Ships Transverse Stability

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On the method of calculation of ship's transverse stability in regular wavesAlexander D. Pipchenko aa Department of Navigation, Odessa National Maritime Academy, Ukraine

First Published:March2009

To cite this Article Pipchenko, Alexander D.(2009)'On the method of calculation of ship's transverse stability in regular waves',Shipsand Offshore Structures,4:1,9 18

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Ships and Offshore Structures

Vol. 4, No. 1, 2009, 918

On the method of calculation of ships transverse stability in regular waves

Alexander D. Pipchenko

Department of Navigation, Odessa National Maritime Academy, Ukraine

(Received 3 September 2008; final version received 5 September 2008)

This article presents a method of determining a ships transverse stability in regular waves of an arbitrary direction. Themain problem due to which the development of this method was caused, is the observed onset of parametric rolling of theship, directly connected with the fluctuation of stability. For continuous determination of a varying upright moment in roughseas the configuration of immersed part of the hull and its volume should be known. The equation for the waterline draft ineach point of the ships hull situated on the wave profile with arbitrary trim and list was obtained. To solve this equation, thespecific hull form mapping techniques was developed (the ordinate of the ships cross section was presented as a functionof draft). Knowing the configuration of immersed hull part, its volume, position of the centre of buoyancy and the uprightlever can be determined. As an application example of described method, two mathematically calculated situations of heavy

rolling in head and following waves because of stability variation are presented.Keywords: ship; stability; upright moment; parametric rolling

Introduction

A Ships behaviour in seaway significantly depends on the

configuration of the immersed hull part. As a ship moves

through the waves, the water plane area changes continu-

ously, thus the pressure distribution on the immersed hull

part due to buoyancy forces is also changing. This affects

the values of dynamic loads on the hull and the restoring

force parameters. This is called the parametric effect, as the

changing wave profile varies the ships stability parameters,

which can create very violent rolling (parametric reso-nance, significant reduceof transverse stability in following

waves) in certain circumstances (Shin et al. 2004; Clark,

2008). Taking into account the significance of negative

consequences that might be caused by above mentioned

phenomena, the problem of determination of such regimes

of ship motion in which dangerous variation of stability on

the specified sea state may arise is important and topical.

On the other hand, to solve this problem it is necessary to

define how the stability will vary during ship motion in a

seaway.

Analysis of the existing methods

Sufficiently detailed methods of calculation of an upright

moment or upright levers in waves are described in pub-

lications of Boroday and Necvetayev (1982), Voytkunsky

(1985) and Nechaev (1989).

The most comprehensive equation for the transverse

upright moment is presented in Boroday and Necvetayev

(1982). However thedetermination of this is connectedwith

significant computation problems. Another, more practical,

method of static placement on the wave (Voytkunsky,

1985) can be implemented using Vlasovs integral curves,

but this is not always convenient when applying this method

using a computer. In addition, for ships with low length to

breadth ratio and high Froude numbers this method leads

to overestimated values of reduced upright levers on the

wavecrest, causedbyinterferenceof incident anddiffracted

waves, that can not always be estimated applying only the-

oretical methods.

The regression model for determination of the upright

lever for the ship proceeding in following waves on thebasis of series of physical models trials was developed by

Nechaev(1989). In this model the upright lever is presented

as a function of wave parameters, hull form parameters and

the Froude number. This model can be easily applied on

a computer; however, it has some limits. Mainly it regards

length to breadth ratio limitation, the model is adequate for

L/B = 3.2 8 (for the typical Panamax container carrier,

L/B = 262/32.2 300/32.2 = 8.13 9.32) and Froude

number limitation (0.15 0.45). In addition, the hull form

of a specified ship can be considered in the model only ap-

proximately, which can significantly influence the adequacy

of obtained results, especially when calculating parametri-

cal roll of a ship having significant forward-aft asymmetry

that is pitching.

