CHEN Jia, YANG Jianming, SHEN Kunfan, CHANG Zongyu, *, and ZHENG Zhongqiang

Probability Density Analysis of Nonlinear Random Ship Rolling

CHEN Jia1), 2), 3), YANG Jianming2), SHEN Kunfan1), 3), CHANG Zongyu1), 3), *, and ZHENG Zhongqiang1)

1) College of Engineering, Ocean University of China, Qingdao 266100, China 2) Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John’s, NL, A1B3X5, Canada 3) Shandong Provincial Key Laboratory of Ocean Engineering, Qingdao 266100, China

Ship rolling in random waves is a complicated nonlinear motion that contributes substantially to ship instability and capsiz- ing. The finite element method (FEM) is employed in this paper to solve the Fokker Planck (FP) equations numerically for homo- clinic and heteroclinic ship rolling under random waves described as periodic and Gaussian white noise excitations. The transient joint probability density functions (PDFs) and marginal PDFs of the rolling responses are also obtained. The effects of stimulation strength on ship rolling are further investigated from a probabilistic standpoint. The homoclinic ship rolling has two rolling states, the connection between the two peaks of the PDF is observed when the periodic excitation amplitude or the noise intensity is large, and the PDF is remarkably distributed in phase space. These phenomena increase the possibility of a random jump in ship motion states and the uncertainty of ship rolling, and the ship may lose stability due to unforeseeable facts or conditions. Meanwhile, only one rolling state is observed when the ship is in heteroclinic rolling. As the periodic excitation amplitude grows, the PDF concentration increases and drifts away from the beginning location, suggesting that the ship rolling substantially changes in a cycle and its stability is low. The PDF becomes increasingly uniform and covers a large region as the noise intensity increases, reducing the certainty of ship rolling and navigation safety. The current numerical solutions and analyses may be applied to evaluate the stability of a rolling ship in irregular waves and capsize mechanisms.

ship rolling; homoclinic rolling; heteroclinic rolling; finite element method; Fokker Planck equation; probability density function

1 Introduction

The rolling motion of a ship in random waves is con- sidered the most prominent and destructive of its six de- grees of freedom. A potentially extreme roll may dras- tically jeopardize the stability of the ship and, in extreme instances, result in the ship capsizing. The rolling move- ment of a vessel is a complex dynamic process with non- linearity, unpredictability, and nonstationary. Many aca- demics have continuously conducted extensive studies on ship rolling.

For the systems with nonlinear displacement elements under random stimulation, the steady-state Fokker Planck (FP) equations were derived by Caughey (1963), and the exact stationary solutions were further obtained. By sim- ulating the nonlinear rolling of the ship, Dalzell (1973) acquired the data of the maximum value distribution of the motion and compared it to the theoretical results related to the random Gaussian process. He discovered that the theoretical distribution of the linear process is Rayleigh, but not the nonlinear process. Hsieh. (1994) and Lin and Yim (1995) investigated ship stability in random waves by evaluating the rolling response and observed that the rolling motion near the heteroclinic orbit prompts capsiz- ing when the noise elicits the vessel. Through numeric simulation and experiments, Lee (2001) studied ship cap- sizing in regular beam waves, finalized computing based on a fully nonlinear ship rolling model, and incorporated the findings into a feasibility study of the basic assump- tions used in a multi-degree-of-freedom nonlinear dynamic model. High nonlinear rolling and capsizing of a ship in random waves were investigated by Tang. (2004) through the Melnikov criterion, and the phase space flux impacted by damping, wave height, frequency and the nonlinear restoring force arm were also monitored. The capsizing probability of a damaged ship in random beam wind and waves was calculated by Lai. (2022) using the Monte Carlo simulation (MCS), demonstrating that the capsizing probability of the damaged ship was higher than that of the intact ship. For the improvement of the ship sta- bility standard, Dostal and Kreuzer (2011, 2014) scruti- nized the coupled heave pitch roll motion to deduce the analytical formula of the mean first passage time of the critical roll amplitude when the ship sailed in the long crest random wave with arbitrary wave encounter angle and suc- cessfully applied the obtained findings to the ship design.

