Stochastic rolling motion of ships in following seas

The aim of this study is to examine both the stochastic rolling motion of ships in following seas by using analytical and numerical safe basin concept and to show how the concept of safe basin can be utilized for the assessment of the stability of ships. First, the boundary of the analytic safe basins of stochastic rolling motion is obtained by using the Lyapunov function used by Schurz (2009). Second, the safe basins of stochastic rolling motion are obtained numerically by using stochastic Euler (Euler–Maruyama) method. It is concluded that the analytically and numerically obtained safe basins are coherent. Lastly, the practical intact stability criterion for ships in following seaway is developed by proposing a formula given for the maximum width of analytic safe basin.     

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.     

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.     

船舶随浪运动稳性仿真计算

本文利用Ｌｉａｐｕｎｏｖ理论，研究了船舶在规则波浪运动的稳性；利用摄动理论，求解出船舶运动响应；并讨论了船舶横摇与垂荡运动频率、最大横摇角和波浪要素对稳性曲线ＧＺ的影响，以及流体动压力对稳性曲线的修正，从而给计算船舶在随浪中的稳性提供了一种方法。     

Using wavelet transforms to analyze nonlinear ship rolling and heave-roll coupling

The use of wavelet transforms is explored to investigate the nonlinear dynamical characteristics of ship roll and coupled heave-roll motion. The harmonic character, double period character and chaotic character are observed via a time–frequency window of the wavelet transform. Typical wave parameters in different stability regions are considered. Features such as restoring rolling, divergence rolling, steady state and chaotic responses of ship roll are obtained as well. The investigation in this paper not only highlights the feasibility of using wavelet transforms in the analysis of nonlinear dynamic characteristics of ship rolling in waves, but also shows how it could enhance the analysis abilities.     

利用相邻测线重叠区域进行多波束测深横摇运动残差改正

针对多波束测深条带边缘波束易受到姿态和声速等多种误差影响、相对中央波束数据质量较低的问题,本文提出一种利用相邻测线重叠区域对多波束测深数据边缘波束进行横摇运动残差改正的模型,提高边缘波束测深数据的质量。使用沿航向的测深点匹配插值模型,完成中央波束测深点与边缘波束测深点的匹配,得到边缘波束测深误差值;使用横摇运动残差改正模型,实现顾及姿态角的条件下补偿波束入射角。计算实例表明：本文模型能够较为准确地提取边缘波束测深误差值,改正后的海底地形削弱了误差导致的上下起伏,有效地减少了影响边缘波束的多种误差,具有实际的工程应用价值。     

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.     

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.     

On the equations of rolling motion of a ship with a flooded compartment

This paper deals with obtaining the governing equations of rolling motion of a ship with a flooded compartment. The equations of motion are obtained through the variational formulation in the form of Hamilton-Ostrogradskii equation by taking the ship, the fluid in the flooded compartment and the sea as a single mechanical system. Since no specification concerning ships or flooded compartments has been made, the obtained equations are applicable to any sea-going vessel. As an application, the equation of rolling motion of a ship with a prismatic flooded compartment is obtained by choosing a suitable velocity potential for the fluid motion in the compartment.     

平面二维流场中质点运动的可视化

对平面二维流场中质点运动的可视化进行了研究,提出了一种适用于三角形非结构化网格系统中质点运动轨迹的跟踪算法,利用数值计算方法的稳定性限制条件对质点位置的搜索算法进行了优化,极大地降低了算法的时间复杂度,提高了计算效率,采用Shepard插值法求解质点的速度,四阶龙格库塔方法积分质点的运动轨迹,选用Visual Fortran语言实现了质点运动的动态显示,将本文方法与平面二维潮流模型相结合,用于渤海潮流场中质点运动规律的研究,初步得出了渤海中质点运动和水效换的一些规律。     

A set of canonical problems in sloshing, Part I: Pressure field in forced roll—comparison between experimental results and SPH

In this article, impact pressure in the case of shallow water sloshing is investigated experimentally and numerically for forced rolling motion. The maximum values of impact pressures have been found for a frequency lower than the first sloshing frequency. Experimental results are compared with numerical ones obtained using smoothed particle hydrodynamics (SPH). The influence of viscosity and of density re-initialization on the SPH results are discussed. A new method for calculating the pressure on walls with SPH is presented.     

