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Modeling of Ship Roll Dynamics and Its Coupling with Heave and Pitch
In order to study the dynamic behavior of ships navigating in severe environmental conditions it is imperative to develop their governing equations of motion taking into account the inherent nonlinearity of large-amplitude ship motion. The purpose of this paper is to present the coupled nonlinear equations of motion in heave, roll, and pitch based on physical grounds. The ingredients of the formulation are comprised of three main components. These are the inertia forces and moments, restoring forces and moments, and damping forces and moments with an emphasis to the roll damping moment. In the formulation of the restoring forces and moments, the influence of large-amplitude ship motions will be considered together with ocean wave loads. The special cases of coupled roll-pitch and purely roll equations of motion are obtained from the general formulation. The paper includes an assessment of roll stochastic stability and probabilistic approaches used to estimate the probability of capsizing and parameter identification.
Generally, ships can experience three types of displacement motions (heave, sway or drift, and surge) and three angular motions (yaw, pitch, and roll) as shown in Figure 1. The general equations of motion have been developed either by using Lagrange's equation (see, e.g., [1–4]) or by using Newton's second law (see, e.g., [5–7]). In order to derive the hydrostatic and hydrodynamic forces and moments acting on the ship, two approaches have been used in the literature. The first approach utilizes a mathematical development based on a Taylor expansion of the force function (see, e.g., [8–12]). The second group employs the integration of hydrodynamic pressure acting on the ship's wetted surface to derive the external forces and moments (see, e.g., [13–18]). Stability against capsizing in heavy seas is one of the fundamental requirements in ship design. Capsizing is related to the extreme motion of the ship and waves. Of the six motions of the ship, the roll oscillation is the most critical motion that can lead to the ship capsizing. For small angles of roll motions, the response of ships can be described by a linear equation. However, as the amplitude of oscillation increases, nonlinear effects come into play. Nonlinearity can magnify small variations in excitation to the point where the restoring force contributes to capsizing. The nonlinearity is due to the nature of restoring moment and damping. The environmental loadings are nonlinear and beyond the control of the designer. The nonlinearity of the restoring moment depends on the shape of the righting arm diagram.
Abkowitz  presented a significant development of the forces acting on a ship in surge, sway, and yaw motions. He used Taylor series expansions of the hydrodynamic forces about a forward cruising speed. The formulation resulted in an unlimited number of parameters and can model forces to an arbitrary degree of accuracy. Thus, it can be reduced to linear and extended to nonlinear equations of motion. Later, Abkowitz [20, 21], Hwang , and Källströِm and Åström  provided different approaches to estimate the coefficients of these models. Son and Nomoto  extended the work of Abkowitz  to include ship roll motion in deriving the forces and moments acting on the ship. Ross  developed the nonlinear equations of motion of a ship maneuvering through waves using Kirchhoff's  convolution integral formulation of the added mass. Kirchhoff's  equations are a set of relations used to obtain the equations of motion from the derivatives of the system kinetic energy. They are special cases of the Euler-Lagrange equations. The derived equations also give the Coriolis and centripetal forces [27, 28].
Rong  considered some problems of weak and strong nonlinear sea loads on floating marine structures. The weak nonlinear problem considers hydrodynamic loads on marine structures due to wave-current-body interaction. The strong nonlinear problem considers slamming loads acting on conventional and high-speed vessels. Theoretical and numerical methods to analyze wave-current interaction effects on large-volume structure were developed. The theory is based on matching a local solution to a far-field solution. It is known that large-amplitude ship motions result in strongly nonlinear, even chaotic behavior . The current trends toward high-speed and unique hull-form vessels in commercial and military applications have broadened the need for robust mathematical approaches to study the dynamics of these innovative ships.
Various models of roll motion containing nonlinear terms in damping and restoring moments have been studied by many researchers [31–33]. Bass and Haddara [34, 35] considered various forms for the roll damping moment and introduced two techniques to identify the parameters of the various models together with a methodology for their evaluation. Taylan  demonstrated that different nonlinear damping and restoring moment formulations reported in the literature have resulted in completely different roll amplitudes, and further yielded different ship stability characteristics. Since ship capsizing is strongly dependent on the magnitude of roll motions, an accurate estimation of roll damping is crucial to the prediction of the ship motion responses. Moreover, the designer should consider the influence of waves on roll damping, especially nonlinear roll damping of large-amplitude roll motion, and subsequently on ship stability.
Different models for the damping moment introduced in the equation of roll motion were proposed by Dalzell , Cardo et al. , and Mathisen and Price . They contain linear-quadratic or linear-cubic terms in the angular roll velocity. El-Bassiouny  studied the dynamic behavior of ships roll motion by considering different forms of damping moments consisting of the linear term associated with radiation, viscous damping, and a cubic term due to frictional resistance and eddies behind bilge keels and hard bilge corners.