Haddara (1974) suggested an algorithm for calculating the transient and steady-state responses of a ship rolling in an oblique random wave considering its nonlinear damping and restoring force properties, and the subsequent calcu- lations were compared to existing schemes. Taylan (2000) presented a rolling equation with various nonlinear damp- ing and restoring terms to examine the impact of non- linear damping and restoring on ship rolling and solved this equation using the extended Duffing method. The re- sults suggested that significant deviations from the actual ship rolling could occur when the damping and restoring were not properly selected. Maki. (2019) adopted the non-Gaussian PDF considering the nonlinear damping fac- tor to improve the forecasting accuracy of the probability density function (PDF) of an irregular transverse wave rolling and reported that the PDF drawn by this approach was consistent with the MCS result. An improved com- plex nonlinear rolling equation under the condition of time delay was proposed by Yang and Guo (2019), considering nonlinear aspects, such as damping and restoring torque, and this formula was recommended to obtain the relevant parameters of a stable or chaotic state. The global analysis technique was employed by Falzarano. (1992) to in- vestigate the transient nonlinear ship rolling, thus, the ef- fects of water-on-deck, damage, and ice on restoring mo- ment, as well as the size and presence of bilge keels on damping, were all appropriately considered. Focusing on the chaotic phenomenon of ship rolling due to bifurcation and amplitude jump, an enhanced linear filter technique and cumulant-neglect closure are applied by Francescutto and Naito (2004) to analyze the nonlinear behavior and complicated dynamics of ship rolling. Pedi?i? Bu?a and Senjanovi? (2006) studied ship rolling in regular and irregular waves using an uncoupled equation and the har- monic acceleration method and suggested a new ship sta- bility standard based on the nonlinear responses and sur- vival diagrams. To (2012) conducted extensive research on nonlinear stochastic systems, inferred the Markov and non- Markov solutions of the system, and summarized the meth- ods and technologies for analyzing nonlinear systems. The stable and unstable probability densities of the random response of an asymmetric nonlinear system under non- rational spectrum non-white random excitation were stud- ied by Kimura and Morimoto (1998) using the non- Gaus- sian equivalent linearization (EL) method and the moment equation approach, respectively, and the results were found to have good correlation with the simulation results. Maki (2017) investigated the ship rolling motion using the mo- ment technique and the EL method. The generated PDF was found to be congruent with the MCS, and the strategy may be used to test the safety of severe nonlinear rolling. Jiang. (2020) examined the PDF and mean up-cross- ing rate of nonlinear ship rollingexponential polyno- mial closure, which outperformed the EL approach and the MCS. An approximation explanation of ship rolling motion in beam sea was formed by Roberts (1982) using the average method and the Markov principle, and the rolling angle formulation was found to match well with the digital simulation of Dalzell (1973). Liu. (2019) formulated a differential equation for ship rolling under parametric and forced excitations, modeled wave rising as a narrow-band random process, and calculated the PDF of ship rolling using the stochastic averaging approach of the energy envelope and the MCS. An enhanced stochastic averaging procedure was applied by Zhou. (2021) to solve the probability features of extreme rolling of a con- tainer ship under random longitudinal and oblique waves, and the accuracy of the finding was additionally confirmed by the MCS. To assess the stability and capsizing of homo- clinic ship rolling, the path integration (PI) method and the random Melnikov criteria were employed by Liu and Tang (2007) to procure the PDF and the chaotic domain of rolling response, and the effects of nonlinear restoring mo- ment and water-on-deck on ship stability were also de- scribed. Naess and Moe (2000) proposed a new PI method that can effectively improve the performance of assessing the responses of nonlinear vibration systems excited by white or colored noise without significantly increasing cal- culation time and then delivered some examples to illus- trate the reliability of this method. The PI and the random average methods were used by Chai. (2018) to solve the random ship rolling problem. They confirmed that both methods could obtain the response characteristics of the ship; however, the former can provide high precision re- sults while the latter can reduce the dimension of the problem and the difficulty of calculation.