高海况下无人水下航行器的横摇运动分析

为明确高海况下海况等级对无人水下航行器横摇运动的影响,得出满足某型无人水下航行器安全回收的海况条件,对高海况下无人水下航行器的横摇运动进行分析。运用频率响应法,分别在4级、5级和6级海况下,根据海浪谱密度函数和该型航行器的横摇频率响应函数,求得其横摇运动谱密度函数。再运用公式推导,得到其横摇运动的时域函数。使用MATLAB仿真软件对该型航行器进行横摇运动仿真,得到3种海况条件下,其横摇运动的时域函数图像,并验证仿真结果的准确性。仿真结论:高海况下的海况等级对无人水下航行器的横摇幅度有较大影响,满足其安全回收的海况条件为5级海况。     

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.     

不规则波下船舶运动响应特征数值分析

针对不规则波浪作用下Wigley型船的运动响应问题进行了系统的研究,采用统计学方法深入探讨了船舶不规则运动幅值和响应周期的分布规律,并通过傅里叶变换对船舶运动响应进行了频谱特征分析。结果表明,船舶横摇方向与升沉和纵摇方向随机运动的响应特征有显著差异。在升沉与纵摇方向,波浪谱峰频率远离自振频率,前十分之一大振幅运动对应周期离散性较小,基本稳定在波浪谱峰周期附近,但小振幅运动周期分布离散性较大,频谱分析指出船舶升沉与纵摇运动响应频谱在波浪谱峰频率附近出现明显峰值。而在横摇方向,波浪谱峰频率与自振频率相耦合,不同振幅的横摇运动响应周期均稳定在自振周期附近,且周期离散性较小,频谱分析也表明横摇运动响应频谱主要集中于船舶运动自振频率附近。     

Experimental and Numerical Investigations of Ship Parametric Rolling in Regular Head Waves

Parametric rolling is one of five types of the ship stability failure modes as proposed by IMO. The periodic change of the metacentric height is often considered as the internal cause of this phenomenon. Parametric rolling is a complex nonlinear hydrodynamic problem, often accompanied by large amplitude vertical motions of ships. In recent years,the Reynolds-averaged Navier–Stokes(RANS) equation simulations for viscous flows have made great progress in the field of ship seakeeping. In this paper, the parametric rolling for the C11 containership in regular waves is studied both experimentally and numerically. In the experiments, parametric rolling amplitudes at different drafts, forward speeds and wave steepnesses are analyzed. The differences in the steady amplitudes of parametric rolling are observed for two drafts. The effect of the incident wave steepness(or wave amplitude) is also studied, and this supports previous results obtained on limits of the stability for parametric rolling. In numerical simulations, the ship motions of parametric rolling are analyzed by use of the potential-flow and viscous-flow methods. In the viscousflow method, the Reynolds-averaged Navier–Stokes equations are solved using the overset grid method. The numerical accuracies of the two methods at different wave steepnesses are also discussed.     

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.     

水动力作用下管道稳定性的试验研究

利用振荡流水槽在以下两种约束条件下，研究波浪作用下直接铺设于砂质海底的管道失稳临界条件：1）管道两端自由；2）管道可水平、垂直自由运动，但流动受到限制。试验结果表明，管重无量纳数G与管道失稳的临界Fr数之间大致呈线性关系，而当G超过一定数值时，管道是侧向稳定的。管道的约束条件、砂床特性和加载速度对管道稳定性均有影响。     

Estimation of survival probability for a ship in beam seas using the safe basin

The purpose of this paper is to analyze the nonlinear ship roll motion equation and the main parameters that induce ship capsizing in beam seas, estimate the survival probability of a ferry in random seas and to find out a risk assessment method for the ship’s intact stability. A single degree of freedom (1-DOF) dynamic system of ship rolling in beam seas is investigated and the nonlinear differential equation is solved in the time domain by the fourth order Runge-Kutta algorithm. The survival probability of a ferry in beam seas is investigated using the theory of “safe basin”. The survival probability is calculated by estimating erosion of “safe basin” during ship rolling motion by Monte Carlo simulations. From the results it can be concluded that the survival probability of a ship in beam sea condition can be predicted by combining Monte Carlo simulations and the theory of “safe basin”.     

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.