This paper presents the derivation of the equations of motion based on physical grounds. The equations of motion will then be simplified to consider the roll-pitch coupling, which is very critical in studying the problem of ship capsizing. It begins with a basic background and terminology commonly used in Marine Engineering. This is followed by considering the hydrostatics of ships in calm water and the corresponding contribution due to sea waves. An account of nonlinear damping in ship roll oscillation will be made based on the main results reported in the literature. The paper includes an overview of ship roll dynamic stability and its stochastic modes, probability methods used in estimating ship capsizing and parameter identification.
2. Background and Terminology
One needs to be familiar with naval architecture terminology. This includes key stability terms that are used in the design and analysis of navigation vessels and their structure components. A list of the main terms is provided in the appendix. Those terms described in this section are written in italics. The purpose of this section is to introduce the fundamental concept of ship roll hydrostatic stability.
A floating ship displaces a volume of water whose weight is equal to the weight of the ship. The ship will be buoyed up by a force equal to the weight of the displaced water. The metacenter shown in Figure 2(a) is the point through which the buoyant forces act at small angles of list. At these small angles the center of buoyancy tends to follow an arc subtended by the metacentric radius , which is the distance between the metacenter and the center of buoyancy . As the vessels' draft changes so does the metacenter moving up with the center of buoyancy when the draft increases and vice versa when the draft decreases. For small angle stability it is assumed that the metacenter does not move.
The center of Buoyancy is the point through which the buoyant forces act on the wetted surface of the hull. The position of the center of buoyancy changes depending on the attitude of the vessel in the water. As the vessel increases or reduces its draft (drawing or pulling), its center of buoyancy moves up or down, respectively, caused by a change in the water displaced. As the vessel lists the center of buoyancy moves in a direction governed by the changing shape of the submerged part of the hull as demonstrated in Figure 2(b). For small angles, the center of buoyancy moves towards the side of the ship that is becoming more submerged. This is true for small angle stability and for vessels with sufficient freeboard. When the water line reaches and moves above the main deck level a relatively smaller volume of the hull is submerged on the lower side for every centimeter movement as the water moves up the deck. The center buoyancy will now begin to move back towards the centerline.
As a vessel rolls its center of buoyancy moves off the centerline. The center of gravity, however, remains on the centerline. For small roll angles up to , depending on hull geometry, the righting arm is It can be seen that the greater the metacenter height the greater the righting arm and therefore the greater the force restoring the vessel (righting moment) to the upright position one. When the metacenter is at or very near the centre of gravity then it is possible for the vessel to have a permanent list due to the lack of an adequate righting arm. Note that this may occur during loading operations. A worst case occurs when the metacenter is located substantially below the center of gravity as shown in Figure 3. This situation will lead to the ship capsizing. As long as the metacenter is located above the center of gravity, the righting arm has a stabilizing effect to bring the ship back to its normal position. If, on the other hand, the righting arm is displaced below the center of gravity, the ship will lose its roll stability and capsize.
Hydrostatic and hydrodynamic characteristics of ships undergo changes because of the varying underwater volume, centers of buoyancy and gravity and pressure distribution. Another factor is the effect of forward speed on ship stability and motions, particularly on rolling motion in synchronous beam waves. Taylan  examined the influence of forward speed by incrementing its value and determining the roll responses at each speed interval. Various characteristics of the curve for a selected test vessel were found to change systematically.
The roll stability of a ship is usually measured by the stability diagram shown in Figure 4. The diagram shows the dependence of the righting arm on the roll angle (list) and is an important design guide for roll stability.
The roll oscillation of a ship is associated with a restoring moment to stabilize the ship about the -axis given by the expression where is the weight of water of displaced volume of the ship which is equal to the weight of the ship. If the ship experiences pitching motion of angle the righting arm will be raised by an increment . In the case the net roll moment becomes Note that the static stability is governed by the minimum value that the metacenter height, , should have and the shape of the static stability curve with respect to the roll angle. This approach is still being applied in the assessment of stability criterion. The dynamic stability approach, on the other hand, is based on the equation of rolling motion. This involves constructing a model for a ship rolling in a realistic sea. The linear restoring parameters can be easily obtained from ship hydrostatics.
The curve for righting arm, known also as the restoring lever, has been represented by an odd-order polynomial up to different degrees [42–45]. Different representations of the restoring moment have been proposed in the literature. For example, Roberts [46, 47], Falzarano and Zhang , Huang et al. , and Senjanović et al.  represented by the polynomial where , , , and for a damaged vessel, but for an intact vessel. Moshchuk et al.  proposed the following representation: where is the capsizing angle, and the function accounts for the difference between the exact function and .