The finite element method (FEM) is extensively used as a numerical approach to tackle engineering and mathe- matical physics challenges and is crucial in researching the FP equations for nonlinear systems. Langley (1985) dis- covered that the numerical solutions agreed well with the analytical results when the FEM was applied to the Duf- fing oscillator or ship rolling system. Anh and Hieu (2012) evaluated the mean square response of the Duffing oscil- lator under combined harmonic and random excitations by the FEM, implying that the findings were in line with the MCS. The van der Pol and Duffing oscillatory systems triggered by white noise were explored by Kumar and Narayanan (2006), and the stationary and transient solu- tions solved by the FEM were observed in accordance with the exact results. The nonlinear random responses of beams exposed to simultaneous acoustic and thermal loads were investigated by Chen. (1996) with the FEM, and the numerical solutions were compared against the classi- cal continuum solution and found to be similar to experi- mental data. A new FEM was developed by Shiau and Wu (1996) to acquire the PDF of nonlinear systems excited by parameters and external Gaussian white noise, and the efficacy and accuracy of the method were tested by various examples and approaches. Under additive Poisson random excitation, a single-degree-of-freedom system was apprais- ed by Náprstek and Král (2010) using the FEM and other analytical methodologies, and the response and PDF of the system were accomplished and comprehensively examined. Galán. (2007) exploited the FEM for a two-dimen- sional FP matter to build a numerical solution that was co- herent with the experimental and calculated results. By ap- plying various FEMs to two or three-dimensional linear and nonlinear systems, Pichler. (2013) obtained the numer- ical solutions and time histories of the PDFs of each sys- tem. In the static analysis of linear and nonlinear stochas- tic systems, Papadopoulos and Kalogeris (2016) launched the FEM to solve a series of pure advection partial differ- ential equations involved in the evolution of probability density, and the complete probability information of the structure at given positions was precisely obtained. To address a two-dimensional fractional Rayleigh Stokes type issue, the FEM was adopted by Dehghan and Abbaszadeh (2017), and the unconditional stability and convergence estimation of the numerical results, as well as the efficacy and reliability of the approach in comparison to other numerical methods, were considered. Proven to be more advanced than the spectral collocation method, the FEM was employed by Scalera. (2021) to solve the FP equation on the unit sphere, and the effectiveness of this numerical method was further confirmed by the analytical solution. After examining the convergence speed, stability, and performance of the FEM, Honrubia and Aragonés (1986) determined that the FEM was a high-order approx- imation based on numerical data, while El- Gebeily and Shabaik (1994) presented a detailed procedure for address- ing the FP problem using the FEM. Some FP equations with fractional order form were treated by Sun(2022) and Le and Stynes (2021) with the FEM, and the existence, convergence, stability, and error estimates of solutions were all investigated. The authenticity of numerical discoveries was also validated by a vast number of numerical instances and theoretical analyses.

The application of the FEM to obtain the PDF evolution of dynamic systems is advantageous for the in-depth explo- ration and analysis of nonlinear vibration systems. In this paper, the FEM is mainly employed to investigate the non- linear homoclinic and heteroclinic ship rolling in random waves and evaluate the ship stability by obtaining the PDF evolution of ship responses. Random wave stimulation is defined by the combination of periodic and Gaussian white noise excitations. The FP equations associated with the dif- ferential equations for ship rolling are inferred and numer- ically solved using the FEM and the Crank Nicolson time difference approach. The complete transient joint and mar- ginal PDFs of the ship responses are then calculated, ex- posing probability features and the instability mechanism of the nonlinear random ship rolling. Furthermore, the sensitivities of the PDFs to external stimulation strength, such as periodic excitation amplitude and Gaussian white noise intensity, are researched, which may deliver some insights into the investigation of ship stability.

2 Problem Description

2.1 Nonlinear Random Ship Rolling

The differential equation for nonlinear ship rolling in waves may be modeled as

whereandare the linear and cubic damping coeffi- cients that obtained by the least square method (Kreuzer and Wendt, 2000; Wang., 2008).1and3are the linear and nonlinear restoring moment coefficients, respec- tively (Wang., 2008; Jamnongpipatkul., 2011).

Substituting Eqs. (2) and (3) into Eq. (1), the following non-dimensional terms are applied:

whereωis the undamped natural frequency, andis a small parameter that scales the external excitation, which is assumed to be small.