Formulation of the research goal

The goal of this research is development of the method

of calculation of the transverse stability for a ship with

an arbitrary angular attitude and draft in regular waves of

an arbitrary direction, completely taking into account the

ships hull geometry.

ISSN: 1744-5302 print / 1754-212X online

Copyright C 2009 Taylor & Francis

DOI: 10.1080/17445300802402579

http://www.informaworld.com


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10 Alexander D. Pipchenko

The restoring moment MR for the ship arbitrarily situ-

ated on the wave profile can be given as:

MR = g ( , , zh) GZ ( , , zh) (1)

in which ( , , zh), GZ ( , , z

h) is the volume displace-

ment and the upright lever of the ship, depending on the

ships immersion and angular orientation on the wave pro-

file; g acceleration of gravity; density of seawater;

(, , ) vector of ship angular orientation (pitch, roll

& yaw angles); (a, , ) vector of wave parameters

(amplitude, length, spreading direction); zh vertical dis-

placement of the ships centre of gravity.

Knowing the position of the centre of gravity

(CG) (xCG, yCG, zCG) and the centre of buoyancy (CB)

(xCB , yCB , zCB ) an upright lever GZ for a particular list

angle can be found as:

GZ = yCB cos + zCB sin zCG sin (2)

The location of the CG in a body-fixed coordinate system

with a constant loading condition is also constant. At the

same time, location of the CB directly depends on the con-

figuration of the ships hulls immersed part. When the ship

is proceeding in a seaway, its position shifts to the side of

ships inclination with respect to vertical motion displace-

ment, pitch and roll angles. Therefore, to derive from static

formulation of the stability problem, the restoring moment

can always be found knowing the location of the CB.

The equation of ships hullwave profile intersection

curve

There are two coordinate systems that were applied in this

method:

body-fixed coordinates system with the centre Ob that is

the point of intersection of the base plane, the middle

plane and the centre plane of the ship. Axis Xb is positive

forward, Yb is positive to starboard andZb is positive

upwards;

moving coordinates system that is moving with the ships

mean velocity, its horizontal plane is parallel to the calm

water surface and submerged with the ships maximum

draft, the other two planes are perpendicular to it and

shifted with respect of the ships angular orientation.

To find the configuration of the immersed hull part in the

moving coordinates system, at first it is necessary to de-

termine the curve of intersection between the hull and the

wave profile.

When the ship is on an even keel without a constant list,

the deviation of a local draft from its calm water value on

the wave profile can be determined as follows:

T = a cos

e tk xb cos k yb sin

(3)

For the ship with initial list and trim in calm water, a

local deviation from the even keel draft can be written as

follows:

T = xb tg + yb tg. (4)

In equations (3) and (4)

T draft increment;

a wave amplitude;

e frequency of encounter; wave direction relatively to the ship (0 following

waves);

k wave number, k = 2 / and

P(xb, yb, zb) an arbitrary point on the hull surface in

the body-fixed system.

To define the enclosed intersection curve between the

ships hull and the wave profile, lets provisionally divide the

hull on M cross sections (usually 2030) and obtain next

equation for corresponding drafts from port and starboard

side of each cross section in body-fixed coordinate system:

Tspw

Tcw xb tg yb tg

+a cos

e t k xb cos k yb sin

= 0(5)

Tsw, Tp

wport and starboard side drafts in a given cross sec-

tion;

Tcw even keel/calm water draft.

In equation (5), the value ofxb is known for each cross

section. Therefore, (5) is the equation with two unknown

variables Tw andyb, which can be solved using the gradient

method.

Hull form mapping method

To solve equation (5) for an arbitrary value of port and star-

board drafts on the cross sections, it is necessary to obtain

an analytical representation of the hull form. However, the

classical conformal mapping may be ineffective, because

for each new draft the corresponding mapping coefficients

need to be recalculated. The most convenient within the

scope of this problem is to obtain the relation between

cross section draft and its breadth.

