Roll Motion Equation Research Articles

Roll stability catastrophe mechanism of a flooded ship on regular sea waves

Based on a typical one-free-degree ship roll motion equation, the cusp catastrophe model is built including the bifurcation set equation, splitting factor ‘u’ and regular factor ‘v’, where both ‘u’ and ‘v’ are further expressed with typical flooded ship parameters. Then, the roll catastrophe mechanism is analyzed mainly by means ‘u’, under the given parameters of a typical trawler boat. The aim of this research is to reveal the mutagenic mechanism of the roll stability and provide a reference for improving ship roll stability.

Research on Roll Stabilizing Based on Energy Optimization for Autonomous Surface Vehicle

Considering the case of ASV (autonomous surface vehicle) navigating with low speed near water surface, a new method for design of roll motion controller is proposed in order to restrain wave disturbance effectively and improve roll stabilizing performance. Control system design is based on GPC (general predictive control) theory and working principle of zero-speed fin stabilizer. Coupling horizontal motion model of ASV is decoupled, and an equivalent transfer function of roll motion is obtained and transformed into a discrete difference equation through inverse Laplace transformation and Euler approximation. Finally, predictive model of GPC, namely, the difference equation of roll motion, is given. GPC algorithm of ASV roll motion is derived from performance index based on roll stabilizing performance and energy consumption used for driving fin stabilizer. In allusion to time-variant parameters in roll motion model, recursive least square method is adopted for parameter estimation. Simulation results of ASV roll motion control show better stabilizing performance and minimized energy consumption improved by self-adaptive GPC.

System Identification of Unmanned Planning Boat Rolling Motion Mode

In this paper, the author took an unmanned planning boat as the object of study and carried out a series of rolling decay ship model test by changing the draft. The author established nine kinds of mathematical model of rolling decay motion model system identification by using different damping and righting moment and established the optimization calculation of the objective function based on the principle of system identification. Then the author adapted the genetic algorithm of system identification program which is based on the Visual Basic 6.0 and got 15 kinds of identification programs. By doing research on the first three cycles of the series of rolling angular velocity curve and identifying respectively the resulting 15 kinds of identification programs, the author confirmed the feasibility of the adapted program. Comparing different drafts and the initial roll angle identification results, the author found a reasonable hydrostatic roll motion equation of the unmanned planning boat in the case of different drafts and the initial roll angle, and made a preliminary analysis.

Examination of the stability of trawlers in beam seas by using safe basins

The focus of this study is to examine the transverse stability of BSRA trawlers in beam seas using both numerical and analytical safe basin concepts and also shows how the concept of safe basin can be utilized for the assessment of the stability of trawlers. In the first part of this study, the nonlinear ship roll motion equation and the main parameters that induce ship capsizing in beam seas are analyzed using the concept of numeric safe basin. Then, the safe operating conditions of trawlers such as maximum allowable wave height and wind velocity are obtained for given loading conditions. In the second part of this study, analytic safe basins of nonlinear rolling motion are determined for given operating and loading conditions using the Lyapunov direct method. It is observed from the comparisons of analytically and numerically obtained safe basins that the analytically obtained safe basins are coherent with numerically obtained safe basins but beside this coherency analytically obtained safe basins are more conservative.

Nonlinear ship rolling motion subjected to noise excitation

The stochastic nonlinear dynamic behavior and probability density function of ship rolling are studied using the nonlinear dynamical systems approach and probability theory. The probability density function of the rolling response is evaluated through solving the Fokker Planck Equation using the path integral method based on a Gauss-Legendre interpolation scheme. The time-dependent probability of ship rolling restricted to within the safe domain is provided and capsizing is investigated from the probability point of view. The random differential equation of ships' rolling motion is established considering the nonlinear damping, nonlinear restoring moment, white noise and colored noise wave excitation.

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”.