Under normal circumstances, the ship rolling belongs to a heteroclinic roll, which reflects the large-scale roll motion of the ship. At this stage, Eq. (3) is used to fit thecurve of the ship, where1and3are positive values. A damaged ship or that with water on the deck has a neg- ative metacentric height, indicating that the ship rolling is the vibration near the loll angle (Pawowski and Tuzcu, 2009; Jamongpipatkul., 2011), and the ship rolling exhibits homoclinic vibration characteristics. At this point, the form of the Eq. (3) can also be achieved by curve fitting the GZ of the ship, where1and3are negative values. The homoclinic and heteroclinic rolling equations for the ship may then be rewritten as:

The random wave is complex and Gaussian in distribu- tion, which may be regarded as a combination of periodic and Gaussian white noise excitations (Lin and Yim, 1995; Liu and Tang, 2007; Jamnongpipatkul., 2011):

whereandare the amplitude and frequency of the periodic excitation, respectively.() is the Gaussian white noise excitation with

where[ ] is the expectation operator,is the spectral density of the Gaussian white noise, and() is the Dirac delta function.

2.2 Fokker Planck Equations for Nonlinear Random Ship Rolling

The corresponding Ito equations are

, (8)

where() is a standard Wiener process.

Based on the Markov process theory, the transition PDF of the random ship rolling is governed by the FP equation, which may be written as

where=(,|0,0) is the transition joint PDF that satisfies the following criteria:

The initial criterion:

The boundary criterion at infinity:

and the normalization criterion:

3 Finite Element Method and Crank Nicolson Approach

In the phase space of ship rolling, Ωdenotes the solution area (Fig.1), which was split intobilinear rectangular elements, and each element (Fig.2) comprises four nodes. The PDF for the ship rolling may be then considered as a sum of shape functions using the Galerkin projection rule (Zienkiewicz., 2005). Furthermore, the shape func- tions are selected such that the PDF value within the-th element may be represented considering the nodal values:

The shape functions of bilinear rectangular elements are:

where theuandvare the abscissas and ordinates of the-th node, respectively.

Fig.1 Mesh with bilinear elements.

Fig.2 Rectangular element in natural coordinate.

Substituting Eq. (14) into Eq. (9), the residual error can be obtained:

whereis the total nodes number. Letting the value of its projection on the weight function to 0, the lowest residual error may be derived:

whereis the joint PDF vector, and

A time difference methodology, known as the Crank Nicolson method is used to perform the numerical analysis of Eq. (18):

where Δis the time difference and 0 ≤≤ 1.

4 Numerical Results and Analyses

In the following analysis, the Gaussian distribution, in which PDF is a single attraction or peak, is adopted to de- scribe the initial probability distribution for the nonlinear random ship rolling, and the parameters in Eq. (5) are1= 0.1321,2= 0.02656,3= 0.9018, and0.15 according to Jamnongpipatkul. (2011). In the FEM, the solu- tion region of the phase space of ship rolling is set to [?2, 2] × [?2, 2], and each direction is divided into 100 ele- ments and 101 nodes. The MCS is also employed to ver- ify the accuracy and effectiveness of the FEM. Figs.3 and 4 respectively show the transient marginal PDFs of the ship rolling angle and angular velocity computed by the FEM and MCS at two random transients. The cumulative distribution functions (CDFs) of ship rolling angle and angular velocity determined by the FEM and MCS at a random transient are shown in Fig.5. These figures reveal that the marginal PDFs and the CDFs obtained by the FEM and MCS are consistent at any transient, indicating that the FEM has great accuracy and reliability in solving ship rolling problems, which allows the approach to study the following homoclinic and heteroclinic ship rolling.

4.1 Numerical Solutions of Homoclinic Ship Rolling

The transient joint PDFs and their projections of the homoclinic ship rolling under periodic excitation and Gaus- sian white noise in two periods are obtained by the FEM. Fig.6 shows that the attraction domain of the joint PDF rapidly changes from one in the beginning to two, form- ing two independent peaks, while the value and shape of the joint PDF always maintain a periodic state. This find- ing indicates the presence of two states in the ship rolling, as the two peaks of the joint PDF are not connected, the ship can only have one rolling state for any defined initial conditions in the phase plane. The joint PDFs of the non- linear ship rolling at= 42.10 s (about 6.5 times the exci- tation period) and= 45.34 s (approximately 7 times the excitation period) are displayed in Figs.6(b) and 6(c), re- spectively. It is clear that their values are the same, their shapes regularly fluctuate, and they are symmetrical along the longitudinal axis.

Fig.3 Marginal PDFs of ship rolling angle with H = 0.15, D = 0.03. (a), t = 35 s; (b), t = 50 s.

Fig.4 Marginal PDFs of ship rolling angular velocity with H = 0.15, D = 0.03. (a) t = 35 s; (b) t = 50 s.

