A Semi-Analytical Method for the PDFs of A Ship Rolling in Random Oblique Waves

The PDFs (probability density functions) and probability of a ship rolling under the random parametric and forced excitations were studied by a semi-analytical method. The rolling motion equation of the ship in random oblique waves was established. The righting arm obtained by the numerical simulation was approximately fitted by an analytical function. The irregular waves were decomposed into two Gauss stationary random processes, and the CARMA (2, 1) model was used to fit the spectral density function of parametric and forced excitations. The stochastic energy envelope averaging method was used to solve the PDFs and the probability. The validity of the semi-analytical method was verified by the Monte Carlo method. The C11 ship was taken as an example, and the influences of the system parameters on the PDFs and probability were analyzed. The results show that the probability of ship rolling is affected by the characteristic wave height, wave length, and the heading angle. In order to provide proper advice for the ship''s manoeuvring, the parametric excitations should be considered appropriately when the ship navigates in the oblique seas.     

The nonlinear rolling response of a vessel including chaotic motions leading to capsize in regular seas

The rolling motion of a ship has been successfully modelled using a semi-empirical nonlinear differential equation by a number of researchers. Experimental data has been used to model nonlinear damping and righting lever characteristics and comparison with observed behaviour has been reasonably good. The present article describes a numerical, phenomenological approach to analyse this type of behaviour. The stability of the periodic motion, and in particular the possibility of capsize, is explored with reference to qualitative prediction techniques. The appearance of chaotic motions in regular beam seas is a new feature which should be of interest to designers. The inability of traditional quantitative methods, such as the perturbation technique, to detect chaos is a further justification for using numerical simulation guided by dynamical systems theory.     

On the identification of ship coupled heave–pitch motions using neural networks

This paper outlines a procedure for the derivation of the differential equations describing the free response of a heaving and pitching ship from its stationary response to random waves. The coupled heave–pitch motion of a ship in random seas is modelled as a multi-dimensional Markov process. The partial differential equation describing the transition probability density function, known as the Fokker-Planck equation, for this process is derived. The Fokker-Planck equation is used to derive the random decrement equations for the coupled heave–pitch motion. The parameters in these equations are then identified using a neural network approach. The method is validated using numerical simulations and experimental results. The experimental data was obtained using an icebreaker ship model heaving and pitching in random waves. It is shown that the method produces good results when the system is lightly damped. An extension for using this method to identify couple heave–pitch motion in realistic seas is suggested.     

On the effectiveness of constant coefficients roll motion equation

As known, the rolling motion characteristics, amplitudes and accelerations, greatly influence the ability of a ship to operate and survive in bad weather. On the other hand, traditional computer codes for seakeeping calculations fail the forecasting of large amplitude rolling. There is a great need of using semi-empirical damping models and coefficients. This stresses the importance of campaigns of measurements as described in the paper, to get a deeper insight into the physical-mathematical modelling of the different contributions to rolling equation.Experimental tests on nonlinear rolling in a regular beam sea of a Ro-Ro ship model have been conducted by varying both the wave steepness and the wave frequency. The use of a parameter estimation technique, based on the least squares fitting of the stationary numerical solution of the nonlinear rolling motion differential equation, allowed to obtain informations on the damping model and on the linear and nonlinear damping coefficients. These exhibit a quite strong dependence on frequency that reduces the efficiency of constant coefficients rolling equation to simulate large amplitude nonlinear rolling. The results indicate that a good quality prediction model of nonlinear rolling cannot be based on constant coefficients time domain simulations. These can infact lead to incorrect estimates of rolling amplitudes even when the parameters have been obtained through high level parameter estimation procedures based on experimental data. The analysis indicates also a marked dependence of the effective wave slope coefficient on wave amplitude. The introduction of both these dependences on the rolling equation allows to reproduce the experimental results with great accuracy even at large amplitudes.     