Effect of Forward Speed on Ship Rolling and Stability

Ship stability is evaluated at zero speed most of the time. Majority of the stability criteria is also based on the behavior of ships at standstill. However unlike fixed offshore platfoims, ships are on the move due to their nature of operation. Therefore a ship's hydrostatic and hydrodynamic characteristics undergo changes because of the varying underwater volume, centers of buoyancy and gravity and pressure distribution. This work deals with the effects of forward speed on ship stability and motions, particularly on rolling motion in synchronous beam waves. An equation of nonlinear roll motion is chosen to calculate roll responses of a test vessel in beam seas. The speed of advance is incremented and roll responses are determined at each speed interval. Various characteristics of GZ curve for the selected test vessel are altered systematically to observe their effects on roll responses along with the forward speed. Several computer programs are also employed to handle colossal mathematical manipulations.

Estimation of coefficients of the equation of nonlinear roll motion for fishing boats by improved energy method and genetic algorithm

The present paper proposes two estimation methods for estimating not only the coefficients of roll damping moment but also the coefficients of roll restoring moment without calculation of hydrostatics. The first method is an energy method which involves expanding Proude’s method [5] to the case for which the restoring moment is nonlinear. The second method can identify the coefficients of arbitrary form of nonlinear damping and restoring moments by genetic algorithm. The advantage of these methods is that identification of a highly precise coefficient is very easy compared to other conventional methods. This advantage is most important in applications to practical problems. These methods are first applied to numerical simulation data and then to real free-decay data obtained from scale models.

Survival analysis of fishing vessels rolling in rough seas

A new approach to the problem of predicting the safety of vessels rolling in rough seas is described. It is based on the state of the art in nonlinear dynamics of a singledegree-of-freedom system. The random wave excitation depends on sea state, vessel speed and direction of wave propagation. The differential equation of rolling motion is integrated by the harmonic acceleration method. The procedure is illustrated for the case of a typical fishing vessel. The roll response of an intact and damaged vessel is presented in the time and frequency domain. The fractal erosion of the safe basin in the initial-value plane is analysed. Finally, these results are used to determine the probability of a vessel9s survival as a function of sea state, vessel speed and heading angle.

Bifurcations in ship rolling: experimental results and parameter identification technique

The steady rolling response in a regular beam sea of a 1 : 50 scale model of a destroyer in the bare hull condition has been studied. The restoring moment of the hull features strongly softening characteristics. This has led to the experimental evidence of bifurcations with jump in amplitude and phase during the tests at two different wave frequencies. The high quality measurements allowed the analysis of the gathered data through a single-degree-of-freedom roll motion equation. Exact numerical solutions have been used to obtain reliable values of the coefficients of the mathematical model to be used for the roll motion simulation. Approximate perturbative solutions have then been used to obtain a description of the domains of attraction of the system. An analysis of the stability of the steady state solutions in the range of frequencies where bifurcations and jumps were detected coupled with the analysis of the change of shape of the domains of attraction with the frequency allowed the differences found between the two physical situations to be explained.

Analysis and validation of ad hoc nonlinear roll motion models

To study the subject of nonlinear roll motion, which is the most difficult one to formulate, a number of mathematical models are used. These models emphasize different aspects of the motion and concentrate on particular characteristics of the phenomenon. Most of the time nonlinearities are introduced to the equation of motion through restoring and damping moments. In general, it is assumed that linearity of the inertia and exciting moment terms are not violated. In this study, an ad hoc nonlinear equation of nonlinear roll motion describing a ship’s rolling in beam waves at the resonance condition has been analyzed. By changing the nonlinear damping and restoring terms in the equation, ten different expressions are derived and solved by a form of harmonic balance method. Validity of the estimated nonlinear roll amplitudes obtained from the derived expressions are investigated through comparison with the experimental results of three ship models, which have been published previously.

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.

