Application of the extended Melnikov's method for single-degree-of-freedom vessel roll motion

Traditionally, when using Melnikov's method to analyze ship motions, the damping terms are treated as small. This is typically true for roll motion but not always true for other and/or multiple degrees of freedom. In order to apply Melnikov's method to other and/or multiple-degree-of-freedom motions, the small damping assumption must be addressed. In this paper, the extended Melnikov method is used to analyze ship motion without the constraint of small linear damping. Two roll motion models are analyzed here. One is a simple roll model with nonlinear damping and cubic restoring moment. The other is the model with biased restoring moment. Numerical simulations are investigated for both models. The effectiveness and accuracy of this method is demonstrated.     

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.     

Experimental investigation of the lift component of roll damping

The lift force and turning moment acting on a model towed obliquely to the direction of motion have been measured. Two models were used; one of them was tested fitted with and without a rudder. These measurements were used to determine the magnitude of the lift coefficient and the point of application of the transverse force acting on the model. The data were then used to determine the lift component of the roll damping moment. It has been found that the equivalent linear damping coefficient due to lift is a nonlinear function of the forward speed of the ship.     

Empirical calculation of roll damping for ships and barges

For a large floating structure in waves, the damping is computed by the linear diffraction/radiation theory. For most degrees of freedom, this radiation damping is adequate for an accurate prediction of the rigid body motions of the structure at the wave frequencies. This is not particularly true for the roll motion of a long floating structure. For ships, barges and similar long offshore structures, the roll damping is highly nonlinear. In these cases the radiation damping is generally quite small compared to the total damping in the system. Moreover, the dynamic amplification in roll may be large for such structures since the roll natural period generally falls within the frequency range of a typical wave energy spectrum experienced by them. Therefore, it is of utmost importance that a good estimate of the roll damping is made for such structures. The actual prediction of roll damping is a difficult analytical task. The nonlinear components of roll damping are determined from model and full scale experiments. This paper examines the roll damping components and their empirical contributions. These empirical expressions should help the designer of such floating structures. The numerical values of roll damping components of typical ships and barges in waves and current (or forward speed) are presented.     

Solution of the nonlinear roll model by a generalized asymptotic method

Mathematical modeling of the nonlinear roll motion of ships is one subject widely dealt with in nonlinear ship dynamics. This paper investigates setting up a form of nonlinear roll motion model and developing its periodic solution by the generalized Krylov–Bogoliubov asymptotic method in the time domain. In this model, nonlinearities are introduced through damping and restoring terms. The restoring term is approximated as a third-order odd polynomial whereas the quadratic term is favored to represent the nonlinear damping. The ship is assumed to be under the influence of a sinusoidal exciting force. Although the method is expressible to contain any order of the perturbing term, a single degree is chosen to avoid cumbersome mathematical complexity. In order to improve the solution a first-order correction term is also included. Moreover, a numerical example is carried out for a small vessel in order to validate the solution scheme.     

A study of the angle dependence of roll damping moment

The angle dependence of the roll damping moment is investigated by analysing experimentally obtained free roll decay records. Two ship models were used with and without bilge keels, also results with forward speed were obtained. The analysis indicate strong angle dependence and explains why the quadratic and cubic velocity dependent damping moments are successful in many cases.     

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.     

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.     

Recovering the functional form of the nonlinear roll damping of ships from a free-roll decay experiment: An inverse formulism

The purpose of this paper is to identify the functional form of the nonlinear roll damping for a particular ship based on an experiment. The problem of damping identification is formulated as an integral equation of the first kind. However, the solution of the problem lacks stability properties, due to the ill-posedness of the first-kind integral equation. To resolve this problem, a stabilization technique (known as a regularization method) is applied to the present problem of the identification of nonlinear damping. The identified results for nonlinear roll dampings are compared with those from a conventional roll identification method. The findings of the present study are validated by the direct comparison of experimental data on free-roll decay motion with the numerically simulated results.     

Parameter identification of nonlinear roll motion equation for floating structures in irregular waves

In order to predict the roll motion of a floating structure in irregular waves accurately, it is crucial to estimate the unknown damping coefficients and restoring moment coefficients in the nonlinear roll motion equation. In this paper, a parameter identification method based on a combination of random decrement technique and support vector regression (SVR) is proposed to identify the coefficients in the roll motion equation of a floating structure by using the measured roll response in irregular waves. Case studies based on the simulation data and model test data respectively are designed to validate the applicability and validity of the identification method. Firstly, the roll motion of a vessel is simulated by using the known coefficients from literature, and the simulated data are used to identify the coefficients in the roll motion equation. The identified coefficients are compared with the known values to validate the applicability of the identification method. Then the roll motion is predicted by using the identified coefficients. The prediction results are compared with the simulated data, and good agreement is achieved. Secondly, the model test data of a FPSO are used to identify the coefficients in the roll motion equation. Then the random decrement signature of the roll motion is predicted by using the identified coefficients and compared with that obtained from the model test data, and satisfactory agreement is achieved. From this study, it is shown that the identification method can be effectively applied to identify the coefficients in the nonlinear roll motion equation in irregular waves.     

