On unstable ship motions resulting from strong non-linear coupling

In this paper, the modelling of strong parametric resonance in head seas is investigated. Non-linear equations of ship motions in waves describing the couplings between heave, roll and pitch are contemplated. A third-order mathematical model is introduced, aimed at describing strong parametric excitation associated with cyclic changes of the ship restoring characteristics. A derivative model is employed to describe the coupled restoring actions up to third order. Non-linear coupling coefficients are analytically derived in terms of hull form characteristics.The main theoretical aspects of the new model are discussed. Numerical simulations obtained from the derived third-order non-linear mathematical model are compared to experimental results, corresponding to excessive motions of the model of a transom stern fishing vessel in head seas. It is shown that this enhanced model gives very realistic results and a much better comparison with the experiments than a second-order model.     

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.     

Stability of small fishing vessels in longitudinal waves

The dynamic stability of fishing vessels in longitudinal regular waves is investigated, both analytically and experimentally. In particular, the influence of stern shape on the parametric stability of fishing vessels is studied. Vessels TS and RS have very similar main characteristics, but their sterns are different. Although their linear responses are comparable, both analytical and experimental investigations indicate substantial differences in their dynamic stability in longitudinal regular waves. Strong resonances are found for the vessel with the deep transom. The analytical method takes into consideration the effects of the heave and pitch motions and wave passage and shows good agreement with experimental results. Stability limits are obtained for different conditions and are used as an aid in the discussion of the results obtained in the tests when relevant parameters are changed, such as wave amplitude and frequency, metacentric height and roll damping moment.     

Analysis of roll motion and stability of a fishing vessel in head seas

The present paper describes an investigation on the relevance of parametric resonance for a typical fishing vessel in head seas. Results for different Froude numbers are discussed based on experimental, numerical and analytical studies.The first region of resonance is investigated. Distinct wave amplitudes are considered. Some intense resonances are found to occur. The paper compares the experimental results with non-linear time simulations of the roll motion. Very good agreement is found, even when large motions take place.Finally, in order to analyze the experimental/numerical results, analytic consideration is given to distinct parameters affecting the dynamic process of roll amplification. The influence of heave, pitch, wave passage effect, speed and roll restoring characteristics are discussed.     

Parametric Resonance Analyses for Spar Platform in Irregular Waves

The parametric instability of a spar platform in irregular waves is analyzed. Parametric resonance is a phenomenon that may occur when a mechanical system parameter varies over time. When it occurs, a spar platform will have excessive pitch motion and may capsize. Therefore, avoiding parametric resonance is an important design requirement. The traditional methodology includes only a prediction of the Mathieu stability with harmonic excitation in regular waves. However, real sea conditions are irregular, and it has been observed that parametric resonance also occurs in non-harmonic excitations. Thus, it is imperative to predict the parametric resonance of a spar platform in irregular waves. A Hill equation is derived in this work, which can be used to analyze the parametric resonance under multi-frequency excitations. The derived Hill equation for predicting the instability of a spar can include non-harmonic excitation and random phases. The stability charts for multi-frequency excitation in irregular waves are given and compared with that for single frequency excitation in regular waves. Simulations of the pitch dynamic responses are carried out to check the stability. Three-dimensional stability charts with various damping coefficients for irregular waves are also investigated. The results show that the stability property in irregular waves has notable differences compared with that in case of regular waves. In addition, using the Hill equation to obtain the stability chart is an effective method to predict the parametric instability of spar platforms. Moreover, some suggestions for designing spar platforms to avoid parametric resonance are presented, such as increasing the damping coefficient, using an appropriate RAO and increasing the metacentric height.     

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.     

Designing against parametric instability in following seas

We investigate the characteristics of parametric instability when very large variations of restoring between the wave trough and the wave crest are taking place, creating a restoring that is alternating between negative (or nearly negative) and strongly positive values. The possible ways to consider the nonlinearities in damping and in restoring are discussed in detail. The boundary separating parametric instability from pure-loss is identified. In depth studies are carried out to ascertain the practical relevance of the parametric mechanism. Instability regions are identified in terms of transient motions, rather than in terms of the customary asymptotic stability chart associated with Mathieu's equation. A basis for comparing parametric roll behavior for different representations of restoring is established. Asymmetric variation laws and bi-chromatic waves are considered.     