3. Heave-Pitch-Roll Equations of Motion
Consider a ship sitting in its static equilibrium position with a submerged volume . During its motion, its instantaneous submerged volume is , and the difference in the submerged volume is . The inertial frame of axes is with unit vectors , , and along -, -, and -axes, respectively. On the other hand, the body frame that moves with the ship is with unit vectors , , and along -, -, and -axes, respectively. Figure 5 shows the instantaneous buoyant center located at point and the corresponding instantaneous force is . The weight of the ship is . In this case the instantaneous restoring hydrostatic force is The restoring moment is the resultant between the moments of weight and instantaneous buoyancy where is the volume of the infinitesimal prism of height , , is the center of mass location from , , , , is the vertical coordinate of the center of gravity, , and is the vertical coordinate of the center of gravity of the submerged volume, and are the coordinates of the elemental prism in the instantaneous plane with respect to the inertial frame . Substituting these parameters in (3.2) gives The elemental prism height can be written in terms of the heave displacement of the origin above the water level, the pitch, , and roll, , angles as The volume variation is The above summations are dependent on , , and . They represent the following geometric properties: In this case, one may write the volume variation in the form Note the above summations could have been replaced by integrals. The instantaneous restoring hydrostatic force given by (3.1) takes the form In scalar form, the absolute value of the restoring force is The summations in (3.3) can also be written in terms of (3.4) as where Introducing (3.10) into (3.3) and writing the result in the absolute and scalar form give Note that the geometrical parameters (3.6) and (3.11) depend on the instantaneous displacements of the ship . These properties may be expanded in multivariable Taylor series around the average position, that is, , , , and are the geometric properties evaluated at the average plane of floatation. Note that the variation of first moment of area about the -axis is dependent on an odd order of roll angle. That dependence does not exist in variations of other geometrical parameters. Paulling and Rosenberg  showed that the dependencies of the heave and pitch coefficients on roll are of even order, while the coefficients in roll due to heave and pitch are odd.
The restoring hydrodynamic force and moments given by (3.8) and (3.3) take the form In achieving the above equations use has been made of the following equalities verified by Neves and Rodríguez :
3.1. Wave Motion Effects
The influence of incident sea waves of arbitrary direction along the hull is to change the average submerged shape defined by the instantaneous position of the wave. These waves exert external forces and moments in heave, roll, and pitch in addition they introduce an additional restoring forces and moments. For the case of head sea, Neves and Rodríguez [11, 12] considered the Airy linear theory in representing longitudinal waves (along x-axis) defined by the expression  where is the wave amplitude, is the wave number, is the wave frequency, is the wave length, is the gravitational acceleration, is the wave incidence, and is the encounter frequency of the wave by the ship when the ship advances with speed .
Note that expressed by (3.4) should read The contributions of longitudinal waves to the restoring force, , and the restoring moments, and , obtained using Taylor series expansion about the average position up to third-order terms are given by the expressions [11, 12] where the derivatives of the above equations are given by the following expressions:
3.2. Ships Roll Damping
The surface waves introduce inertia and drag hydrodynamic forces. The inertia force is the sum of two components. The first is a buoyancy force acting on the structure in the fluid due to a pressure gradient generated from the flow acceleration. The buoyancy force is equal to the mass of the fluid displaced by the structure multiplied by the acceleration of the flow. The second inertia component is due to the added mass, which is proportional to the relative acceleration between the structure and the fluid. This component accounts for the flow entrained by the structure. The drag force is the sum of the viscous and pressure drags produced by the relative velocity between the structure and the flow. This type of hydrodynamic drag is proportional to the square of the relative velocity.
Viscosity plays an important role in ship responses especially at large-amplitude roll motions in which the wave radiation damping is relatively low. The effect of the bilge keel on the roll damping was first discussed by Bryan . Hishida [53–55] proposed an analytical approach to roll damping for ship hulls in simple oscillatory waves. The regressive curve of the roll damping obtained from the experiments by Kato  has been widely used in the prediction of ship roll motions. Since amplitudes and frequencies are varying in random waves, the hydrodynamic coefficients are time-dependent and irregular. Several experimental investigations have been conducted to measure the effect of bilge keels on the roll damping (see, e.g., [57–66]).
It was indicated by Bishop and Price  that existing information on the structural damping of ships is far from satisfactory. It cannot be calculated and it can only be measured in the presence of hydrodynamic damping, whose nature and magnitude are also somewhat obscure. Yet it is very important. Much less is known about antisymmetric responses to waves, either as regards the means of estimating them or the appropriate levels of hull damping. Vibration at higher frequencies, due to excitation by machinery (notably propellers), is limited by structural damping to a much greater extent than it is by the fluid actions of the sea. Damping measurements at these frequencies therefore give more accurate estimates of hull damping. The damping moment of ships is related to multiplicity of factors such as hull shape, loading condition, bilge keel, rolling frequency, and range of rolling angle. For small roll angles, the damping moment is directly proportional to the angular roll velocity. But with increasing roll angle, nonlinear damping will become significant. Due to the occurrence of strong viscous effects, the roll damping moment cannot be computed by means of potential theory. Himeno  provided a detailed description of the equivalent damping coefficient and expressed it in terms of various contributions due to hull skin friction damping, hull eddy shedding damping, free surface wave damping, lift force damping, and bilge keel damping. The viscous damping is due to the following sources.(i)Wave-making moment, .(ii)Skin-friction damping moment, .(iii)The moment resulting from the bare hull arising from separation and eddies mostly near the bilge keels, .(iv)Lift damping moment due to an apparent angle of attack as the ship rolls, .(v)Bilge-keel damping moment, .
Damping due to bilge keels can be decomposed into the following components.(i)Bilge keels moment due to normal force, .(ii)Moment due to interaction between hull and bilge keel, .(iii)Modification to wave making due to the presence of bilge keels, .
