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DESCRIPTIONPaper on ship stability in dynamic environment
Ocean Engineering 30 (2003) 13051317www.elsevier.com/locate/oceaneng
Numerical simulation of ship stability fordynamic environment
S. Surendran , J. Venkata Ramana ReddyDepartment of Ocean Engineering, Indian Institute of Technology Madras, Chennai 600 036, India
Received 28 March 2002; accepted 15 July 2002
The prediction of ship stability during the early stages of design is very important from thepoint of vessels safety. Out of the six motions of a ship, the critical motion leading to capsizeof a vessel is the rolling motion. In the present study, particular attention is paid to the perform-ance of a ship in beam sea. The linear ship response in waves is evaluated using strip theory.Critical condition in the rolling motion of a ship is when it is subjected to synchronous beamwaves. In this paper, a nonlinear approach has been tried to predict the roll response of avessel. Various representations of damping and restoring terms found in the literature areinvestigated. A parametric investigation is undertaken to identify the effect of a number ofkey parameters like wave amplitude, wave frequency, metacentric height, etc. 2002 Elsevier Science Ltd. All rights reserved.
Keywords: Stability; Capsizing; Metacentric height; Beam sea; Roll motion
Stability against capsizing in heavy seas is one of the fundamental requirementsin ship design. Capsizing is related to the extreme motion both of ship and waves.Rolling of a ship in rough environment may be influenced by many factors. Theycan be divided into three main situations; beam sea, following and quartering seaconditions. In the present study, the problem of ship safety has been studied withregard to the rolling motion of a ship in beam waves.
Bhattacharyya (1978) discussed rolling motion of a ship and the devices for roll
Corresponding author.E-mail addresses: firstname.lastname@example.org (S. Surendran); email@example.com (J. Venkata Ramana Reddy).
0029-8018/03/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved.doi:10.1016/S0029-8018(02)00109-9
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A44 added mass moment of inertia in rollAfv area under GZ curve up to the angle of fvB breadth of the vesselB44 damping moment coefficientBL linear roll damping moment coefficientBN nonlinear roll damping moment coefficientB(f ,f) nonlinear damping momentbN nondimensional nonlinear damping termbL nondimensional linear damping termC44 restoring moment coefficientCb block coefficientD depth of the vesselGM metacentric heightGZ righting armH wave heightI44 transverse mass moment of inertia of the vesselKB vertical position of the center of buoyancy above the baselineKG vertical position of the center of gravity above baselineKM vertical position of the metacenter above baselineL wave lengthLBP length between perpendicularsLOA length overallM0 amplitude of the wave exciting momentT draughtw wave frequencywe encountering frequencywf natural frequency of the roll motionf relative roll angle volume of displacement weight displacementam maximum wave slopel nondimensional inertia termfv angle of vanishing stabilityf angular velocityf angular acceleration
damping. Dalzell (1978) discussed about the representation of damping in differentnonlinear forms. Odabasi and Vince (1982) concentrated on the roll response of aship under the action of sudden excitation. They studied the importance of roll damp-ing on the response of a ship. Vassalos et al. (1985) explained stability criteria for
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semisubmersible stability. Lewis (1988) concentrated on rolling dynamics taking intoaccount the wave and other environmental effects. Witz et al. (1989) investigatedthe roll response of a semisubmersible model with an inflectional restoring moment.Zborowski and Taylan (1989) studied the small vessels roll motion stability reservefor resonance conditions. De Kat and Paulling (1989) investigated motions and cap-sizing of ships in severe sea conditions. Francescutto (2000) studied the problem ofship safety with regard to the stability and rolling motion of ships in beam waves.Taylan (2000) investigated the effect of nonlinear damping and restoring in shiprolling. Chakrabarti (2001) explained various types of damping associated with rol-ling. He contributed empirical relationships for the calculation of roll damping.
2. Formulation of the problem
For the purpose of analysis, only the significant motion pertaining to stability andcapsizing, namely roll motion has been considered. This simplification can be justi-fied by the reasoning that the vessel capsize is strongly influenced by the roll motion.In addition, among the three transverse coupled motions, only roll has restoringforces and exhibits strong resonant motions. Hence, roll motion can be consideredto be the most important in the stability analysis of a vessel.