In-flight flexure and spin lock-in for antitank kinetic energy projectiles

A time-dependent analysis, coupling the rod vibration equation and pure roll motion equation, is formulated to numerically simulate the in-flight bending behavior of antitank kinetic energy projectile rods. The projectile is modeled as undergoing continuous, simulated planar pitching motion with the aerodynamic, spin, and structural damping forces included. The main parameter affecting the projectile spin and rod deflection responses is the fin torque producing the spin. Detailed spin, deflection, maximum stress, pitching angle, and rod shape histories are obtained along the projectile trajectory. Various steady-state spin cases are studied to simulate the effect of damaged fins. The spin lock-in phenomenon at the first lateral flexing natural frequency is captured by the present model for rods with simulated damaged fins. Any deformable projectile attempting, because of damaged fins, to spin past that frequency was found to lock-in to spinning at that frequency. Both suband supercritical spin cases are computed. Predictions of the rod shape at any instant and under actual flight conditions are made.

Numerical simulation of a ship capsizing in irregular waves

The capsize boundary in the initial value plane and excitation plane for an intact and a damaged ship are determined. The differential equation of rolling motion and the slope spectrum of irregular wave excitation are simplified. The time integration is performed by the harmonic acceleration method. The influence of random phase angles of irregular waves on fractal capsize boundary is explored.

Estimation of Ship Roll Parameters in Random Waves

The problem of estimating the parameters in an equation of roll motion from roll measurements only, taken in an irregular sea, is discussed. A single degree of freedom equation of motion is assumed, with a wide-band stochastic input and with a linear-in-the-parameters representation of both the damping and restoration terms. A method based on the Markov property of the energy envelope process, associated with the roll motion, is developed which enables all the relevant parameters to be estimated. The method is validated by applying it to some simulated data, for which the true parameters are known.

The Dynamics of Marine Vehicles With Inflectional Righting Moment Curves

This paper investigates modifications to the dynamics of a marine vehicle caused by a point of inflection in its hydrostatic righting moment curve. Such inflections are shown to yield multiple equilibrium angles due to static over turning moments and to the occurrence of a classical fold catastrophe. Both frequency domain analyses and numerical solutions of the equation of roll motion are used to evaluate the influence of such inflections on the vessel’s dynamics. The analyses demonstrate that inflectional righting moment curves can yield unexpected roll amplitude jump phenomena close to resonance. The paper concludes with a discussion of the effects of such behavior on vessel hull designs.

Computer-aided simulations of large amplitude roll motions of ships in waves and of dynamic stability

This paper deals with accounting for nonlinearities in damping and restoring terms in the development of a simplified but fundamental equation of roll motion. For the restoring term, this includes coupling with heave motion, a constant bias, and regular wave excitation. Both analytical-numerical and computer-aided methods are used to study roll response and the stability of the formulated nonlinear differential equation.

A theory for vortex shedding from the keels of marine vehicles

This paper presents a discrete vortex-shedding method for predicting the damping forces experienced by a floating marine vehicle responding at and around roll resonance. The method utilizes a Schwartz-Christoffel transformation, the temporal flow development being calculated to yield vortex positions, pressures and overall forces on the vessel hull. The effect of the velocity term in Bernoulli's equation on hull forces is calculated by integrating pressure and confirmed using the Blasius formula. A functional form of the vortex-induced moment amplitude is deduced from the theory and applied to a frequency-domain equation of roll motion for the vessel. Comparison of theory with test data indicates that the vortex-shedding theory does predict the effects of this phenomenon on the roll motions of a specific hull shape. An extensive review of previous work in this area is also presented.

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

Abstract 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.

Stability of Fishing Vessels with Water on Deck: A Review

A thorough literature survey of research into the effect of water on deck on fishing vessel stability revealed that four predominant analysis techniques have been employed. The first, a static approach, is used to judge vessel stability by predicting the pseudo-static angle of heel, the heel angle about which the vessel rolls when water is trapped on deck. The second approach is a dynamic one in which the equation of motion in roll is written and the water on deck is treated as a roll-inducing moment. The third approach employs probabilistic analysis to determine the likelihood of having water on deck, usually presented as a function of freeboard and sea state. A hydrodynamic analysis, applied in the research of Dillingham, models the geometry of the water on deck as a function of time and inputs the resulting forces and moments as an exciting term in the roll motion equation, assuming linear motions. The hydrodynamic approach represents a significant advancement in understanding the phenomenon and is a logical point from which to initiate further research efforts.