On the identification of ship coupled heave–pitch motions using neural networks

This paper outlines a procedure for the derivation of the differential equations describing the free response of a heaving and pitching ship from its stationary response to random waves. The coupled heave–pitch motion of a ship in random seas is modelled as a multi-dimensional Markov process. The partial differential equation describing the transition probability density function, known as the Fokker-Planck equation, for this process is derived. The Fokker-Planck equation is used to derive the random decrement equations for the coupled heave–pitch motion. The parameters in these equations are then identified using a neural network approach. The method is validated using numerical simulations and experimental results. The experimental data was obtained using an icebreaker ship model heaving and pitching in random waves. It is shown that the method produces good results when the system is lightly damped. An extension for using this method to identify couple heave–pitch motion in realistic seas is suggested.     

Numerical simulation of ship stability for dynamic environment

The prediction of ship stability during the early stages of design is very important from the point of vessel’s safety. Out of the six motions of a ship, the critical motion leading to capsize of a vessel is the rolling motion. In the present study, particular attention is paid to the performance of a ship in beam sea. The linear ship response in waves is evaluated using strip theory. Critical condition in the rolling motion of a ship is when it is subjected to synchronous beam waves. In this paper, a nonlinear approach has been tried to predict the roll response of a vessel. Various representations of damping and restoring terms found in the literature are investigated. A parametric investigation is undertaken to identify the effect of a number of key parameters like wave amplitude, wave frequency, metacentric height, etc.     

The effect of nonlinear damping and restoring in ship rolling

Many researchers have studied a wide range of nonlinear equations of motion describing a ship rolling in waves. In this study, a form of nonlinear equation governing the motion of a rolling ship subjected to synchronous beam waves is suggested and solved by the generalized Duffing's method in the frequency domain. Various representations of damping and restoring terms found in the literature are investigated and their solutions are analyzed by the above-mentioned method. Comparative results of nonlinear roll responses are obtained for four distinct vessel types at resonance conditions which constitute the worst situation. The results indicate the importance of roll damping and restoring, when constructing a nonlinear roll model. An inappropriate selection of damping and restoring terms may lead to serious discrepancies with reality, especially in peak roll amplitudes.     

Stability and capsizing analysis of nonlinear ship rolling in wind and stochastic beam seas

Considering the actual seaway condition, stability and capsizing of nonlinear ship rolling system in stochastic beam seas is of significant importance for voyage safety. Safe zone are defined in the phase space plan of the unperturbed Hamilton system to qualitatively distinguish ship motions as capsize and noncapsize. Capsize events are defined by solutions passing out of the safe zone. The probability of such an occurrence is studied by virtue of the random Melnikov function and the concept of phase space flux. In this paper, besides conventional wave excitation, the effect of wind load is also taken into account. The introduction of wind load will lead to asymmetry, in other words, it transforms the symmetric heteroclinic orbits into asymmetric homoclinic orbits. For asymmetric dynamical system, the orbital analytic solutions and its power spectrum are not readily available, and the technique of discrete time Fourier transformation (DTFT) is used. In the end, as verification of theoretical critical significant wave height, capsizing probability contour diagram is generated by means of numerical simulation. The contour diagram shows that these analytical methods provide reliable and predictive results about the likelihood of a vessel capsizing in a given seaway condition.     

Capsizing Probability of Dead Ship Stability in Beam Wind and Wave for Damaged Ship

The International Maritime Organization has developed the second-generation intact stability criteria. Thus, damage stability criteria can be established in the future. In order to identity the capsizing probability of damaged ship under dead ship condition, this paper investigates two methods that can be used to research the capsizing probability in time domain, which mainly focus on the nonlinear righting lever GZ curve solution. One method subjects the influence of damaged tanks on the hull shape down to the wind and wave, and the other method is consistent with the real-time calculation of the GZ curve. On the basis of one degree of freedom rolling equation, the solution is Monte Carlo method, and a damaged fishery bureau vessel is taken as a sample ship. In addition, the results of the time-domain capsizing probability under different loading conditions are compared and analyzed. The relation of GM and heeling angle with the capsizing probability is investigated, and its possible reason is analyzed. On the basis of combining the time-domain flooding process with the capsizing probability calculation, this research aims to lay the foundation for the study of capsizing probability in time domain under dead ship condition, as well as provide technical support for capsizing mechanism of dead ship stability and damage stability criteria establishment in waves.     

