Abstract
Parametric roll motion is a phenomenon that occurs within seconds, and it can reach high roll degrees. If the periodic stability of a ship changes, it may cause roll angles over 25 degrees, threatening the safety of the crew and the ship. This also threatens the operational skills of the ship. To investigate this phenomenon, a navy combatant model DTMB 5512 is selected. The model also has a bilge keel. Himeno’s method was employed to calculate damping coefficients based on roll decay experiments (from literature) conducted at various model speeds and initial angles. This approach facilitated the extraction of both linear and non-linear damping coefficients from experimental data. Additionally, extinction coefficients were also obtained. Maxsurf stability software was utilized to compute GM values and generate the GZ curve. A Runge-Kutta method implementation in Python programming enabled numerical analysis, comprising a total of 240 simulations across 10 wave heights and 24-speed scenarios. For each scenario, the maximum roll angle was determined. It was observed that roll angles increased notably when the encounter frequency approached twice the natural roll frequency. Based on the analysis findings, maximum roll angles did not exceed 25 degrees, indicating that the DTMB 5512 model is not vulnerable to parametric roll resonance.
1. Introduction
Parametric roll motion is a rapid phenomenon characterized by high roll angles (exceeding 25 degrees) occurring within seconds. Research on parametric roll began in the 1930s, marked by seminal theoretical analyses by Watanabe [1]and Kempf [2]. Subsequent studies integrated nonlinear damping and Mathieu-type equations, notably by Kerwin [3] and Paulling and Rosenberg [4]. Experimental investigations by Paulling [5] in 1972 further contributed to understanding this phenomenon. The practical significance of parametric roll gained prominence in the 1990s following accidents, such as the damage to a post-panamax C11 type cargo ship in 1998, which prompted detailed publications emphasizing its importance [6]. Bulian [7] explored the nonlinear damped 1-degree-of-freedom motion associated with the parametric roll. The International Maritime Organization (IMO) included parametric roll in the second-generation stability criteria for ship safety [8], while recent studies continue to investigate various aspects of parametric roll [9, 10, 11]. Parametric roll analysis plays a critical role in maritime safety guidelines established by classification societies such as ClassNK [12] and ABS [13], as well as the updated guidelines from the IMO in 2020 [14]. The metacentric height (GM) is a pivotal factor in parametric roll analysis, varying with different loading conditions. Recent research has also explored computational fluid dynamics solutions tailored to specific vessel types, including fishing boats [15].
In this study, we conducted numerical parametric roll analysis for the DTMB 5512 model. Damping coefficients were derived using roll decay data specific to the DTMB 5512, as detailed by Irvine et al. [16]. Both linear and nonlinear damping coefficients were considered, adhering closely to IMO guidelines [17]. The restoring term GZ was modelled using a seventh-degree equation, while GM was approximated using a cosine function. The numerical analysis was executed utilizing the Runge-Kutta method.
2. Physical Background
Parametric roll can occur when the ship length and the wavelength of encountered waves are closely aligned, and the wave encounter frequency is twice the ship natural roll frequency. Changes in transverse inertia in the head or following seas lead to fluctuations in GM (metacentric height).
Ships typically have a wider beam at upper decks, especially in cargo ships, where the underwater portion is streamlined compared to the above-water structure. Consequently, when the wave trough is amidships, the ship experiences greater inertia, enhancing its stability. Conversely, when the wave crest is amidships, the ship encounters less inertia, resulting in decreased stability. The variation in GM mentioned above is influenced by factors such as wavelength, wave height, and the wave position. Figures 1 and 2 illustrate the submerged portions of the ship at different wave positions.
Figures 3 and 4 depict the submerged waterplane area of the ship at various positions relative to the wave. These perspectives should be considered together for a comprehensive evaluation. As the wave moves past the ship, the relationship with width becomes crucial, significantly as the height varies at different points along the ship.
The variation of these two variables relative to the wave’s position defines the changes in GM.
Parametric roll is characterized by periodic changes influenced by the relationship between the ship’s natural roll frequency and the encounter frequency of waves. Figure 5 illustrates a scenario where the encounter frequency is twice the ship’s natural roll frequency. Initially, at the zero-roll degree point with the wave crest position, the ship experiences zero roll angle. As the wave progresses, the ship reaches its maximum roll angle with the wave in a trough position. The ship’s stability is enhanced during this phase due to a greater restoring moment. Subsequently, as the ship continues to roll, the wave position returns to a crest. At this stage, the ship’s stability decreases, leading to increased rolling compared to earlier stages, the ship reaches its maximum roll angle again when the wave returns to a trough position. Consequently, stability improves once more. This cyclic pattern of roll and stability changes is characteristic of parametric roll phenomena, influenced by the dynamic interplay between wave conditions and the ship’s response.
