prof.surendran paper.pdf

Ocean Engineering 30 (2003) 13051317www.elsevier.com/locate/oceaneng

Technical Note

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

Abstract

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

1. Introduction

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: [email protected] (S. Surendran); [email protected] (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

1306 S. Surendran, J. Venkata Ramana Reddy / Ocean Engineering 30 (2003) 13051317

Nomenclature

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

1307S. Surendran, J. Venkata Ramana Reddy / Ocean Engineering 30 (2003) 13051317

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)


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


w2f GM

(I44 A44), (6a)

m3 4w2ff2v

3AfvGMf2v1, (6b)m5

3w2ff4v

4AfvGMf21, (6c)bL

BL(I44 A44), (6d)

bN BN

(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:

C1 d(GZ)

df GM, (7a)

C3 4f4v

(3AfvGMf2v), (7b)

C5 3f6v

(4AfvGMf2v). (7c)

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


Fig. 1. Body plan of a RORO ship.

Fig. 2. Isometric view of a RORO ship.

based on strip theory. For the analysis of linear rolling motion of a RORO ship, arange of wave heights varying from 2 to 8 m have been considered. Eq. (1) wassolved using the fourth order RungeKutta method. Fig. 4 shows a typical plot of rollangle against time for a particular wave height of 6 m and for different frequencies ofwave, ranging from 0 to 2.40 rad/s, in regular sinusoidal waves, in beam sea con-dition. Fig. 5 shows the plot of roll transfer function (RTF) against encounter fre-quency of wave (we). RTF is defined as the ratio of roll amplitude to the maximumwave slope. The roll response of the floating body is at a maximum in the lowerfrequency range. The resonance can be noticed at an encounter frequency value of0.4 rad/s. It is interesting to note that the present simulation agrees very well withthe method suggested by Bhattacharyya (1978). The published literature gives aclosed form solution to this linear dynamic problem.


Table 1Principal particulars of a RoRo ship

S. no. Principal particulars Symbol

1. Length overall LOA 192.60 m2. Length between perpendiculars LBP 177.60 m3. Breadth B 28.00 m4. Depth D 18.00 m5. Draught T 7.50 m6. Displacement 22012 t7. Transverse metacentric height GM 2.66 m8. Vertical center of gravity KG 13.98 m9. Vertical center of buoyancy KB 4.40 m10. Block coefficient Cb 0.60

Fig. 3. Restoring arm curve of a RORO ship.

Table 2Comparison between IMO criteria and the RORO ship

Norm IMO code The RORo ship

A 0.055 mrad 0.145 mradB 0.09 mrad 0.265 mradC 0.03 mrad 0.12 mradD 0.03 mrad 0.12 mradE 25 40F 0.15 m 2.66 m

A, area under GZ curve until 30; B, area under GZ curve until 40; C, area under GZ curve between30 and 40; D, maximum righting arm; E, angle of maximum stability; F, metacentric height.


Fig. 4. Time series plots of roll motion in beam sea condition (wave height of 6 m).


Table 3Principal particulars of test vessel

S. no Principal particulars Symbol

1 Length between perpendiculars LBP 29.83 m2. Breadth B 5.53 m3. Draught T 2.01 m4. Displacement D 200 t5. Transverse metacentric height GM 0.35 m6. Vertical center of gravity KG 2.30 m7. Vertical position of metacenter above the keel KM 2.65 m8. Block coefficient Cb 0.60

Fig. 5. Linear roll response for regular waves.

Table 3 shows the principal particulars of a test vessel taken from Ref. Zborowskiand Taylan (1989). Eq. (5) was solved using the fourth order RungeKutta method.Fig. 6 shows a typical plot of roll angle against time for a particular wave heightof 2.20 m and for different frequencies of wave, ranging from 0 to 2.40 rad/s, inregular sinusoidal waves, in beam sea condition. The encounter frequency variationinfluences the roll response of the ship. Fig. 6 shows the results for various fre-quencies of encounters. All are nonlinear roll responses, which are highly sensitiveto the initial conditions. Although single wave is considered for the purpose of com-parison, the method is found to be versatile for any conditions. Fig. 7 shows theresults as per present approach. It can be seen that the published results and thepresent results are matching. The resonance occurs at an encounter frequency valueof 0.92 rad/s.