深吃水半潜式平台参数共振研究

深吃水半潜式平台(deep draft semi-submersible,DDS)作为一种新型海洋结构物,既继承了传统半潜式平台的优点,又改善了传统半潜式平台垂荡运动性能差的不足。但随着吃水的增加,DDS纵摇运动参数共振也成为一个不容忽视的问题。当DDS纵摇运动固有周期和垂荡运动周期满足一定关系时,纵摇运动将发生参数共振,纵摇角度将会显著增大。研究推导了DDS纵摇运动方程,并简化为标准的马修方程,运用希尔无穷行列式法求解马修方程,得到了含阻尼的马修稳定性图谱,并结合具体算例讨论了不同参数对DDS纵摇运动参数共振问题的影响。研究表明:深吃水半潜式平台的参数共振,是设计人员在设计之初必须考虑的问题;通过合理选取平台的系统参数,可以有效避免纵摇运动参数共振现象的发生。     

Bounded and unbounded coupled transverse response of parametrically excited vertical marine risers and tensioned cable legs for marine applications

The paper deals with the non-linear dynamic response in the transverse direction of vertical marine risers or a tensioned cable legs subjected to parametric excitation at the top of the structure. The dynamic model contains both elastic and bending effects. The analytical approach reveals that the dynamic lateral response is governed by effects originated from the coupling of modes in transverse direction. The mathematical model is being treated numerically by retaining a sufficient number of transverse modes. Numerical results are given for specific case studies and refer both to the time histories of the lateral response for all modes of motion, and to the corresponding power spectral densities obtained through FFT. The numerical predictions are suitably plotted and discussed. The calculations concern both the undamped and the damped dynamic system. The damping in the system is a non-linear Morison type term, which describes the effect of the hydrodynamic drag. Both coupled and uncoupled equations are treated and points as well as regions of coupled and uncoupled stability and instability are defined. It is shown that the impacts originated from the coupling, evaluate new instabilities for the respective undamped system. The numerical results obtained through FFT of the time histories, provide qualitative conclusions for the features of the dynamic response for the modes of motions considered. Special attention has been paid to the effect of the hydrodynamic drag for the parametric excitation frequencies that guide the dynamic system to lie within a region of coupled instability.     

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.     

水下悬浮隧道锚索稳定性分析

为研究水下悬浮隧道锚索的稳定性,建立锚索在涡街激励作用下振动的数学方程,采用伽辽金法对其进行化简,锚索振动的稳定性通过Lyapunov指数法进行判断,分析了锚索动静张力比的大小、悬浮隧道系统的阻尼比和锚索参数激励频率对锚索振动稳定性的影响。计算结果表明,锚索振动失稳的范围取决于锚索的动静张力比、锚索的阻尼比和频率比(锚索参数激励频率与锚索1阶模态固有频率的比值);当频率比为1和2左右时,随着锚索动静张力比的增大和阻尼比的减小,锚索逐渐从稳定状态转变为不稳定状态,且锚索的不稳定区域逐渐增大。     

Head-wave parametric rolling of a surface combatant

Complementary CFD, towing tank EFD, and nonlinear dynamics approach study of parametric roll for the ONR Tumblehome surface combatant both with and without bilge keels is presented. The investigations without bilge keels include a wide range of conditions. CFD closely agrees with EFD for resistance, sinkage, and trim except for Fr>0.5 which may be due to free surface and/or turbulence modeling. CFD shows fairly close agreement with EFD for forward-speed roll decay in calm water, although damping is over/under predicted for largest/smaller GM. Most importantly CFD shows remarkably close agreement with EFD for forward-speed parametric roll in head waves for GM=0.038 and 0.033 m, although CFD predicts larger instability zones at high and low Fr, respectively. The CFD and EFD results are analyzed with consideration ship motion theory and compared with Mathieu equation and nonlinear dynamics approaches. Nonlinear dynamics approaches are in qualitative agreement with CFD and EFD. The CFD and nonlinear dynamics approach results were blind in that the actual EFD radius of gyration k_xx was not known a priori.     