Fig.5 CDFs of the ship rolling angle and angular velocity at t = 35 s with H = 0.15, D = 0.03.

Fig.6 Joint PDFs and their projections of the homoclinic ship rolling with F = 0.1, D = 0.01. (a), t = 38.87 s; (b), t = 42.10 s; (c), t = 45.34 s; (d), t = 48.58 s;

In Fig.7, the transient joint PDFs and their projections of the nonlinear homoclinic ship rolling with different peri- odic excitation amplitudes in a period are presented by the FEM to investigate the influence of periodic excitation on ship rolling. When the periodic stimulation amplitude is increased to 0.25, Fig.7 shows that the joint PDF con- stantly maintains an attraction domain, and its form and value exhibit evident periodicity and symmetry during sub- sequent motion. Furthermore, the connection channel be- tween the two peaks of the joint PDF becomes visible as the periodic excitation amplitude increases. This phenom- enon indicates that the random jump probability of the ship between the two homoclinic rolling states increases, even- tually reducing the stability of the ship and even leading to capsizing under some uncertain and complex conditions.

Fig.7 Joint PDFs and their projections of the homoclinic ship rolling with F = 0.25, D = 0.01. (a), t = 43.72 s; (b), t = 45.34 s; (c), t = 46.96 s; (d), t = 48.58 s.

Certain traits are distinctive in the marginal PDF. Figs. 8 and 9 show the marginal PDFs of the ship rolling angle and angular velocity in a period computed by the FEM, respectively, with various lines denoting different periodic excitation amplitudes. The two figures indicate that the marginal PDFs of the rolling angle and angular velocity quickly reached a periodic state with changing periodic excitation amplitudes. The heights for both peaks of the marginal PDFs of the ship rolling angle fluctuate with time, as illustrated in Fig.8, indicating that the likelihood of the two rolling states varies as well. When the excitation amplitude is 0.1, the marginal PDF of the rolling angle maintains a high peak on the left and a low one on the right during the quarter and double cycles, suggesting that the probability of performing the rolling state of the left plane is larger than that of the right one in the phase plane comprising the rolling angle and angular velocity. By con- trast, the marginal PDF of the rolling angle exhibits distinct features as the periodic excitation amplitude rises. For ex- ample, increasing the amplitude to 0.25 causes regular heights shifting of the two peaks of the marginal PDF of the ship rolling angle, with the period matching the period- ic stimulation. Furthermore, the height difference between the two peaks of the marginal PDF reduces, and the channel between them expands as the excitation amplitude grows. That is, the ship rolling is likely to jump between the two states, resulting in a worsening of ship stability. Unlike the bimodal distribution of the marginal PDF of the rolling angle, the marginal PDF of the rolling angular ve- locity preserves the shape of a single peak throughout the ship rolling process when the excitation amplitude is small, and the trajectory of this peak is regularly symmetrical around the origin, as shown in Fig.9. The height of the peak drops and its position deviates from the origin as the external excitation amplitude increases from 0.1 to 0.25. Overall, the joint and marginal PDF diagrams demonstrate that a large excitation amplitude produces a short and broad PDF, causing the uniform distribution of probability den- sity across the solution region, which implies that the un- certainty of ship motion is increased.

The transient joint PDFs and their projections of the homoclinic rolling with varying noise levels are shown in Fig.10 to explore the effect of white noise intensity on homoclinic rolling. The figure shows that the joint PDF of the ship rolling always maintains an area of attraction, resulting in two connected peaks throughout the rolling process, while the value and shape of the joint PDF always retain a periodical state in the sequential rolling, implying that the ship rolling has two states; thus, the ship motion will randomly alternate between them. Furthermore, as il- lustrated in Fig.10, increasing the noise level to 0.05 re- duces the joint PDF value while increasing the visibility of the channel between the peaks, this phenomenon increas- es the unpredictability of ship rolling and tendency to ran- dom jumps, resulting in poor ship stability and an increas- ed likelihood of capsizing.

Fig.8 Marginal PDFs of homoclinic ship rolling angle with D = 0.01. (a), t = 43.72 s; (b), t = 45.34 s; (c), t = 46.96 s; (d), t = 48.58 s.