The damping components and are linear, while , , and are nonlinear. The linear and nonlinear damping moments can be expressed as follows: A pseudospectral model for nonlinear ship-surface wave interactions was developed by Lin et al. . The algorithm is a combination of spectral and boundary element methods. All possible ship-wave interactions were included in the model. The nonlinear bow waves at high Froude numbers from the pseudospectral model are much closer to the experimental results than those from linear ship wave models. One of the main problems in modeling ship-wave hydrodynamics is solving for the forcing (pressure) at the ship boundary. With an arbitrary ship, singularities occur in evaluating the velocity potential and the velocities on the hull. Inaccuracies in the evaluation of the singular terms in the velocity potential result in discretization errors, numerical errors, and excessive computational costs. Lin and Kuang [70, 71] presented a new approach to evaluating the pressure on a ship. They used the digital, self-consistent, ship experimental laboratory (DiSSEL) ship motion model to test its effectiveness in predicting ship roll motion. It was shown that the implementation of this roll damping component improves significantly the accuracy of numerical model results. Salvesen  reported some results pertaining numerical methods such as large amplitude motion program (LAMP) used to evaluate hydrodynamic performance characteristics. These methods were developed for solving fully three-dimensional ship-motions, ship-wave-resistance and local-flow problems using linearized free-surface boundary conditions. Lin et al.  examined the capabilities of the 3D nonlinear time-domain Large Amplitude Motion Program (LAMP) for the evaluation of fishing vessels operating in extreme waves. They extended their previous work to the modeling of maritime casualties, including a time-domain simulation of a ship capsizing in stern quartering seas.
The damping characteristics of a variety of ship shapes and offshore structures undergoing roll oscillation in the presence of ocean waves have been assessed by Chakrabarti . Chakrabarti  relied on empirical formulas derived from a series of model experiments reported by Ikeda  and Ikeda et al. [76–78]. These experiments were performed on two-dimensional shapes. The damping roll moment is nonlinear and may be expressed by the expression [40, 74] The first term is the usual linear viscous damping, the second is the quadratic damping term originally developed by Morison et al. . It is in phase with the velocity but it is quadratic because the flow is separated and the drag is primarily due to pressure rather than the skin friction. Sarpkaya and Isaacson  provided a critical assessment of Morison's equation, which describes the forces acting on a pile due to the action of progressive waves. The third term is cubic damping. The total damping may be replaced by an equivalent viscous term in the form where is the equivalent damping coefficient. This coefficient can be expressed in terms of the nonlinear coefficients as where is the wave frequency and is the amplitude of the ship roll angle.
Dalzell  replaced the nonlinear damping term by an equivalent smooth nonlinear polynomial given by where is the maximum amplitude of roll velocity. The numerical coefficients were estimated by using least-square fitting.
Haddara  employed the concept of the random decrement in the damping identification of linear systems. He extended the concept of the random decrement for a ship performing rolling motion in random beam waves. Wave excitation was assumed to be a Gaussian white noise process. The equations were used to identify the parameters of the nonlinear roll damping moment. Wu et al.  conducted an experimental investigation to measure the nonlinear roll damping of a ship in regular and irregular waves.
3.3. Ship Inertia Forces and Moments
The inertia forces and moments in heave, roll, and pitch motions are mainly due to the ship mass and mass moment of inertia and the corresponding added mass terms. These are well documented in Neves and Rodríguez  and are given in the form where is the ship mass, and are the ship mass moment of inertia about roll and pitch axes, is the hydrodynamic added mass in heave, is the hydrodynamic added inertia in heave due to pitch motion, inertia and are the hydrodynamic added polar mass moment of inertia about the ship roll and pitch axes, respectively, and is the added inertia in pitch due to heave motion. The added inertia parameters may be evaluated using the potential theory as described by Salvesen et al.  and Meyers et al. .
3.4. Governing Equations of Motion
Applying Newton's second law, the equations governing heave-roll-pitch motion may be written in a form.
The heave equation of motion is The roll moment equation of motion taking into account the beam sea hydrodynamic wave excitation moment, , is The pitch moment equation of motion taking into account the beam sea hydrodynamic wave excitation moment, , is where and are linear damping coefficients associated with heave and pitch motions, respectively. , , and are the external excitations due to sea waves. One can extract from the above three equations the coupled roll-pitch equations of motion or the purely roll equation of motion. Nayfeh et al.  described two different mechanisms that cause roll instabilities in ships. An approximate solution based on the method of multiple scales was presented together with different simulations using the Large Amplitude Motions Program (LAMP) code to determine linear parameters of the heave, pitch, and roll response. A methodology for nonlinear system identification that combines the method of multiple scales and higher-order statistics was also proposed.
3.4.1. Coupled Roll-Pitch Equations of Motion
Considering the coupled roll-pitch equations of motion, (3.27) and (3.28) take the form Note that the nonlinear coupling terms may result in nonlinear internal resonances among pitch and roll motions (see, e.g., [2, 3]).
