Application of extended state space to nonlinear ship rolling

It is well known in the field of marine hydrodynamics that the added mass, damping and wave exciting forces are functions of frequency (Newman, 1977. Marine Hydrodynamics. MIT Press, Cambridge). Although most previous studies of nonlinear ship rolling motion have assumed that these forces do not vary with frequency, in this study the frequency dependent added mass and damping coefficients are approximated in the time domain with extended state space variables. Using numerical time simulation (integration), the extended state space model is compared to the constant coefficient model with a constant frequency forcing and the results for two constant value approximations of the added mass and damping are compared to the extended state space model with a multiple component pseudo random forcing.     

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.     

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.     

Subharmonic oscillations in nonlinear rolling

In this paper the rolling motion of a ship is examined with particular regard to the possibility of obtaining oscillations which are subharmonic to the excitation frequency. Three different mechanisms are found to be responsible for this phenomenon, the importance of which has already been recognized in the context of ship stability. The first is related to a strong symmetric or non-symmetric nonlinearity in the righting arm. The second is linked to the harmonic composition of sea waves and the third to the well known parametric excitation caused by coupling between different ship motions in a following sea.The onset of subharmonics is related to a threshold value for the excitation strongly depending on damping. The more appropriate analytical methods for a theoretical study of each mechanism are suggested.     

A multiscale analysis of nonlinear rolling

A new analysis of nonlinear rolling carried out by the multiscale perturbation method is herewith presented. The behaviour of a ship in a regular beam sea is considered and approximate analytical solutions in three nonlinear resonance regions are obtained. These concern the transient and the steady state roll oscillations. The latter fits in well with a previous one obtained through the averaging method and with the results of the numerical simulation.The obtained results appear to be particularly convenient due to their major mathematical simplicity. Moreover, they allow a simple estimation of the maximum roll amplitudes predictable for a given excitation intensity.     

Coupled analysis of nonlinear sloshing and ship motions

A coupled numerical model considering nonlinear sloshing flows and the linear ship motions has been developed based on a boundary element method. Hydrodynamic performances of a tank containing internal fluid under regular wave excitations in sway are investigated by the present time-domain simulation model and comparative model tests. The numerical model features well the hydrodynamic performance of a tank and its internal sloshing flows obtained from the experiments. In particular, the numerical simulations of the strong nonlinear sloshing flows at the natural frequency have been validated. The influence of the excitation wave height and wave frequency on ship motions and internal sloshing has been investigated. The magnitude of the internal sloshing increases nonlinearly as the wave excitation increases. It is observed that the asymmetry of the internal sloshing relative to still water surface becomes more pronounced at higher wave excitation. The internal sloshing-induced wave elevation is found to be amplitude-modulated. The frequency of the amplitude modulation envelope is determined by the difference between the incident wave frequency and the natural frequency of the internal sloshing. Furthermore, the coupling mechanism between ship motions and internal sloshing is discussed.     

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.     

A nonlinear model of parametric rolling stabilization by anti-roll tanks

The present paper describes a mathematical model in which the fluid motion inside a U-tank is nonlinearly coupled to the heave, roll and pitch motions of the ship. The main purpose of the investigation is centred on the control of roll motion in the case of parametric resonance in longitudinal waves. A transom stern small vessel, known to be quite prone to parametric amplification, is employed in the study. Four tank designs are employed in order to study the influence of tank mass, tank natural frequency and tank internal damping on the control of parametric rolling at different head seas conditions. Additionally, the influence of the vertical position of the tank is also investigated. The main results are presented in the form of limits of stability, with encounter frequency and wave amplitudes as parameters. Distinct dynamical characteristics are discussed and conclusions are drawn on the relevant parameters for the efficient control of the roll amplifications in head seas.     

Stochastic inverse modeling of nonlinear roll damping moment of a ship

This work presents an application of stochastic inverse method for the determination of nonlinear roll damping moment of a ship at zero forward speed. Nonlinear roll damping moment was estimated from the measured dynamic responses through stochastic inverse model. It is shown that this method enables nonlinear characteristic of the roll damping to be estimated without any assumption on its form of nonlinearity, including its confidence intervals given noisy data. The accuracy and practicability were assessed with laboratory tests related to both free-decay and forced rolling motions. The estimation results of the nonlinear damping moment show good agreement in all cases.     

物探船拖缆姿态及拉力预报

针对海洋地震物探船布设探测网络实际工况的要求,提出一种拟合数值计算与模型实验结果预报拖缆姿态及拉力的方法,应用此方法进行多方案比较研究,获得优化的拖缆－扩展器－检测电缆组合探测系统。经海上拖航实验验证,此种预报方法查可行,预报精度完全能满足工程实用要求。     

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.     