To obtain such a relation the neural networks theory

(Haykin, 2006) can be applied. For the individual cross


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Ships and Offshore Structures 11

section, the followingneural approximation wasdeveloped:

f (zb) =

Nn=1

vn th

zb

Tcw wn + bn

;

yb = max (0, f (zb)) max (Bs /2) ,

ifzmin < zb < Hf byb = 0, ifzb < zmin orzb > Hf b

(6)

In the equations above,

N neuron size, n = 1 , . . . N , mainly, N= 35;

w ,v ,b weight coefficients;

Hf b freeboard height or the height of the highest

deck superstructure in the cross section (if it is necessary to

define deck structures such as cargo holds hatches, accom-

modation, forecastle, etc.);

zmin appropriate, minimal elevation of a cross section

above the base plane.

The results of the ships hull cross section approxima-

tion using different mapping techniques, including one de-veloped by the author, neural mapping by draft, and by

parameter, are shown in Figure 1.

Solving the equation of ships hullwave profile

intersection curve

Using neural approximation (6), port and starboard side

drafts of each cross section can be found from equation (5)

using the gradient method.

Taking as draft, initial value Tw(0) = Tcw, each next

iteration it can be changed in the next manner:

Tspw (n) = Tsp

w (n 1)+ E(n) (7)

E(n) = Tspw (n)

Tcw xb tg yb tg

+a cos

e tk xb cos k yb sin

(8)

E output error; learning rate and

N number of iterations.

If = 0.5, the convergence of equation (5) to zero

and, consequently, the value ofTw can be obtained forn =

7, . . . , 20.

Finding the CB location

When the values of port and starboard drafts on each cross

section are obtained, using neural approximation (6), the

ensemble of points Sg= [Xg YgZg] of the immersed hull

surface on the wave profile by M cross sections in thebody-fixed coordinate system can be calculated. As secant

planes should be located in a moving coordinate frame, to

converse Sg next rotation matrix, excluding the yaw angle,

should be used (Fossen, 2002):

Rm =

cos() sin () sin () sin () cos ()0 cos () sin ()

sin () cos () sin () sin () cos ()

(9)

As the result the next ensemble of points in the moving

coordinate system will be obtained: Sm = Sg Rm.

The immersed part of the ships hull calculated by theabove mentioned method is shown in Figure 2.

0

2

4

6

8

10

12

0 5 10 15 20

Cross section

Lewis method

N-parameter

conformal

Neural with

parameter

Neural by draft

0

2

4

6

8

10

12

0 0.5 1 1.5 2 2.5 3

T= 5 m

zb, m zb,m

yb, m yb, m

Figure 1. Analytical mapping results for forward (left) and middle (right) cross sections of Panamax container carriers hull.


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Figure 2. Configuration of the immersed hull part: (a) calm water, even keel; (b) calm water with list and trim; (c) ship is on the waveprofile with list and trim.

As ordinates of conversed cross sections from the port

and starboard sides are different and the hull form lost its

port-starboard symmetry, to find the position of the CB two

methods can be applied. The first is to recalculate mapping

coefficients for the transformed cross sections and then to

calculate appropriate static moments and volume of the

immersed hull part, from which the CB coordinates can beeasily foundusing following equations (Voytkunsky, 1985):

xCB = Myz /V; yCB = Mxz /V; zCB = Mxy /V

V =

L/2L/2

zmaxzmin

(yR yL) dz dx (10)

Myz =

L/2L/2

zmaxzmin

(yR yL) x dz dx (11)

Mxz =

L/2L/2

zmaxzmin (y

R)2 (yL)2 dz dx (12)

Mxy =

L/2L/2

zmaxzmin

(yR yL) z dz dx (13)

Where yR andyL are y coordinates of thecorresponding

right and left contours of the cross section.

To simplify mathematical expressions the cross-

sectional coordinates in the moving system are given with-

out subscript m.

However, this method demands high computational ef-

forts. A comparatively faster way is to find the CB as the

intersection point of the three planes dividing the immersedpart of the ships hull into four parts of equal volume using

the gradient method. Taking into account that the vector of

buoyancy force is perpendicular to the water surface, it can

be concluded that horizontal secant plane is parallel to the

water surface and vertical and transverse secant planes are

perpendicular to it.