Stochastic inversion for the roll gyradius second moment mass property in ships at full-scale and model-scale

In current Naval Architecture practice, employing static considerations is an important and necessary step in assessing ship stability and seakeeping properties (e.g. inclining experiments, load line regulations, range of stability calculations). However, damaged vessels and vessels operating in heavy weather or in conditions where topside icing is a concern may require an additional assessment of stability that considers dynamic effects. Within such contexts, the actual (i.e. current) second moment properties of the vessel mass become very important in the associated equations of motion for a given ship. One such critical second mass moment property is the roll gyradius, as it is closely related to the occurrence of capsizing. The present paper furnishes a means for reckoning the actual roll gyradius of a given ship operating within a seaway. The approach hinges on the formulation and solution of a stochastic inverse problem that leverages existing seakeeping software against the shipboard inertial measurement unit (IMU) telemetry. The method is demonstrated at full-scale and validated at model scale.     

Vortex Patterns and Roll Damping on Various Cross Sections of Ships

A numerical study is presented on roll damping of ships by solving Navier-Stokes equation.Two-Dimensional unsteady incompressible viscous flow around the rolling cylinders of various ship-likecross sections are numerically simulated by use of the computational scheme previously developed by theauthors.The numerical results show that the location of the vortices is very similar to the existing experi-mental result.For comparison of vortex patterns and roll damping on various ship-like cross sections,vari-ous distributions of shear stress and pressure on the rolling ship hull surface are presented in this paper.Itis found that there are two vortices around the midship-like section and there is one vortex around the foreor stern section.Based on these simulation results.the roll damping of a ship including viscous effects iscalculated.The contribution of pressure to the roll moment is larger than the contribution of frictionalshear stress.     

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.     

Impulse control method for hydraulic propulsion system used in 3500 m work-class ROV

In terms of the nonlinear characteristics of hydraulic propulsion system used in 3500 m rated work-class ROV (remotely operated underwater vehicle), the paper improved the responsiveness of the hydraulic propulsion system by adding an impulse signal to the input end of the system. Because the maximum static damping moment provided from water is much larger than the dynamic damping moment, it results in large dead zone of thrust during the startup process of the hydraulic propulsion system. The dead zone of thrust caused by static damping moment can be effectively reduced by adding a specific impulse signal to the input end of the propulsion system. The results of numerical simulations and underwater experiments show that using this method, the nonlinear characteristics of the hydraulic propulsion system have been significantly improved.     

Influence of wave approach angle on TLP's response

The current study focuses on the response analysis of triangular tension leg platform (TLP) for different wave approach angles varying from 0° through 90° and its influence on the coupled dynamic response of triangular TLPs. Hydrodynamic loading is modeled using Stokes fifth-order nonlinear wave theory along with various other nonlinearities arising caused by change in tether tension and change in buoyancy caused by set down effect. Low frequency surge oscillations and high frequency tension oscillations of tethers are ignored in the analysis. Results show that wave approach angle influences the coupled dynamic response of triangular TLP in all degrees of freedom except heave. Response in roll and sway degrees of freedom are activated which otherwise are not present in TLP's response to unidirectional waves. Pitch and roll responses are highly stochastic in nature indicating high degree of randomness. Variation in surge, sway and heave responses are nonlinear and are not proportional to change in wave height for the same period.     

CFD approach to roll damping of ship with bilge keel with experimental validation

The most common method of reducing roll motion of ship-shaped floating systems is the use of bilge keel which act as damping elements. The estimation of the damping introduced by bilge keel is still largely based on empirical methods. The present work adopts the CFD approach to the estimation of roll damping, both without and with bilge keel and validates the results with experiments conducted in a wave flume. Specifically, free oscillation tests are conducted at model scale to obtain roll damping, both by experiments and CFD simulation and reasonably good comparisons are obtained. The experiments also include PIV study of the flow field and attempt has been made to correlate the measured flow field with that obtained by CFD. The CFD methodology has the potential to determine rationally the size and orientation of bilge keels in design with reasonably accurate estimate of the additional roll damping that it provides to ship's roll motion.     

Calculation of a capsizing rate of a ship in stochastic beam seas

This study introduces a method of calculating a capsizing rate of a ship. The phenomenon ‘capsizing’ is described as a jump of local equilibrium point from that near the upright position of a ship to what describes the upside-down attitude of the capsized ship; the rate of occurrence of such jumps was calculated. The potential function corresponding to the roll restoring moment have two potential wells located at the roll displacement angle 0 and 180°, respectively. A nonlinear Fokker–Planck equation for the joint probability density function of roll angle and velocity was solved. The excitation to the ship was assumed to be a combination of a regular harmonic wave and a white noise process.     

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.