Design of passive anti-roll tanks for roll stabilization in the nonlinear range

The best way of reducing roll motion is by increasing roll damping. Bilge keels are the most common devices for increasing roll damping. If more control is required, anti-roll tanks and fins are used. Tanks have the advantage of being able to function when the ship is not underway. Our objective is to develop design procedures for passive tanks for roll reduction in rough seas. This paper focuses on the design of passive U-tube tanks. The tank-liquid equation of motion is integrated simultaneously with the six-degree-of-freedom (6DOF) equations of the ship motion. The coupled set of equations is solved by using the Large Amplitude Motion Program ‘LAMP’, which is a three-dimensional time-domain simulation of the motion of ships in waves. The unstabilized and stabilized roll motions of a S60-70 ship with forward speed and beam waves have been analyzed. For high-amplitude waves, the unstabilized roll angle exhibits typical nonlinear phenomena: a shift in the resonance frequency, multi-valued responses, and jumps. The performance of a S60-70 ship with a passive tank is investigated in various sea states with different encounter wave directions. It is found that passive anti-roll tanks tuned in the linear or nonlinear ranges are very effective in reducing the roll motion in the nonlinear range. The effect of the tank damping, frequency, and mass on the tank performance is studied. Also, it is found that passive anti-roll tanks are very effective in reducing the roll motion for ships having a pitch frequency that is nearly twice the roll frequency in sea states 5 and 6.     

Estimation of extreme slamming pressures using the non-uniform Fourier phase distributions of a design loads generator

Slamming pressures are predicted using a nonlinear ship motion program whose input is an ensemble of short wave trains tailored to produce a large, linear pitch response. These short wave trains are calculated via a design methodology that first creates short time series containing a specified, large ship response and then back-calculates the incident wave trains using linear systems theory. The background simulations and theory used to create these short time series are presented here. Monte Carlo simulation of moderately rare events of a random process indicate the random Fourier component phase PDFs are non-uniform, non-identically distributed, and dependent on the rarity of the target event. These PDFs are modeled using a single parameter, Modified Gaussian distribution and used to generate design time series with a given expected value at a specific time. To predict rare events without resorting to Monte Carlo simulation, the parameters of the Modified Gaussian distributions are calculated via characteristic function comparison. The characteristic functions compare a target PDF calculated from extreme value theory to a PDF based on a discrete Fourier representation of the stochastic process with non-uniform component phases. The comparison to extreme value theory helps to quantify the risk associated with rare events.     

Complex dynamics and chaotic responses in the time domain simulations of a floating structure

The motion of a moored semi-submersible is modelled using a semi-empirical nonlinear ordinary differential equation. Time domain simulations, guided by the qualitative theory of dynamical systems, illustrate the extremely complex nature of typical responses in regular seas. The importance of nonlinearities is emphasised together with the implications for floating structures in general.     

Improved prediction of the threshold of surf-riding of a ship in steep following seas

An extended version of Melnikov's method is implemented in order to predict more accurately the threshold of global surf-riding for a ship operating in steep following seas. The key advantage of the proposed method is that it overcomes the limitation of small damping and/or small forcing that are intrinsic to the implementation of the standard Melnikov's method. A reference ITTC ship is used here by way of example and the result is compared to that obtained from standard analysis as well as numerical simulations. Because of the primary drawback of the extended Melnikov's method is the inability to arrive at a closed form equation, in this work the authors arrive at a “best fit” approximation to the extended Melnikov numerically predicted result.     

Studies on Stochastic Parametric Roll of Ship with Stochastic Averaging Method

The paper studies the parametric stochastic roll motion in the random waves. The differential equation of the ship parametric roll under random wave is established with considering the nonlinear damping and ship speed. Random sea surface is treated as a narrow-band stochastic process, and the stochastic parametric excitation is studied based on the effective wave theory. The nonlinear restored arm function obtained from the numerical simulation is expressed as the approximate analytic function. By using the stochastic averaging method, the differential equation of motion is transformed into Ito's stochastic differential equation. The steady-state probability density function of roll motion is obtained, and the results are validated with the numerical simulation and model test.     