Lets define theextreme values of thehulls offsetsLX=

max(Xm); LY= max(Ym); LZ= max(Zm).

The coordinates of each cross section should be divided

into left- andright-side coordinates with the division point

situated in the minimum value ofzm.

Positions of the variable secant planes can be given as

follows:

LX = max (min (Xm) , LX HX) (14)

LY = max (min (Ym) , LYHY) (15)

LZ

= max (min (Zm) , LZ HZ) (16)Hn = Hn1 + V (17)

Where H step size for the gradient method;

V difference between volumes truncated by variable

secant plane;

n corresponding step number.

The vertical axis in the point ym= 0 cannot be used

to perform the integration as the left or right side of the

inclined hull may cross it. Therefore, the integration can be

performed relative to any arbitrary vertical axis that isnt

crossed by the inclined ships hull. Lets situate this axis in

the point LYthat is always the extreme starboard point. Forthe general case the cross-sectional area can be calculated

as follows:

A

L =

zLmaxzLmin

(LY yL)dz (18)

A

R =

zRmaxzRmin

(LY yR)dz (19)

A

R A

L, A

A

L, A

= A

L A

R

V = L/2

L/2

(A

L A

R)dx (20)

Volumes created separately by each variable secant

plane can be calculated by performing the double integra-

tion using the trapezium method.

1. YZ-plane. For this secant plane volumes aheadVfwdand

behindVaft it should be calculated. Cross-sectional areas

can be calculated using equations (18) and (19). The

increments of the volumes by ships length dependently

on LX position can be given as follows:

if xi > LXandxi1 LX


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dVfwd= (A

i + A

i+1) (xi xi1)/2 (21)

ifxi LX andxi1 < LX

dVaft = (A

i +A

i+1) (xi xi1)/2 (22)

if xi > LXandxi1 < LX

A

LX = A

i1 +(xi LX

) (A

i A

i1)

(xi xi1)(23)

dVfwd= (A

i + A

LX) (xi LX

)/2 (24)

dVaft = (A

i1 + A

LX) (LX xi1)/2 (25)

where i cross-section number.

Vfwd=

i

dVfwd; Vaft =

i

dVaft; (26)

V = Vfwd Vaft

2. XZ-plane. For this secant, plane volumes from the left

VL and from the right VRof it should be calculated. To

obtain the volumes increments the cross-sectional areas

from the left and from the right of the secant plane

should be calculated. These areas can be given as (see

Figure 3)

follows:

A = A

LY + A

RY

A

LY = A

LL A

RL (27)

A

RY = A

LR +A

LLR A

RRL A

RR

The increments of the cross-section areas by ships draft

dependently on LY position for the left-side offsets can

be given as follows:

if yLj1 < LY

andyLj LY

dA

LR =2LY yLj y

Lj1

zLj zLj1

/2 (28)

dA

LLR = (LY LY

) (zLj zLj1) (29)

if yLj1 < LY

and yLj > LY

zLY

j = zLj1 +

LY

yLj1

zLj z

Lj1

yLj yLj1 +

(30)dA

LR =2LY LY

yLj

zLj zLY

j

/2 (31)

+

LY LY

zLY

j zLj1

dA

LL = (LY

yLj1) (zLY

j zLj1)/2 (32)

Figure 3. Determination of the cross-sectional areas to the left and to the right of the secant plane LY.


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Figure 4. The roll angles of m/v Hanjin Pretoria (. . . .) and the GMvalues, calculated using corresponding roll periods ().

ifyLj1 > LY

andyLj < LY

dA

LR =2LY LY

yLj1

zLY

j zLj1

/2

+(LYLY

) (zLj zLY

j ) (33)

dA

LL =

LY

yLj

zLj z

LY

j

/2 (34)

if yLj1 > LY and yLj > LY

dA

LR =2LY yRj y

Rj1

zRj zRj1

/2 (35)

The calculation for right-side offsets should be per-

formed in a similar manner.