According to Luthy [11], the following conditions are required for parametric roll to occur:
- Wavelength should be close to the length of the ship,
- GM value should be affected as much as possible by the
interaction of the hull of the ship with the wave profile,
- Encounter frequency should be twice the natural roll
frequency of the ship,
- Head or following seas,
- Insufficient damping.
3. The Method of Analysis
3.1. General Information About Analysis
The main parameters of DTMB 5512 (Iowa scale) is given in Table 1 [16].
In many studies, observing different GM values for DTMB 5512 is possible. The reason of the difference is caused by the baseline selection. In this study, Figure 6 is used as the baseline, representing the lowest point of the sonar dome.
The equation of roll motion can be found in various forms in the literature. As a basis, the Mathieu-type (the changing GM in time) in Equation 1 can be used if the ship is sailing in longitudinal seas and there is no wave roll moment:
Here B 44 is the linear or linearized damping coefficient and ΔGM(t)ϕ is the linear restoring moment. These linear terms can be used at the lower roll degrees, but parametric roll resonance reaches high roll degrees, and this formula can not be used in this situation. Also, the Mathieu equation can only indicate whether parametric rolling starts or not. At higher roll degrees, the solution goes to infinite. Therefore, the Mathieu equation is insufficient to find the final degree of parametric roll resonance. Therefore, a non-linear equation can be used to better express the dynamics at higher roll degrees. After introducing the non-linear terms, the Equation 2 becomes the following:
3.2. Restoring Term
The restoring term is expressed as Equation 3 [9]. As seen, GZ is a function that has different parameters. GM max and GM min are maximal and minimal instantaneous values of GM for several wave crest positions along the ship hull as stated in Belenky et al., [18].
Also, wave height is defined as follows :
h j = 0.01 jL , where j = 0,1, 2,3…9,10
Table 2 shows GM m and GM a values for different wave heights calculated with the help of the Maxsurf Stability [19]. The wave heights listed in the Table 2 are 0.065 meters less than the actual values. This is because the baseline has been chosen as the reference point. The distance of 0.065 meters is the gap between the baseline and the orange line seen in Figure 6.
To make it easier to express the GM changing with the position of the wave, the encounter frequency and time are used. An example result is shown in Figure 7. The curves in Figure 7, are calculated for DTMB 5512 and show an example cosine approximation for DTMB 5512.
To find a, b, c, and d coefficients, the GZ (Restoring moment arm) graph was obtained with the help of the Maxsurf Stability software. Then GZ values were divided by GM T value. Using Matlab, a curve was fitted for GZ / GM T values by the form in Equation 3. The fitted curve can be seen in Figure 8. The a, b, c, and d coefficients obtained as a result of curve fitting are as in Table 3.
3.3. Mass Moment of Inertia and Hydrodynamic Added Inertia
As stated in Equation 6, a restoring term and model natural roll period are needed to find inertia and added inertia. The restoring term was calculated with the help of
Equation 7. When the roll decay results of the model are examined, different natural roll periods are observed at various speeds and initial angles. The natural roll period was calculated at all speeds and initial angles. Table 4 shows a natural roll period with the average of these values was obtained.
T ϕ = 2π √ _ I 44 + A 44 _ C 44 (6)
C 44 = Δ GM (7)
The average inertia of the ship was found using the average natural period and restoring term.
I 44 + A 44 = 2.322 kg m 2
3.4. Coefficients of Non-linear Representation of Roll Damping
Equation 8 is the general formula for the nonlinear damping coefficients. In this study, the B 2 term will be ignored since the quadratic damping coefficient will not be used. In the rest of the study, B L stands for B 1 as it represents the linear coefficient of roll damping and B NL stands for the B 3 as cubic coefficient of roll damping.
To determine the damping coefficients, roll decay test data were analyzed across all speeds and initial angles of 10°, 15°, and 20°. Given the intention to incorporate coefficients of non-linear representation of roll damping in the study, Himeno's method [20] was employed. This method is suited explicitly for extracting non-linear damping characteristics from experimental data, ensuring accurate characterization of the ship’s damping behavior during roll decay tests. In experimental data, as in Figure 9, the red points where the roll degree of the model peaks are determined. Using Equations 9 and 10, the difference between the peak points and, after that, the averages of the peak points are calculated (Table 5).