3.4.2. Roll Equation of Motion
The prediction of ship stability during the early stages of design is very important from the point of a vessel's safety. Of the six motions of a ship, the critical motion leading to capsize is the rolling motion. Thus for studying roll stability in beam seas one should consider the nonlinear roll equation In formulating the roll equation in beam seas one should realize that the hydrodynamic roll moments on the ship are dependent on the relative motion of ship and wave, rather than upon the absolute roll motion. In a beam sea the relative roll is defined as , where is the local wave slope in a long-crested regular beam sea. In this case, the nonlinear equation of roll motion may be written in the form  where is the roll added inertia and is the bias moment created by several sources such as a steady beam wind, a shift of cargo, water, or ice on deck. Setting , (3.31) takes the form (see, e.g., ) Wright and Marshfield  solved (3.32) for small nonlinear restoring moment and small linear and cubic damping near the resonance frequency using three different approximate techniques: perturbation method, averaging method, and harmonic balance. Lin and Salvesen  presented an assessment of the Large Amplitude Motion Program (LAMP) for evaluating ship performance in extreme seas. The study included a time domain simulation of a ship capsizing in beam seas. It was shown that capsizing can happen due to dynamic effects even for ships that satisfy the minimum righting arm requirement. Surendran and Reddy [89, 90] evaluated the performance of a ship in beam seas using strip theory. The critical condition in the rolling motion of a ship is when it is subjected to synchronous beam waves (i.e., the encounter frequency coincides with the wave frequency). They considered various representations of damping and restoring terms to identify the effect of wave amplitude, wave frequency, and metacentric height (was represented by a quintic polynomial).
Contento et al.  reported some results of experimental tests on nonlinear rolling in a regular beam sea of a Ro-Ro ship model by varying both the wave steepness and the wave frequency. They adopted a parameter estimation technique based on the least squares fitting of the stationary numerical solution of the nonlinear rolling motion differential equation. It was possible to extract information 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. The analysis indicates also a marked dependence of the effective wave slope coefficient on wave amplitude. The effect of the excitation modeling on the fitting capability of the nonlinear roll motion equation to experimental data was studied by Francescutto et al. . Several frequency dependent and constant effective wave slope coefficients were derived for five different scale models corresponding to different ship typologies by a parameter identification technique. Later, Francescutto and Contento  studied the steady rolling response in a regular beam sea of a scale model of a destroyer in the bare hull condition. In view of the softening characteristics of the restoring moment, bifurcations with jump in amplitude and phase at two different wave frequencies were observed experimentally. Exact numerical solutions were used to obtain reliable values of the coefficients of the mathematical model to be used for the roll motion simulation.
Mahfouz  presented a robust method for the identification of linear and nonlinear damping and restoring parameters in the equation describing the rolling motion of a ship using only its measured response at sea. The parameters were identified using a combination of the random decrement technique, auto- and cross-correlation functions, a linear regression algorithm, and a neural-network technique. The proposed method would be particularly useful in identifying the nonlinear damping and restoring parameters for a ship rolling under the action of unknown excitations effected by a realistic sea.
3.5. Memory Effect
Note that the previous formulation did not account for the hydrodynamic memory effect. The hydrodynamic load due to the ship motion is a function of its frequency of oscillation. When the ship oscillates, waves will be generated on the free surface. As time increases, these waves will propagate outward from the body, but they continue to affect the fluid pressure and hence the body force for all subsequent times . In the time domain, this force or moment can be represented by a convolution integral of the impulse response function as outlined by Cummins , that is, where indicate surge, sway, and yaw, respectively. is the ship velocity along the axis , is the ship added mass, and is the retardation function and can be expressed in terms of the velocity potential function as where is the ith component of the normal vector of the surface element . Chung and Bernitsas  evaluated these forces in details.
A component of this force initiated at a certain moment continues to attribute its influence on the system for a period of time. This is referred to as the hydrodynamic memory effect . It was indicated that calculating this effect in the time domain is very time consuming. Tick  represented the convolution integral by a set of recursive differential equations with constant coefficients. These coefficients are determined by curve fitting the added mass and damping in the frequency domain. This method was used for estimating the memory effect on ship maneuvering by McCreight , a single-point mooring tanker by Jiang et al.  and Sharma et al. , and other motions by Schmiechen [101, 102].
4. Ship Roll Dynamic Stability
Traditional ship stability analysis compares the vessel righting arm curve to a standard or to a steady wind heeling moment (see, e.g., [103, 104]). Modern analysis methods of ship stability are based on analyzing the vessel's roll motion response either by simulation or using modern methods of dynamical systems. One method of analyzing nonlinear dynamical systems used by numerous researchers is the analysis of the so-called safe basin . This method consists of numerically integrating a grid of initial conditions in order to determine which initial conditions will lead to bounded motions (safe basin) and which will yield unbounded motions (i.e., capsizing). There are two cases that are generally analyzed and these are the unbiased and biased cases. The unbiased case involves the symmetric ship while the biased case is for an asymmetric ship. The asymmetry may be due to a steady wind moment, cargo shifting, or an asymmetric ice accretion.