Fig.9 Marginal PDFs of homoclinic ship rolling angular velocity with D = 0.01. (a), t = 43.72 s; (b), t = 45.34 s; (c), t = 46.96 s; (d), t = 48.58 s.

Fig.10 Joint PDFs and their projections of the homoclinic ship rolling with D = 0.05, F = 0.1. (a), t = 43.72 s; (b), t = 45.34 s; (c), t = 46.96 s; (d), t = 48.58 s.

The marginal PDFs of the homoclinic rolling angle and angular velocity with varied white noise intensities are res- pectively shown in Figs.11 and 12, wherein different line types stand for various noise levels. The marginal PDFs of the rolling angle and angular velocity promptly reached a periodic state with varying noise intensities. Fig.11 re- veals that different noise intensities result in differential height changes in the two marginal PDF peaks at any given time. When the white noise intensity is 0.01, the marginal PDF of the rolling angle always has a prominent peak on the left and a low one on the right during the quarter and double cycles. By contrast, the heights of the two peaks fluctuate regularly when the noise intensity is 0.05, indi- cating that different white noise intensities result in a dif- ferent probability of the two rolling states. Furthermore, Fig.12 shows that the marginal PDF of the rolling angular velocity always keeps a peak with periodic symmetry re- garding the origin and that the larger the white noise in- tensity, the lower the height of the peak, and the greater the deviation from its origin. Additionally, the high noise intensity lowers the PDF height and expands the distribu- tion range, which is particularly unfavorable for predict- ing ship rolling.

4.2 Numerical Solutions of Heteroclinic Ship Rolling

Fig.13 shows transient joint PDFs and their projections of the heteroclinic rolling of the ship under periodic and white noise excitation in two periods. The attraction domain of the joint PDF, according to the figure, always has one attraction domain and a single peak during the entire het- eroclinic rolling, and the value and shape of the joint PDF are always in a periodic condition. Figs.13(b) and 13(c) illustrate the joint PDF of heteroclinic rolling at= 42.10 s (about 6.5 times the excitation period) and= 45.34 s (around 7 times the excitation period), respectively. The PDFs in the two figures show that their values are identical, their forms vary regularly, and they are symmetrical along the longitudinal axis. Unlike homoclinic rolling, the find- ing indicates that the heteroclinic rolling of the ship has only one definite state for any beginning condition on the phase plane, indicating that the heteroclinic rolling of the ship is more stable than homoclinic rolling. To investigate the impact of periodic excitation amplitude on heteroclinic rolling, Fig.14 depicts the transient joint PDFs and their projections of the response of homoclinic ship rolling with increasing periodic excitation amplitude in a period. When the periodic excitation amplitude grows to 0.35, as shown in Fig.14, the shape and periodicity of the joint PDF are not modified. Notably, the value or the peak height of the joint PDF increases and the form of the joint PDF shifts away from its initial location throughout this period.

Figs.15 and 16 show the marginal PDFs of the hetero- clinic rolling angle and angular velocity in a period ob- tained by the FEM, respectively, with different line types standing for different periodic excitation amplitudes. Re- gardless of the magnitude of the periodic stimulation, the two figures demonstrate that the marginal PDF of rolling angle and angular velocity immediately finds a periodic shape and value. Furthermore, the positions of the mar- ginal PDF of the rolling angle and angular velocity devi- ate from the original or initial position when the excita- tion amplitude increases. The difference lies in the varia- tion of the marginal PDF of roll angle and angular veloc- ity with the magnitude of the periodic excitation. More specifically, the marginal PDF of the angle increases throughout the cycle as the excitation intensity increases. However, the increased amplitude varies over time, where- as the marginal PDF value of angular velocity increases at quarter and three-quarter cycle moments and falls at half and double cycle times. This finding implies that the ship rolling speed dramatically fluctuates when the exte- rnal periodic stimulation grows, posing a serious hazard to the crew, equipment, and potentially the stability of the ship.

Fig.11 Marginal PDFs of the homoclinic ship rolling angle with F = 0.10. (a), t = 43.72 s; (b), t = 45.34 s; (c), t = 46.96 s; (d), t = 48.58 s.

Fig.13 Joint PDFs and their projections of the heteroclinic ship rolling with F = 0.15, D = 0.05. (a), t = 38.87 s; (b), t = 42.10 s; (c), t = 45.34 s; (d), t = 48.58 s.