The factors that influence roll response of different vessels are listed below:
the ratio between the natural period of rolling and the encountering period ofwave;
the shape of the hull, its stability, total weight and buoyancy; the wave steepness, H/L, where H and L are wave height and length, respectively; the damping efficiency of the underwater parts of the hull; the encountering speed of wave.
There are two ways of thinking with regard to the possible approach to the shipstability in wave.
Using the available linear ship motion theory. Using the nonlinear theoretical model.
2.1. Equation of roll motionlinear approach
One of the main reasons of ship capsizing in waves is loss of stability in rollmotion. A simplified analytical roll response model is assumed for the vessel whensubjected to regular sinusoidal waves. For a ship in regular beam sea, the rollingmotion can be simulated by a single degree of freedom second order differentialequation of the general form
(I44 A44)f B44f C44f M0coswet. (1)
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In this paper, linear equations of motion are solved for solutions at small angle ofroll. For larger angles, nonlinearity plays significant role in roll dynamics.
2.2. Equation of roll motionnonlinear approach
One of the problems associated with modeling nonlinear systems is the difficultyin establishing which of the nonlinear components are critical. The nonlinear modelinvolves two forms of nonlinearities:
the damping the restoring moment.
Assumptions made in the formulation of nonlinear rolling motion equation:
No coupling exists between roll and any other degrees of freedom. The added mass moment of inertia is approximately constant with frequency and,
therefore, the total inertia is constant. Forcing is harmonic.
A typical equation of nonlinear roll motion can be expressed as (Taylan, 2000)(I44 A44)f B44(f ,f) GZ(f) w2eamI44coswet. (2)
Eq. (2) is a relevant expression for roll motion prediction. In the present study,three different nonlinear damping and nonlinear restoring terms are considered. Theexpressions defining the damping are as follows:
B1(f) BLf BNf f , (3a)B2(f,f) BLf BNf2f , (3b)B3(f) BLf BNf3. (3c)
Cubic and quintic expressions are the most favorable descriptions for restoring,but it is not usual to come across a seventh degree polynomial. In general, higherdegree polynomials are avoided due to their relatively cumbersome manipulationsin the solution procedure. Let us consider an equation of nonlinear roll motion withB1 type damping and quintic restoring (Taylan, 2000)
(I44 A44)f BLf BNf f (C1f C3f3 C5f5) (4) w2eamI44coswet.
If Eq. (4) is divided throughout by (I44 A44), and the values of coefficients C1,C2, C3 are substituted, respectively. It takes the form
f bLf bNf f w2ff m3f3 m5f5 lw2eamcoswet, (5)where
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(I44 A44), (6a)
BL(I44 A44), (6d)
(I44 A44). (6e)
The righting arm curve is formulated as GZ C1f C3f3 C5f5 quintic poly-nomials. Coefficients of the polynomials are determined by static and dynamiccharacteristics of the GZ curve such as metacentric height, GM; angle of vanishingstability, fv; and area under the curve, Afv as follows:
df GM, (7a)
Eqs. (7a)(7c) are solved after a careful study of the righting arm curve for a parti-cular loading condition of the ship.
3. Results and discussions
For the purpose of analysis of linear and nonlinear rolling motion of ships, twovessels that differ in hydrostatic and stability characteristics have been considered.Figs. 1 and 2 show the body plan and isometric view of a RORO ship. Table 1shows the principal particulars of a RORO ship. The stability characteristics of theRORO ship, viz. GZ, GM, vanishing angle of stability and area under GZ curveare determined by well known Kryloves method. Fig. 3 shows the GZ curveobtained based on Kryloves method. Table 2 shows the comparison between theIMO criteria and the RORO ship. The curve of statical stability obtained based onKryloves method satisfies the IMO standards. Table 3 shows the principal particularsof a vessel taken from Zborowski and Taylan (1989).
The added mass moment of inertia of the vessel in rolling is assumed to be 20%of the mass moment of inertia. The damping moment coefficient has been calculated
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