The upper bound of the wave excitation amplitude, beyond which the ship becomes dynamically unstable, is governed by both the damping factor and the excitation frequency. This is reflected by the reduction of the safe basin area with the excitation amplitude. The stability fraction known in the literature as the normalized safe basin area or Safe Basin Integrity Factor [106–109]. It is usually obtained by estimating the ratio of the area of the stable region in the phase plane (area of the safe basin) to the total area encompassed by the homoclinic orbit, which is defined to be the safe basin in the absence of excitation. The stability fraction is strongly dependent on the excitation amplitude. For excitation amplitudes less than a critical value, governed by the excitation frequency, there is no erosion at all for the safe basin. Above this critical value of the excitation amplitude, the area of the safe basin shrinks and the stability fraction drops. As the excitation frequency changes so does the critical excitation amplitude.
Froude  observed that ships have undesirable roll characteristics when the frequency of a small free oscillation in pitch is twice the frequency of a small free oscillation in roll. It was Paulling and Rosenberg  who formulated the analytical modeling of the coupled roll-pitch motion. This coupled motion is described by a set of nonlinear equations. If the nonlinear effect of roll is neglected, the pitch equation of motion is reduced to a linear differential equation, which is free from roll motion terms. When pitch equation is solved, its response appears as a coefficient to the restoring moment in the roll equation of motion in the well-known Mathieu equation. Nayfeh et al.  analyzed the nonlinear coupling of the pitch and roll modes in regular seas when their frequencies are in the ratio of two to one. When the frequency of encounter (excitation frequency) is near the pitch frequency, the pitch mode is excited if the encountered wave amplitude (excitation amplitude) is small. As the excitation amplitude increases, the amplitude of the pitch mode increases until it reaches a critical small value. As the excitation amplitude increases further, the pitch amplitude reaches a saturated value, and all of the extra energy is transferred to the roll mode.
4.1. Stochastic Roll Stability
Sea waves are not sinusoidal and may cause severe or dangerous ship motions. The probabilistic theory of ship dynamics was documented by Price and Bishop  and Lloyd . If the nonlinear effect of roll is neglected, the pitch equation of motion is reduced to a linear differential equation, which is free from roll motion terms. When the pitch equation is solved, its response appears as a coefficient to the restoring moment of the roll motion, and the roll equation of motion is reduced to the Mathieu equation where is the roll angle, represents the pitch angle which is assumed to be a random stationary process, represents the wave random excitation, is a linear damping factor, is the natural frequency of the ship roll oscillation, and is a small parameter.
The analysis of the ship roll stability in an irregular sea was addressed by some investigators (see, e.g., [113–115]). The stochastic stability and response of the ship roll motion in random seas have been predicted analytically using the stochastic averaging method [46, 47, 49, 116]. Roberts  analyzed (4.1) for the stochastic stability and statistical response moments using the stochastic averaging method. The sample stability condition of the roll angle amplitude was obtained in the form where is the power spectral density of the random pitch process at frequency . Condition (4.2) reveals that the onset of instability is not affected by the forcing excitation . If this excitation is removed, the probability density function of the response degenerates into a delta function. The stability conditions of the first moment of the response amplitude is The stability condition of the second moment is These stability conditions apply only for the case of random sea waves in the absence of ice effects. Haddara  obtained the autocorrelation function of the roll motion in irregular seas.
4.2. Probabilistic Roll Dynamics
In addition to the modes of stochastic stability outlined in the previous subsection, it is important to examine the ship probabilistic description in random seas. It is of great importance to estimate the probability of capsizing. Equally important is to identify ship's parameters in roll motion. Different probability approaches have been found very effective in studying these issues. For example, the path integral technique was applied to the roll nonlinear motion of a ship in irregular waves by Kwon et al. . The exciting moment due to irregular waves was modeled as a nonwhite noise. Both damping and nonlinear restoring functions were included with the equivalent white-noise intensity. Lin and Yim  developed a stochastic analysis to examine the properties of chaotic roll motion and capsize of ships subjected to a periodic excitation with a random noise disturbance. They used a generalized Melnikov method to provide an upper bound on the domain of the potential chaotic roll motion. The associated Fokker-Planck equation governing the evolution of the probability density function of the roll motion was numerically solved by the path integral solution procedure to obtain joint probability density functions in state space. A chaotic response was found to take place near the homoclinic and heteroclinic orbits. The heteroclinic model emulates symmetric vessel capsize and the homoclinic model represents a vessel with an initial bias caused by water on deck. It was found that the presence of noise enlarges the boundary of the chaotic domains and bridges coexisting attracting basins in the local regimes. The probability of capsize was considered as an extreme excursion problem with the time-averaged probability density function as an invariant measure. In the presence of noise, the numerical results revealed that all roll motion trajectories that visit the regime near the heteroclinic orbit will eventually lead to capsize.
Another version of the path integration approach based on the Gauss-Legendre quadrature integration rule was proposed by Gu . It was applied for estimating the probability density of the nonlinear roll motion of ships in stochastic beam seas. The ship roll motion was described by a nonlinear random differential equation that includes a nonlinear damping moment and restoring moment. The results include the time evolution of the ship response probability density as well as the tail region, which is very important for the system reliability analysis. Gu  derived an approximate stationary probability density function and stationary mean out-crossing rate of the response of nonlinear roll-motion subjected to additive stochastic white noise excitations.