Fig.14 Joint PDFs and their projections of the heteroclinic ship rolling with F = 0.35, D = 0.05. (a), t = 43.72 s; (b), t = 45.34 s; (c), t = 46.96 s; (d), t = 48.58 s.

Fig.15 Marginal PDFs of the heteroclinic ship rolling angle with D = 0.05. (a), t = 43.72 s; (b), t = 45.34 s; (c), t = 46.96 s; (d) t = 48.58 s.

Fig.16 Marginal PDFs of the heteroclinic ship rolling angular velocity with D = 0.05. (a), t = 43.72 s; (b), t = 45.34 s; (c), t = 46.96 s; (d), t = 48.58 s.

Fig.17 depicts the transient joint PDFs and their projec- tions of heteroclinic rolling when the white noise intensity is 0.08, from which the influence of white noise intensity on heteroclinic rolling can be examined. The figure reveals that when the noise intensity reaches 0.08, the value and shape of the joint PDF of ship rolling always maintain the state of periodic motion in continuous rolling, but the shape of the joint PDF is low and short and occupies a large phase space, indicating a reduction in the certainty and safety of ship rolling. The marginal PDFs of the hetero- clinic rolling angle and angular velocity with different white noise densities are shown in Figs.18 and 19, re- spectively, and varied noise intensities are represented by different line types. The two figures show that despite the fluctuation in noise intensity, the marginal PDF of rolling angle and angular velocity always maintains a periodic symmetrical peak around the origin, and the form of the PDF peak becomes low and short as the level of white noise increases. This result shows that the motion state of heteroclinic rolling is lightly concentrated under the in- fluence of high-intensity white noise.

5 Conclusions

The joint and marginal PDFs of two nonlinear random ship motions, namely homoclinic and heteroclinic rolling, are explored in this paper under the combined action of periodic and Gaussian white noise excitations. The FP equations corresponding to the two rolling motions are established, the transient joint PDFs and marginal PDFs for the FP equations are numerically solved using the FEM, and the efficacy and effectiveness of this approach are also validated by the MCS. Furthermore, the impacts of varying periodic excitation amplitudes and random excitation in- tensities on the homoclinic and heteroclinic ship rolling are investigated from a probabilistic standpoint, which may provide some references for ship stability research and cap- size processes.

In the research of homoclinic rolling, when the excita- tion amplitude or the white noise intensity is modest, the ship rolling has only one of two motion states, and the ship has certain stability or predictability at this moment. When the periodic excitation amplitude or the noise inten- sity grows, the joint and marginal PDFs of the ship rolling become low and broad, indicating that the probability of a random jump between two rolling states increases, the predictability of the ship motion decreases, and the ship may lose stability or possibly capsize in some uncertain circumstances. Unlike the homoclinic motion, the hetero- clinic ship rolling has only one motion state despite the fluctuation of the periodic amplitude or noise intensity, showing that the ship under heteroclinic rolling is more stable than that under homoclinic rolling. The joint and marginal PDFs of the heteroclinic rolling move away from the initial center as the periodic excitation amplitude grows, implying that the ship roll substantially changes in a cycle, which weakens the stability of the ship. The PDFs of the heteroclinic rolling occupy a large area and are uniformly distributed in the phase space as the noise intensity in- creases. Thus, the possibility of any motion state in the phase space is comparable, the uncertainty of ship motion is high, and the safety of the ship is insufficient.

Fig.17 Joint PDFs and their projections of the heteroclinic ship rolling with D = 0.08, F = 0.25. (a), t = 43.72 s; (b), t = 45.34 s; (c), t = 46.96 s; (d) t = 48.58 s.

Fig.18 Marginal PDFs of the heteroclinic ship rolling angle with F = 0.25. (a), t = 43.72 s; (b), t = 45.34 s; (c), t = 46.96 s; (d), t = 48.58 s.

Fig.19 Marginal PDFs of the heteroclinic ship rolling angular velocity with F = 0.25. (a), t = 43.72 s; (b), t = 45.34 s; (c), t = 46.96 s; (d) t = 48.58 s.

Acknowledgements

This study is supported by the National Natural Science Foundation of China (Nos. 52088102, 51875540).

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? Ocean University of China, Science Press and Springer-Verlag GmbH Germany 2023

. E-mail: zongyuchang@ouc.edu.cn

(Edited by Xie Jun)