系泊船非线性波浪力时域计算:二维模型

为找到具有工程实用价值的港口系泊船波浪力的时域计算方法,建立了在港口中存在系泊船时非线性波浪力时域计算的垂直二维耦合模型:用Boussinesq方程计算船的两侧的外域,用欧拉方程计算船底面下的内域,两域在交界面处的连接条件是流量连续和压力相等.将复平面内的边界元方法应用于所研究问题,对耦合模型进行了验证.进行了相关模型实验,实验结果与数值计算结果比较表明这两种数值计算模型都具有满意的精度,但耦合模型的计算效率要远远高于边界元方法的计算效率.本耦合模型的数学处理简单,可适用于工程计算.     

On the nonlinear hydrodynamic forces for a ship advancing in waves

In this paper, using a second-order steady-state approach and a three-dimensional (3D) pulsating source distribution method derives the nonlinear hydrodynamic forces on a ship advancing in waves. The nonlinear hydrodynamic forces considered here consist of the mean lateral drifting force and the added resistance, which can be expressed as products of the ship-motion responses, the radiation potential, diffraction potential and the incident-wave potential. All related velocity potentials applied in the calculations are in 3D form. The Series 60 and Marine ship hulls are used for numerical calculations and the results are compared with existing experimental data and two-dimensional (2D) solutions. The comparisons show that the results obtained in the paper generally agree with experimental data well. It is also found that the nonlinear hydrodynamic forces obtained based on the present 3D source distribution methods are indeed improved in some calculations compared with the 2D method, especially for the mean lateral drifting force.     

A 3D fully coupled analysis of nonlinear sloshing and ship motion

This study investigates the coupling effects of six degrees of freedom in ship motion with fluid oscillation inside a three-dimensional rectangular container using a novel time domain simulation scheme. During the time marching, the tank-sloshing algorithm is coupled with the vessel-motion algorithm so that the influence of tank sloshing on vessel motions and vice versa can be assessed. Several factors influencing the dynamic behavior of tank–liquid system due to moving ship are also investigated. These factors include container parameters, environmental settings such as the significant wave height, current velocity as well as the direction of wind, wave and flow current acting on the ship. The nonlinear sloshing is studied using a finite element model whereas nonlinear ship motion is simulated using a hybrid marine control system. Computed roll response is compared with the existing results, showing fair agreement. Although the two hull forms and the sea states are not identical, the numerical result shows the same trend of the roll motion when the anti-rolling tanks are considered. Thus, the numerical approach presented in this paper is expected to be very useful and realistic in evaluating the coupling effects of nonlinear sloshing and 6-DOF ship motion.     

Application of the state space model to the system of wave energy conversion Analytical method for wave forces on single oscillating rectangular buoy

A new analytical method is proposed to analyze the force acting on a rectangular oscillating buoy due to linear waves. In the method a new analytical expression for the diffraction velocity potential is obtained first by use of the eigenfunction expansion method and then the wave excitation force is calculated by use of the known incident wave potential and the diffraction potential. Compared with the classical analytical method, it can be seen that the present method is simpler for a two-dimensional problem due to the comparable effort needed for the computation of diffraction potential and for that of radiated potential. To verify the correctness of the method, a classical example in the reference is recomputed and the obtained results are in good accordance with those by use of other methods, which shows that the present method is correct.     

An extended time-dependent numerical model of the mild-slope equation with weakly nonlinear amplitude dispersion

In the present paper, by introducing the effective wave elevation, we transform the extended ellip- tic mild-slope equation with bottom friction, wave breaking and steep or rapidly varying bottom topography to the simplest time-dependent hyperbolic equation. Based on this equation and the empirical nonlinear amplitude dispersion relation proposed by Li et al. (2003), the numerical scheme is established. Error analysis by Taylor expansion method shows that the numerical stability of the present model succeeds the merits in Song et al. (2007)’s model because of the introduced dissipation terms. For the purpose of verifying its performance on wave nonlinearity, rapidly vary- ing topography and wave breaking, the present model is applied to study: (1) wave refraction and diffraction over a submerged elliptic shoal on a slope (Berkhoff et al., 1982); (2) Bragg reflection of monochromatic waves from the sinusoidal ripples (Davies and Heathershaw, 1985); (3) wave transformation near a shore attached breakwater (Watanabe and Maruyama, 1986). Comparisons of the numerical solutions with the experimental or theoretical ones or with those of other models (REF/DIF model and FUNWAVE model) show good results, which indicate that the present model is capable of giving favorably predictions of wave refraction, diffraction, reflection, shoaling, bottom friction, breaking energy dissipation and weak nonlinearity in the near shore zone.