Subsequently,

VL =

L/2L/2

A

LYdx ; VR =

L/2L/2

A

RYdx ; (36)

V = VL VR

3. XY-plane. For this secant plane volumes below VD and

above VR should be calculated. Corresponding cross-

sectional area increments for the left-side offsets should

be calculated as follows:

if zLj1 < LZ

and zLj LZ

dA

LD =2LY yLj y

Lj1

zLj zLj1

/2 (37)

ifzLj1 < LZ and zLj > LZ

yLZ

j = yLj1 +

LZ

zLj1

yLj yLj1 +

zLj zLj1

(38)

dA

LD =2LY yLZ

j yLj1

LZ

zLj1

/2(39)

dA

LU =2LY yLZ

j yLj

zLj LZ

/2 (40)

if z

L

j1 LZ

and z

L

j > LZ

dA

LU =2LY yLj y

Lj1

zLj zLj1

/2 (41)

The calculation for right-side offsets should be per-

formed in a similar manner.

Subsequently:

VU =

L/2L/2

A

Udx ; VD =

L/2L/2

A

Ddx ; (42)

V = VU VD

Theconvergenceof thevolumes difference to zero when us-

ing the gradient method ( = 105) can be reached already

at n 510. As a result, we can obtain the CB location in

the moving coordinate system.

Taking into account the orthogonality of the rotation

matrix Rm, we can obtain the location of the CB in body-

fixed coordinate system:

CB (xCB, yCB, zCB) = CB (xm, ym, zm) RTm (43)


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Figure 5. Static stability curves and corresponding metacentric heights of container carrier Hanjin Pretoria. Key: o calm water; on three wave troughs; on three wave crests.

Observation and estimation comparison: Calculated

examplesTo check the adequacy of the method the estimated results

calculated for the Panamax container carrier m/v Hanjin

Figure 6. Ships stability in waves: (a) upright lever as function of time and list angle, length overall (LOA) = 200 m, B = 30.5 m, =170 m, = 40, H = 10 m andGM= 1.17 m; (b) upright lever as function of relative wave direction and list angle (on the wave crest),LOA= 200 m, B = 30.5, = 120 m, H = 8 m andGM= 1.17 m.

Pretoria were compared with the observations carried out

by the author. Ship parameters: L = 262 m, B = 32.2 m(L/B = 8.75), T = 10.5 m, V = 21 knots, GRT= 50252 m.

Ship was heading in following waves, Tmean = 8 s, h1/3 =


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Table 1. Transverse stability variation: Estimated and observed results.

The upright levers

List angle , 0 5 10 20 30 40 50 60On the trough L, m 0 0.098 0.224 0.623 1.13 1.47 1.32 0.81Calm water L, m 0 0.064 0.198 0.595 1.102 1.45 1.32 0.86

On the crest L, m 0 0.034 0.173 0.582 1.08 1.37 1.26 0.89Natural roll period,TR Wave period of encounter, Te

22 s 21 s

Estimated Observed

GMe GM+e GM

e GM GM

+ GM

0.7645 1.109 0.4141 0.7379 1.062 0.4141

3 m, = 307. The estimated metacentric heightGMe =

0.868 m.

The wave parameters were defined by the radar visually

and were compared with the corresponding meteorological

wave analysis charts.

The instant value of the metacentric height was defined

as function of instant rolling period, using formula (IMO,

2002):

GM=

2 C B

TR

2(44)

C = 0.373+ 0.023(B/ T) 0.043(L/100) (45)

In equation (43): L, B, T waterline length, mean draft

and moulded breadth.

The rolling angles of the ship and the instant valuesof the metacentric height calculated by equation (44) are

shown in Figure 4.

