitude for each section and the average roll amplitude as a linear equation (Eq. (8)). From this relationship, the non-dimensional damping ratio is determined through the slope (q) and y-intercept (p) (Fig. 4).
The roll damping coefficient (b) in Eq. (1) is expressed as the sum of the linear damping coefficient (b1), which is proportional to the velocity of the structural motion, and the quadratic damping coefficient (b2), which is proportional to the square of the velocity (Eq. (9)). The linear and quadratic damping coefficients can be determined from the slope and y-intercept of the trend line, as given by Eqs. (10) and (11), respectively.
The kinetic energy lost through the linear damping caused by the surface friction that results from the interaction between the floating structure and the surrounding fluid and the quadratic damping caused by the nonlinear effect of vortex shedding is expressed as the equivalent linear damping moment coefficient (be) (Eq. (12)). This parameter is divided by the critical damping coefficient to calculate the non-dimensional roll damping ratio ζEQ (Eq. (13)).
3.2.3 Logarithmic decrement method
The logarithmic decrement method (Kim and Park, 2015; Inman, 2013) utilizes the logarithmic decrement (δi ), which is the natural logarithm of the ratio between the peaks (ϕi ) measured through the free roll decay test (Eq. (14)). The roll damping ratio, ζL is calculated using logarithmic decrement through a linear regression method, and the roll damping ratio, ζMean is determined by averaging the logarithmic decrement.
The roll damping ratio ζL, which utilizes the linear regression technique, is calculated through the slope (s) and y-intercept (r) of the trend line after representing the relationship between the roll damping ratio ζi calculated for each section of roll motion (Eqs. (16) and (17)) and the average amplitude (Fig. 5) and obtaining the trend line (Eq. (18)). As roll damping ratio (ζi ) calculation methods by section are different when positive and negative peaks are used and when double peaks are used, they are expressed in Eqs. (16) and (17), respectively.
ζMean calculated using the average of the logarithmic decrement, is determined by applying the average value of the logarithmic decrement for each section of roll motion (Eq. (15)) as δi in Eqs. (16) and (17). Table 6 shows the parameters used for each roll damping ratio calculation method.
Among the roll damping ratio calculation methods employed in this study, three methods, excluding ζEX, utilized all positive, negative, and double peaks of the roll motion. This approach aimed to minimize discrepancies in the calculations. A representative value was determined by averaging the damping ratios obtained from each peak. Table 7 presents the calculation results derived from each peak when the initial roll angle was set to 5°.
4. Experiment Results
Fig. 6 shows the time history of the roll motion measured from the free roll decay test in calm water for various initial roll angles. The initial roll angles measured through the test were 5.54°, 10.89°, 16.89°, and 22.43°. The results were utilized for the calculation of the roll natural period (Troll) and roll damping ratio (ζEX, ζEQ, ζL, and ζMean ).
4.1 Roll Natural Period
Table 8 presents the roll natural period results calculated using both the FFT method and the peak-to-peak method for eight rolling periods. For the FFT method, the roll natural period increased from 1.02 s to 1.18 s as the number of analysis data points increased from 210 to 216. The roll natural period converged to 1.18 s when the number of data points exceeded 214. Notably, the roll natural period showed no significant variation with changes in the initial roll angle. This stability can be attributed to the improved frequency resolution achieved through the zero-padding technique, as shown in Table 5. The roll natural period determined using the peak-to-peak method ranged from 1.18 s to 1.19 s, depending on the initial roll angle. The difference between the two methods was minimal, at approximately 0.1 s (0.84%). Both methods yielded comparable results, demonstrating their consistency in evaluating the roll natural period.
According to Baniela (2008), the roll natural period varies during a free roll decay test due to changes in the transverse metacentric height, which depends on the roll motion of the floating structure. However, this variation in transverse metacentric height is minimal when the initial roll angle is small (10° or less). In tests conducted with initial roll angles of 16.89° and 22.43°, which exceeded 10°, the initial roll angle was observed to decrease from 16.89° to 9.42° and from 22.43° to 10.55° by the second rolling period, as illustrated in Figs. 6(c) and 6(d).