Numerical modeling on hydrodynamic performance of a bottom-hinged flap wave energy converter

The hydrodynamic performance of a bottom-hinged flap wave energy converter(WEC) is investigated through a frequency domain numerical model.The numerical model is verified through a two-dimensional analytic solution,as well as the qualitative analysis on the dynamic response of avibrating system.The concept of "optimum density" of the bottom-hinged flap is proposed,and its analytic expression is derived as well.The frequency interval in which the optimum density exists is also obtained.The analytic expression of the optimum linear damping coefficient is obtained by a bottom-hinged WEC.Some basic dynamic properties involving natural period,excitation moment,pitch amplitude,and optimum damping coefficient are analyzed and discussed in detail.In addition,this paper highlights the analysis of effects on the conversion performance of the device exerted by some important parameters.The results indicate that "the optimum linear damping period of 5.0 s" is the most ideal option in the short wave sea states with the wave period below 6.0 s.Shallow water depth,large flap thickness and low flap density are advised in the practical design of the device in short wave sea states in order to maximize power capture.In the sea state with water depth of 5.0 m and wave period of 5.0 s,the results of parametric optimization suggest a flap with the width of 8.0 m,thickness of 1.6 m,and with the density as little as possible when the optimum power take-off(PTO) damping coefficient is adopted.     

Nonlinear analysis of traffic jams in an anisotropic continuum model

This paper presents our study of the nonlinear stability of a new anisotropic continuum traffic flow model in which the dimensionless parameter or anisotropic factor controls the non-isotropic character and diffusive influence. In order to establish traffic flow stability criterion or to know the critical parameters that lead, on one hand, to a stable response to perturbations or disturbances or, on the other hand, to an unstable response and therefore to a possible congestion, a nonlinear stability criterion is derived by using a wavefront expansion technique. The stability criterion is illustrated by numerical results using the finite difference method for two different values of anisotropic parameter. It is also been observed that the newly derived stability results are consistent with previously reported results obtained using approximate linearisation methods. Moreover, the stability criterion derived in this paper can provide more refined information from the perspective of the capability to reproduce nonlinear traffic flow behaviors observed in real traffic than previously established methodologies.     

Experimental and numerical hydrodynamic analysis of a stepped planing hull

This work addresses the experimental and numerical study of a stepped planing hull and the related fluid dynamics phenomena typically occurring in the stepped hull in the unwetted aft body area behind the step. In the last few years, the interest in high-speed planing crafts, with low weight-to-power ratios, has been increasing significantly, and, in such context, naval architects have been orienting toward the stepped hull solution. Stepped planing hulls ensure good dynamic stability and seakeeping qualities at high speeds. This is mainly due to the reduction of the wetted area, which is caused by the flow separation occurring at the step. This paper presents the experimental results of towing tank tests in calm water on a single-step hull model, which is the first model of a new systematic series. The same flow conditions are analyzed via Reynolds Averaged Navier-Stokes (RANS) and Large Eddy Simulations (LES), with different moving mesh techniques (overset/chimera and morphing grid), performed at different model speeds. The numerical results are in accordance with experimental data, and overset/chimera grid is found to be the best approach between the analyzed ones. The flow patterns obtained numerically through LES on a refined grid appear similar to the ones observed in towing tank investigations through photographic acquisitions. These flow patterns are dominated by a rather complex 3D arrangement of vortices originating from air spillage at both sides of the step. The understanding of these phenomena is important for the effectiveness of stepped hull designs.     

Influence of yaw-roll coupling on the behavior of a FPSO: An experimental and numerical investigation

An inconvenience in the experimental set-up of a FPSO in regular waves highlighted occurrence of parametric-roll events promoted by yaw-roll coupling and motivated a combined physical and numerical analysis on the relevance of this phenomenon on the roll resonance, as well as on the water shipping. The model tests examine the ship in head- and bow-sea waves in the zone of the first parametric resonance. Numerically, it is adopted a 3D Domain-Decomposition (DD) strategy combining a weakly-nonlinear potential-flow solver based on the weak-scatterer theory with a shallow-water approximation for the shipped liquid and with a bottom-slamming solution. Detailed comparisons against these and other seakeeping experiments validated the numerical method in its different aspects with global success.At first, a 2-dof equivalent linearized yaw-roll coupled system is examined and the measurements are used to estimate hydrodynamic coefficients required to complete the mathematical model of the problem. Then the DD method is applied to verify the instability occurrence and compared against the experiments. From the analysis, the parametric-roll instability does not occur if all nonlinearities in the roll restoring load are not accounted for. However the amplitude of the resonant roll is affected by the coupling with the other degrees of freedom. Especially the coupling with yaw tends to increase the steady-state roll amplitude. It also affects the water shipping with the trend in reducing its severity for the vessel, this is opposite to the influence of the parametric roll in head-sea waves on the water on deck, as documented in Greco et al. (2014) [4].     