After calculating ϕ m and Δϕ values, a curve fitting was performed using Matlab software. In Figure 10, the curve fitting using peak values can be seen. With the equation obtained with the curve, Equation 8 can be equated. This can be seen in Equation 11. The transformation in Equation 7 is used for the restoring term (C 44). With the modifications made to the Equation 11, and B L, B NL are obtained in Equation 12 and Equation 13.
The process in Figure 10 was performed for each speed and initial angle. Table 6 shows the values of k 1 and k 3.
The relation of the extinction coefficients can be seen in Equation 14 and 15. "a and c" are called the decay coefficients (obtained from free-roll test). In our equation, they correspond to k1, k3. With the help of this equation, the relation between ship speed and extinction coefficients can be seen in Table 7. The relationship between decay coefficients, coefficients of the non-linear representation of roll damping, and extinction coefficients can be examined in detail in the ITTC documents [21].
After these calculations, a scatter plot was drawn with speed on the x-axis and linear (B L) or non-linear
(B NL) damping coefficients on the y-axis. As can be seen in Figures 11 and 12, a 2nd order curve fitting was performed for the linear coefficient, while a 1st order curve fitting was performed for the non-linear damping coefficient. Thus, the relationship between the damping coefficients and the forward speed was revealed for DTMB 5512.
3.5. Numerical Analysis Background
Runge-Kutta method was used for numerical analysis. The code was written in Python language to perform the analysis. The algorithm can be seen in Figure 13. The inputs of the algorithm are as follows:
- Speed
Since the ship will be examined with the head and the following waves, therefore cos(φ w) = 1 or
cos(φ w) = - 1, respectively.
ω e = ω − k V cos (φ w) (16)
Equation 16 shows how the encounter frequency is calculated for the speed the ship has. φ w refers to the angle of encounter. Therefore, in Table 8, i ≤ 12 refers to following waves, while i > 12 refers to head waves. The speed coefficients in the table express the ratio with service speed. The service speed of the ship is assumed to be Fn= 0.41. With the help of Equation 17, the ship’s service speed at Fn= 0.41 is equal to 2.2434 (m/s).
Fn = V _ √ _ g L
V s = 2.2434 m / s (17)
- Wave height
Wave height as an input determines the values of GM m and GM a in the equation. More information on wave heights is in section 3.2.
- Initial roll amplitude and roll velocity
The initial roll angle and roll velocity for the analysis are as follows:
ϕ(0) = 0.0872 rd (5deg)
ϕ ˙ (0) = 0 rd / s
- Time
The time interval for numerical analysis was determined as 0.01 seconds.
- Runge-Kutta Method
In the numerical analysis, the 4th-order Runge-Kutta method was used to calculate the instantaneous roll angles and velocities.
4. Results and Discussion
4.1. Validation
The experimental results were compared with those derived from the Runge-Kutta method to assess the proximity of the calculated damping coefficients to experimental values [16]. The equation was structured to depict free-damped motion. The numerical analysis using the Runge-Kutta method was conducted for different initial roll angles and vessel speeds. Figures 14 to 21 show that the numerical results closely match the experimental data obtained under the same initial conditions. This agreement between experimental and numerical results shows that the coded Runge Kutta method gives reliable outcomes.
4.2. Results
Table 9 on the subsequent pages displays the wave heights, ship speeds, encounter frequencies, and maximum roll angles (degrees). Additionally, motion graphs depicting these unstable states can be found in from Figure 21 to Figure 26. Please note that following figures show the head sea cases.
5. Conclusion
Parametric roll resonance significantly impacts the operational capabilities of naval combatants. Therefore, in the current study, an analysis of parametric roll resonance was conducted for the DTMB 5512 model. This analysis involved determining both linear and non-linear damping coefficients, as well as extinction coefficients. GM values for the restoring term were established based on varying wave heights, and coefficients of GZ were also calculated. Using the Runge-Kutta method for analysis, maximum roll angles were observed to reach approximately 14 degrees. Although this angle does not meet the IMO threshold of 25 degrees for considering parametric roll, it is crucial to recognize that such roll angles still pose significant risks and should be carefully evaluated.
NOMENCLATURE
LPP : Length between perpendiculars
LWL: Length of waterline
BWL: Maximum moulded breadth at design waterline
CB: Block coefficient
T: Draught
KG: Centre of gravity above moulded base or keel
GM: Transverse metacentric height
λw: Wave length
Fn: Froude Number
ωe: Encounter frequency
ωϕ : Natural roll frequency