The number of rolling periods required to influence the roll natural period due to the transverse metacentric height was found to be one. Consequently, this influence had a negligible impact on the total number of rolling periods (eight) used for evaluating the roll natural period (Troll).
4.2 Roll Damping Ratio
Table 9 and Fig. 7 present the roll damping ratio, calculated using roll amplitude values of ±1.0° or larger. Except for ζMean, the roll damping ratio was approximately 2% for an initial roll angle of 5.54° and reached a maximum of 16.08% for an initial roll angle of 22.43°. A linear increase in the roll damping ratio was observed as the initial roll angle increased, which appears to be attributed to the heightened nonlinearity of roll motion caused by the larger initial roll angles. This nonlinearity of roll motion is likely induced by the formation of eddies, generated as the fluid flow passes and separates at the edges of the model, particularly when the midship section coefficient of the model approaches 1. The ζMean values ranged from 2.76% to 4.60% as the initial roll angle varied from 5.54° to 22.43°, indicating that ζmean is relatively insensitive to changes in the initial roll angle. Furthermore, when the initial roll angle was 5.54°, the variation in the roll damping ratio was less than 1%, with results ranging from 2.05% to 2.89%, depending on the evaluation method. However, at an initial roll angle of 22.43°, the results ranged from 3.89% to 16.08%, leading to a substantial difference in the roll damping ratio, up to 12.19%, depending on the evaluation method employed. The pronounced disparity in the roll damping ratio, particularly at larger initial roll angles, arises because the logarithmic decrement at small magnitudes of roll motion has a dominant influence on the calculation of ζMean . As ζMean is derived from the average logarithmic decrement, the effect of the logarithmic decrement from small roll amplitudes becomes more pronounced as the roll amplitude decreases.
A previous study employing a model with a high midship section coefficient (CM: 0.98 to 0.99) reported a maximum roll damping ratio of 12.4% at an initial roll angle of 36.28° in the absence of a bilge keel, and 15.9% at an initial roll angle of 16.89° when a bilge keel was present. BV (2011) indicated a linear roll damping ratio ranging from 4% to 8% for vessels with a high midship section coefficient and without bilge keels, such as tankers and bulk carriers. Given that the floating structure analyzed in this study lacks a bilge keel, the roll damping ratio observed at a roll angle of 22.43° appears to be relatively high, with the exception of ζMean . This can be attributed to the limitations inherent in existing damping ratio calculation methods, such as the relative decrement method and the linear-quadratic decrement method. As noted by Bassler et al. (2010) and Sun and Shao (2019), these methods are primarily designed for evaluating roll motion at small angles and are less effective in capturing the nonlinear effects that arise at larger initial roll angles.
To address these limitations, further research is essential to account for the nonlinear damping effects inherent in each methodology. Specifically, investigating the nonlinear characteristics of the roll damping ratio by visualizing and quantitatively analyzing eddy formation and eddy intensity is necessary, achieved by examining the fluid flow field and flow separation around the model through PIV-based experimental methods. Additionally, numerical simulations using computational fluid dynamics (CFD) for viscous flow analysis are required to provide deeper insights into these nonlinear phenomena.