The interceptor hydrodynamic analysis for controlling the porpoising instability in high speed crafts

A well-known instability of the high-speed planing crafts is the porpoising instability. This instability involves periodic, coupled heave/pitch oscillations possibly experienced in a planing vessel at high speeds. The porpoising can be controlled by using external devices. Interceptors are vertical blades installed symmetrically at the aft of the craft and have been introduced as a trim control appendage. Here, based on numerical methods and Savitsky porpoising theory, the effects of hydrodynamic interceptors on the porpoising control are investigated. Using computational fluid dynamics, the pressure distribution created by interceptor and its effects on porpoising are computed and then discussed. To model the flow around the vessel model, the Reynolds Average Navier Stokes (RANS) equations are applied. The work deals with craft with and without an interceptor at different heights. A dynamic grid mode involving two degrees of freedom is used. The results show that the interceptor causes an intense pressure at the stern bottom. It also decreases the trim and resistance of the vessel and increases the lift force coefficient which directly affects the porpoising instabilities. Based on the results, the interceptor can completely control the porpoising phenomenon.     

Hydrodynamic and dynamic analysis to determine the directional stability of an underwater vehicle near a free surface

A theoretical methodology to determine the open-loop directional stability of a near-surface underwater vehicle is presented. It involves a solution of coupled sway and yaw equations of motion in a manner similar to that carried out for surface ships. The stability derivatives are obtained numerically through simulation of motions corresponding to planar motion mechanism (PMM) model tests. For the numerical simulation, a boundary-integral method based on the mixed Lagrangian-Eulerian formulation is developed. The free-surface effect on the vehicle stability is determined by comparing the results with that obtained for vehicle motion in infinite fluid. The methodology was used to determine the stability of the Florida Atlantic University’s Ocean EXplorer (OEX) AUV. The presence of the free surface, through radiation damping, is found to suppress unsteady oscillations and thereby enhance the directional stability of the vehicle. With effects of free surface, forward speed, location and geometry of rudders, location of the center of gravity etc. all being significant factors affecting stability, a general conclusion cannot be drawn on their combined effect on the vehicle stability. The present computational methodology is therefore a useful tool to determine an underwater vehicle’s stability for a given configuration and thus the viability of an intended mission a priori.     

Experimental studies of a damaged ship section in forced heave motion

This work documents a detailed series of experiments performed in a wave flume on a thin walled prismatic hull form. The model consists of a rectangular opening located on the side. The length of the model is slightly smaller than the flume breadth to achieve two-dimensional (2D) behavior in the experiments. Forced oscillatory heave tests in calm water have been carried out by varying the model-motion parameters and examining both intact and damaged conditions. Video recordings, measurements of the wave elevation inside the damaged compartment and of the force on the model were performed in all the experiments. The effect of damage opening in the model on hydrodynamic loads is examined by comparing with an intact section. A theoretical analysis is used to explain the behavior of added mass and damping coefficients in heave for a 2D damaged section. The presented results demonstrate occurrence of sloshing and piston mode resonances in the tests and their influence on the hydrodynamics loads of a damaged ship. Detailed physical investigations are presented at these resonance frequencies for the damaged section. Effect of filling level in the damage compartment, damage-opening length and air compressibility in the airtight compartment is examined. Nonlinear effects are documented and appear dominant, especially, for lowest filling level where we have shallow-water depth conditions in the damaged compartment. Resonance phenomena that can lead to significant local loads are identified for the shallow water condition. Air compressibility in the airtight compartment and floodwater act as a coupled system and influence inflow/outflow of floodwater in the compartment. It has a significant effect on local floodwater behavior in the damaged compartment.