Yim et al. [122, 123] developed an analytical approach for the identification of ship parameters and calibration of their prediction capability using experimental results. They examined a three-degree-of-freedom fully coupled roll-heave-sway model, which features realistic and practical high-degree polynomial approximations of rigid body motion relations, hydrostatic and hydrodynamic forces and moments. System parameters of the model were identified using physical model test results from several regular wave cases. The predictive capability of the model is then calibrated using results from a random wave test case. Yim et al.  presented a computationally quasi-two-degree-of-freedom stochastic model describing the coupled roll-heave motions and a stability analysis of barges in random seas. Stochastic differential equations governing the evolution of probability densities of roll-heave and roll responses were derived using the Fokker-Planck formulation. Numerical results of roll responses using direct simulation in the time domain and the path integral solution technique in the probability domain were compared to determine the effects of neglecting the influence of heave on roll motion.
The case of small ships with water on deck subjected to random beam waves described by a periodic force and white noise perturbation was considered by Liu and Yougang  using the path integral solution. The random Melnikov mean square criterion was used to determine the parameter domain for the ship's stochastic chaotic motion. The evolution of the probability density function of the roll response was calculated by solving the stochastic differential equations using the path integral method. It was found that in the probability density function of the system has two peaks for which the response of the system was found to jump from one peak to another for large amplitudes of periodic excitation. Mamontov and Naess  developed a combined analytical-numerical approach referred to as the successive-transition method, which is essentially a version of the path-integration solution and is based on an analytical approximation for the transition probability density. The method was applied to a one-dimensional nonlinear Ito's equation describing the velocity of a ship maneuvering along a straight line under the action of the stochastic drag due to wind or sea waves. It was also used for the problem of ship roll motion up to its possible capsizing. It was indicated that the advantage of the proposed successive transition is that it provides an account for the damping matrix in the approximation.
Haddara and Zhang  developed an expression for the joint conditional probability density function for the ship roll angle and roll velocity in beam seas. The joint probability density function was expressed as a double series in the nondimensional roll angle and roll velocity. Jiang et al.  examined ships capsizing in random beam seas using the Melnikov function and the concept of phase-flux rates. Damping and wave excitation moments were treated as perturbations since they are relatively small compared with inertial effects and hydrostatic righting moments. Safe and unsafe areas were defined in the phase plane of the unperturbed system model to distinguish the qualitatively different ship motions of capsize and noncapsize. They derived expressions for the phase space flux rate. The correlation of phase space flux and capsize was investigated through extensive simulations. It was shown that these analytical tools provide reliable predictive information regarding the likelihood of a vessel capsize in a given sea state. Gu  and Tang et al.  employed the Melnikov function and phase space flux to examine the nonlinear roll motion of a fishing ship in random beam seas. They showed that the phase space flux is monotonically increasing as the significant wave height increases, while the safe basin is decreasing rapidly.
Liu et al.  considered some methods for constructing safe basins and predicting the survival probability of ships in random beam waves. The nonlinear random roll differential equation was numerically solved in the time domain by considering the instantaneous state of ships and the narrowband wave energy spectrum. The safe basins were constructed for safe navigation, and the survival probability of ships was also estimated. In another work, Liu et al.  considered the random differential equation of roll motion in beam seas and the random Melnikov mean-square criterion was used to determine the threshold intensity for the onset of chaos. It was found that ships undergo stochastic chaotic motion when the real intensity of the white noise exceeds the threshold intensity. The stable probability density function of the roll response was found to possess two peaks and the random jump happened in the response of the system for high intensity of the white noise excitation.
Reliability of ship operations under Gaussian or nonGaussian random sea waves deals with the probability that the ship will not capsize. One may estimate the ship reliability in terms of the probabilistic characteristics of the time at which the roll motion first exits from the safe domain. When capsizing is defined by the first exit of response from a safe domain of operation, the reliability is referred to as a first-passage problem. For ships whose response is described by a Markov process, the mean value of the exit time is usually governed by a partial differential equation known as the Pontryagin-Vitt (PV) equation .
The first-passage problem of nonlinear roll oscillations in random seas has been considered by Roberts [46, 47], Cai and Lin , Cai et al. , and Moshchuk et al. [51, 135]. Roberts [46, 47] developed an approximate theory based on a combination of averaging techniques and the theory of Markov processes. His analysis resulted in a simple expression for the distribution of the ship roll angle. Cai et al.  adopted the same modeling and introduced a parametric excitation term. They used the modified version of quasiconservative averaging. Moshchuk et al.  determined the mean exit time of the perturbed ship motion by solving Pontryagin's partial differential equation using the method of asymptotic expansion. It was found that the mean exit time is extremely large for any excitation intensity less than a critical value above which it experiences exponential decay.