Fig. 6. Time series plots of roll motion in beam sea condition (wave height of 2.20 m).


Fig. 7. Nonlinear roll response for regular waves.

Figs. 810 show the variation of roll amplitudes with B1, B2 and B3 type dampinggiven in Eqs. (3a)(3c), respectively, and quintic restoring moment for the ROROship with the principal particulars as per Table 1. The coefficients C1, C3 and C5have been calculated as per Eqs. (7a)(7c), respectively. Fig. 8 shows the nonlinearroll response with B1 damping for a regular sinusoidal wave of height 6 m in beamsea condition. The maximum roll angle is seen to be 17.5 at an encounter frequencyof 0.40 rad/s. This roll angle is well within the stability range of the RORO ship.The interaction with the nonlinear damping and nonlinear restoring moment is seenfrom the response. Fig. 9 shows the variation of roll amplitude with B2 damping.

Fig. 8. Roll amplitude with B1 type damping and quintic restoring moment.




In this case the roll response is raised to 22.5. Damping is seen to be lesser thanthe previous case. Fig. 10 is for B3 damping with quintic restoring moment. Themaximum roll angle is seen to be almost same as the case of B2 damping. Therefore,the effect due to B2 and B3 damping is found to be the same. Further simulationcarried out for higher wave heights shows larger roll responses, but they are wellwithin the range of stability. The dramatic changes in roll response are due to thecross-sectional shape of the ship, bilge keels, etc. They are all dampening the rollmotion. Hence, B1, B2 and B3 justify the representation for damping.


4. Conclusions

Analytical methods are formulated for the roll motion of ship. Linear and nonlinearapproaches are tried. Solutions are obtained using the fourth order RungeKuttamethod. Three types of possible damping are considered based on the literature sur-vey. The restoring arm curve is represented by quintic polynomial. The result of thepresent approach very well agrees with the published results. A number of caseshave been worked out.

References

Bhattacharyya, R., 1978. Dynamics of Marine Vehicles. Wiley, New York.Chakrabarti, S., 2001. A technical note on empirical calculation of roll damping for ships and barges.

Ocean Engineering 28, 915932.Dalzell, J.F., 1978. A note on the form of ship roll damping. Journal of Ship Research 22, 178185.De Kat, J.O., Paulling, J.R., 1989. The simulation of ship motions and capsizing in severe seas. Trans-

actions SNAME 117, 127135.Francescutto, A., 2000. Mathematical modeling of roll motion of a catamaran in intact and damage con-

ditions in beam waves. In: Proceedings of the Tenth International Offshore and Polar EngineeringConference, Seattle, USA, May 28June 2, 2000. SNANE, New Jersey, pp. 362368.

Lewis, N., 1988. Principles of Naval Architecture, third ed. Wiley series.Odabasi, A.Y., Vince, J., 1982. Roll response of a ship under the action of a sudden excitation. Inter-

national Shipbuilding Progress 29, 327328.Taylan, M., 2000. The effect of nonlinear damping and restoring in ship rolling. Ocean Engineering 27,

921932.Vassalos, D., Konstantopoulos, G., Kuo, C., Welaya, Y., 1985. A realistic approach to semisubmersible

stability. Transactions SNAME 93, 95128.Witz, J.A., Albett, C.B., Harrison, J.H., 1989. Roll response of semisubmersibles with nonlinear restoring

moment characteristics. Applied Ocean Research 11, 153166.Zborowski, A., Taylan, M., 1989. Evaluation of small vessels roll motion stability reserve for resonance

conditions. In: SNAME Spring meeting/STAR Symposium, New Orleans, LA.

Numerical simulation of ship stability for dynamic environmentIntroductionFormulation of the problemEquation of roll motion-linear approachEquation of roll motion-nonlinear approach

Results and discussionsConclusions

References