Table 10 and Fig. 8 present the damping ratios as a function of the number of rolling periods. A rolling period of one corresponds to one complete reciprocating roll motion of the floating structure during the free roll decay test. As the roll motion approached the threshold of ±1.0°, the roll damping ratios determined by the four evaluation damping ratio at the first roll motion of the floating structure and the damping ratio at the threshold (±1.0°) was minimal—up to 2.16% for ζEX, ζEQ, and ζL, which employ the linear regression method. In contrast, ζMean decreased by 7.82%, from 11.71% to 3.89%. When the number of rolling periods reached four, the roll amplitude stabilized at approximately 5° (Fig. 6), and the logarithmic decrement reduced by a factor of approximately 4.63. Consequently, the roll damping ratio decreased from 11.71% to 5.96% (Table 10 and Fig. 8(d)). From a roll amplitude of 5.54° to the threshold, the logarithmic decrement decreased by a factor of 1.08, and the roll damping ratio dropped to 3.89%. Due to the characteristics of ζMean, which uses an average value, the influence of the damping ratio calculated for small angles appears to have a greater impact than the logarithmic decrement associated with the initial roll angle. This suggests that the linear regression method yields a more stable damping ratio, even for large initial roll angles, as it effectively accounts for the nonlinear damping effects.
When the roll damping ratio at the minimum number of rolling periods (eight), as recommended by the ITTC regulations, was compared with the damping ratio at the threshold (±1.0°), the difference ranged from 0.08% to 0.15% at an initial roll angle of 5.54° and from 0.39% to 2.21% at 22.43°. Therefore, when the initial roll angle is small (e.g., 10°), using the minimum number of rolling periods (eight) specified by the ITTC regulations as the evaluation standard for the roll damping ratio is reasonable.
5. Conclusions
This study involved a free roll decay test conducted using a model with a midship section coefficient of 0.99 in a two-dimensional wave tank. This test aimed to compare the roll damping ratio (ζEX, ζEQ, ζL, and ζMean ) results of the floating model at varying initial roll angles (5.54°, 10.89°, 16.89°, and 22.43°) and with different roll damping ratio calculation methods.
(1) The number of rolling periods was set to eight for the measurement data in accordance with the ITTC regulations. The roll natural period was calculated using both the FFT and peak-to-peak methods. The roll natural period was evaluated to be 1.18 s using the FFT method and between 1.18 and 1.19 s using the peak-to-peak method, indicating that the results of the two methods were nearly identical. Both methods can be employed to evaluate the roll natural period, provided the minimum number of rolling periods (eight) prescribed by the ITTC regulations is used and the frequency resolution is enhanced through the zero-padding technique in the FFT method.
(2) The difference between the roll damping ratios (ζEX, ζEQ, ζL, and ζMean) obtained using different evaluation methods was less than 1% when the initial roll angle was 5.54° or less and up to 1.62% when the initial roll angle was 10.89° or less. However, when the initial roll angle was 22.43°, the ζMean value showed a maximum difference of 12.19% in the roll damping ratio. This large discrepancy is likely due to the dominant effect of the roll damping ratio for smaller angles (5.54° or less) on the overall roll damping ratio, owing to the nature of the ζMean method, which uses the arithmetic mean of the roll damping ratio based on the logarithmic decrement (δi ).
(3) The difference in ζMean values was more sensitive to the number of rolling periods used in the roll damping ratio evaluation than to the initial roll angle, possibly owing to the dominance of the damping ratio calculated at smaller roll angles.
(4) When the initial roll angle was small (10.89°), the difference between the roll damping ratio evaluated above the threshold and that calculated using the minimum number of rolling periods (eight) based on the ITTC regulations ranged from 0.08% to 0.15%. Therefore, the minimum number of rolling periods can be used as a standard to evaluate the roll damping ratio.
(5) For initial roll angles of 10.89° or less, selecting any of the four roll damping ratio calculation methods does not appear to cause significant issues in evaluating the roll damping ratio. However, when the initial roll angle exceeds 10.89°, choosing a method that accounts for the specific characteristics of the roll damping ratio evaluation methods is important.
To quantify the viscous damping effect caused by eddies, which significantly influences the roll damping phenomenon in floating structures with large roll angles, future research should investigate the nonlinear characteristics of roll damping, achieved by studying the quantitative relationship between the fluid flow field, eddy intensity around the floating structure, and the roll-damping phenomenon using techniques such as PIV or CFD for viscous flow analysis.
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