5. Closing Remarks
The nonlinear dynamic modeling of ship motions in roll, pitch, and heave has been formulated based on physical ground. The formulation has been adopted from the work of Neves and Rodríguez [11, 12]. One can use the coupled nonlinear equations motion to examine only the ship motion in roll oscillations under regular and random sea waves. Other issues related to this modeling deal with the effect of roll damping and hydrodynamic memory effect arising from the ship motion. An overview of the roll dynamic stability under random sea waves has been presented in terms of the sample stability condition and response statistical moments. Equally important are the probability of capsizing and the identification of parameters in roll motion. Different probability approaches have been found very effective in studying these issues such as the path integral technique and the generalized Melnikov method, which provides an upper bound on the domain of the potential chaotic roll motion. This paper has not addressed the interaction of roll dynamics with floating ice and the reader can consult the review article by Ibrahim et al. .
Figure 6 clarifies some of the terminology defined in this Appendix.
Aft: toward the stern of the boat.
Beam: the width of a vessel also a structural component. Both Uses come from the Anglo-Saxon word beam, meaning, “tree”.
Beam Sea: sea coming on the side of the ship.
Bilge: the lower point of inner hull of a ship.
Bow: the forward part of a boat. The word may come from the Old Icelandic bogr, meaning “shoulder.”
Broach: the action of turning a vessel broadside to the waves.
Broadside: presenting the side of the ship.
Buoyancy: the upward push of water pressure, equal to the weight of the volume of water the ship displaces ().
Capsize: to turn over.
Center of Buoyancy (): the geometric center of the submerged hull, acting vertically upward.
Center of Flotation (): the geometric center of the waterline plane, about which the ship trims fore and aft.
Center of Gravity (): the center of all mass of the ship, acting vertically downward.
Displacement Volume (): the volume of the underwater hull at any given waterline.
Displacement (): the weight of water of the displaced volume of the ship, which equals the weight of the ship and cargo.
Draft: the depth of water a boat draws.
Fathom: six feet.
Following Sea: sea coming on the stern.
Forecastle: pronounced “fo'c's'l” and usually now spelled that way. Now the foredeck of a vessel, the term originally referred to a raised and fortified platform at the ship's bow. Used by archers in combat at sea as early as the 13th century.
Freeboard: that part of a ships sides above water, from the Anglo-Saxon framebord, meaning “the frame's side.”
Head: the uppermost or forward-most part of a ship (or of some specific part of a ship, such as the masthead, beakhead, stemhead, or whatever. The bathroom. In the age of sail, the crew was quartered forward in the forecastle, and their latrine was located on the beakhead, over hanging the water (for obvious reasons).
Heading: the direction in which a vessel's bow points at any given time.
Headway: the forward motion of a boat. Opposite of sternway.
Heel: constant roll angle—such as caused by a side wind or turning of the vessel.
Hull: the main body of a vessel.
Keel: the centerline of a boat running fore and aft; the backbone of a vessel.
Knot: a measure of speed equal to one nautical mile (6076 feet) per hour.
Lee: the side sheltered from the wind.
Leeward: the direction away from the wind. Opposite of windward.
Leeway: the sideways movement of the boat caused by either wind or current.
List, Heel, and Roll: it is both a noun and a verb referring to ships upping to one side or the other due to poor trim, shifting cargo, or sinking. The word comes from the Anglo-Saxon lystan, meaning “to lean.” Angular transverse inclinations. List describes a static inclination such as list due to side damage. Heel describes a temporary inclination generally involving motion, such as wind or turning, while roll indicates periodic inclination from side to side such as wave action.
Metacenter (): when the ship is inclined at small angles, the metacenter is the intersection of the buoyant force with the ship centerline. If the metacenter is above the center of gravity then the ship is stable.
Midship: approximately in the location equally distant from the bow and stern.
Nautical Mile: one minute of latitude; approximately 6076 feet: about 1/8 longer than the statute mile of 5280 feet.
Naval Architecture: ship design: especially hull design, overall layout with attention to stability, sea keeping, and strength.
Port: the left side of a boat looking forward.
Quarter: the sides of a boat aft of amidships.
Quarter Sea: sea coming on a boat's quarter.
Reserve Buoyancy: the watertight volume between the waterline and the uppermost continuous watertight deck.
Righting Arm (also Restoring Lever): it is the horizontal distance between the vertical line passing through the buoyant center and the vertical line passing through the ship center of gravity.
Starboard: the right side of a boat when looking forward.
Stern: the rear of any vessel. The word came from the Norse Stjorn (pronounced “Styorn”), meaning “steering.” It is the after part of the boat.
Thwartships: it means across the ship.
Trim: longitudinal tilt. Stern draft, bow draft.
Wake: moving waves, track, or path that a boat leaves behind it, when moving across the waters.
Waterline: a line painted on a hull which shows the point to which a boat sinks when it is properly trimmed.
Way: movement of a vessel through the water such as headway, sternway, or leeway.
Windward: toward the direction from which the wind is coming.
Yaw: to swing or steer off course, as when running with a quartering sea.
This work is supported by a grant from ONR under Award no: N00014-05-1-0040. Dr. Kelly B. Cooper is the Program Director.
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