An investigation on the vertical motion sickness characteristics of a high-speed catamaran ferry

Low frequent motions of vessel may cause motion sickness in rough seas. These undesirable effects induce fatigue of crews during the navigation. The motion sickness is always an important criterion for the high-speed craft design. Modern ferry designs have been marketed with a great emphasis on the seakeeping performance. This research has been carried out by investigating the results on the vertical motion sickness incidence (MSI) study for a 40 m wave-piecing catamaran at seas. The primary purpose of this research is to investigate the vertical motion sickness characteristics of a high-speed catamaran ferry. Two mathematical models, three-dimensional translating–pulsating source distribution technique and three-dimensional pulsating source distribution technique, are used for predicting the vertical acceleration responses of the wave-piecing catamaran in oblique waves. The comparison between numerical predictions and experimental data shows a good agreement except that around the pitch resonance region in FP vertical acceleration motions. Based on the experimental observation, the discrepancies may be caused by the nonlinear effects of centre bow during large pitch motions in waves. The comfort assessments are based on the ISO-2631/1997 standard with the hydrodynamic analysis for determining the acceleration levels in different locations on the vessel. The effects of seating location, wave heading and duration of motion exposure on seasickness are discussed.     

A rational approach to intact ship stability assessment

In this study, asymptotic and total stability of the non-linear free and forced pure rolling motions of a ship are investigated. A ship performing a rolling motion is taken as a dynamical system. Lyapunov's direct method is employed in the analysis. By generating a time-invariant Lyapunov function, conditions and the domain of asymptotic stability are obtained for free rolling motion. Results of the work on “boundedness” and “uniform boundedness” of the solutions of the equation of forced rolling motion, done by Özkan (1977), that is, conditions of total (practical) stability and its domain in the phase-plane are given and illustrated.     

Total (practical) stability of ships

In this paper the “total (practical) stability” concept is introduced to nonlinear forced rolling motion of a ship. This is achieved by employing “boundedness” and “Lyapunov's function” approach. In this respect two new theorems are proved and conditions and domain of Practical Stability are evaluated. The Paper also contains a critical review of the present status of international intact ship stability regulations. A qualitative discussion of oscillatory rolling motion and the capsizing phenomena is presented.     

Installing single-propeller twin-rudder system with less asymmetric maneuvering motions

The inflow characteristics to each one of the rudders of single-propeller twin-rudder system are investigated. It is shown that this inflow is not parallel to ship's centerline. This may result in asymmetric maneuvering characteristic of the ship thereby reducing ship's maneuvering performance. For this purpose, a method of installing single-propeller twin-rudder system is proposed. This method is called “virtual zero rudder angle” arrangement. Here, each one of the twin-rudder is set at an angle corresponding to the inflow to the rudder. The improvement in ship's maneuvering characteristics with “virtual zero rudder angle” arrangement is investigated using experiments and numerical simulations for different ship types. It is shown that this arrangement may also improve ship's propulsion performance.     

On the parametric rolling of ships using a numerical simulation method

This paper has shown a numerical motion simulation method which can be employed to study on parametric rolling of ships in a seaway. The method takes account of the main nonlinear terms in the rolling equation which stabilize parametric rolling, including the nonlinear shape of the righting arm curve, nonlinear damping and cross coupling among all 6 degrees of freedom. For the heave, pitch, sway and yaw motions, the method uses response amplitude operators determined by means of the strip method, whereas the roll and surge motions of the ship are simulated, using nonlinear motion equations coupled with the other 4 degrees of freedom. For computing righting arms in seaways, Grim's effective wave concept is used. Using these transfer functions of effective wave together with the heave and pitch transfer functions, the mean ship immersion, its trim and the effective regular wave height are computed for every time step during the simulation. The righting arm is interpolated from tables, computed before starting the simulation, depending on these three quantities and the heel angle. The nonlinear damping moment and the effect of bilge keels are also taken into account. The numerical simulation tool has shown to be able to model the basic mechanism of parametric rolling motions. Some main characteristics of parametric rolling of ships in a seaway can be good reproduced by means of the method. Comprehensive parametric analyses on parametric rolling amplitude in regular waves have been carried out, with that the complicated parametric rolling phenomena can be understood better.