By GM values obtained from equation (44) the

mean value of the metacentric height can be calculated:

GMmean = 0.7379 m. Calculated mean square deviation of

the metacentric heightGM= 0.3237 m, thus, GM =

0.4141 m, GM+ = 1.062 m. As shownin Figure 4, the main

part of the GM(t) curve is limited by GM. Whereas,

calculation will be carried out on the regular waves with

parameters Tmean = 8 s, h1/3 = 3 m, values ofGM(t) bigger

or smaller, then set limits will be not taken into account as

they are induced by the waves irregularity.

Applying the prescribed method for the defined wave

parameters and ship load conditions, the stability curves in

calm water, on the wave trough and on the wave crest and

corresponding values of GM can be obtained (Figure 5).

The observed data and results of the estimation are given

in Table 1. It can be seen that the discrepancies in GM

calculations are small.

As illustrative example of the variation of the ships sta-

bility in waves, the results of corresponding calculations ofstatic stability curves for Handymax class container carrier

are shown in Figures 67.

In Figures 89 ship dynamics parameters in two situa-

tions calculated using a non-linear ship dynamics model

1.5

1

0.5

0

0.5

1

1.5

0 10 20 30 40 50 60 70 80 90

H= 0 m

H= 4 m

H= 8 m

H= 12 m

,

GZ, m

Figure 7. The static stability curves for the ship located on the wave crest for different wave heights (H wave height). L = 200 m, =170 m, = 0.


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Figure 8. Calculation of a parametric resonance onset in head waves. LOA = 200 m, B = 30.5 m, Dm = 10 m, GM= 1.2 m, TR =19.5 s; H = 10 m, = 240 m, = 180, Te = 10.5 s andVs = 7 knots.

Figure 9. Calculation of the ships motion in following waves. LOA = 200 m, B = 30.5 m, Dm = 10 m, GM= 0.8 m, TR = 26.8 s;H = 12 m, = 100 m, = 0, Te = 37.9 s andVs = 20 knots.


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with above mentioned method applied are shown. First

is severe parametric rolling in head waves, and second is

ships rolling with continuously reduced stability in follow-

ing waves.

Conclusion

This article presents the method of determination of aships transverse stability in regular waves of an arbitrary

direction. The comparison of observed and estimated data

shows the sufficient accuracy of this method.

However, the hydrodynamic aspects of the restoring

moment determination, such as Smiths effect and effect of

ship produced waves and free waves interference should be

deeply studied in further research in conjunction with the

developed calculation method.

Because the effects of parametric rolling and reduc-

tion of stability are significant for modern ships, the anal-

ysis of stability variation problem, especially the detailed

study of the onset conditions of the harmonic and paramet-ric rolling resonances should be perspective trend.

Acknowledgements

The author expresses his appreciation and gratitude to Navigationand Ship Theory chairs of Odessa National Marine Academy for

theoretical support in developing ship dynamics model, especiallyto Dr. V. Bondar for fruitful discussions and to Dr S. Petinov of St.Petersburg Polytechnic University, forhighlighting the importanceof carried out research.

References

Boroday IK, Necvetayev UA. 1982. Ships seaworthiness [In Rus-sian]. Leningrad (USSR): Sudostroenie.

Clark I. 2008. Stability: The parametric effect of waves on a shipsstability. Seaways The Int J NI. p. 2730, Feb 2008.

IMO. 2002. Code on Intact Stability for all types of ships coveredby IMO instruments. Resolution A. 749(18).

Fossen TI. 2002. Marine control systems. guidance, navigationandcontrol of ships, rigs andunderwater vehicles.Trondheim,Norway: Marine Cybernetics

Haykin S. 2006. Neural networks: A comprehensive foundation.2nd ed. NJ: Prentice Hall.

Nechaev UI. 1989. Modeling of the ships stability in waves. Rus-sian: Modern tendencies. Leningrad (USSR): Sudostroenie.

Shin YS, Belenky VL, Paulling JR, Weems KM, Lin WM.2004. Criteria for parametric roll of large containershipsin longitudinal seas. Paper presented at: SNAME AnnualMeeting, Washington (DC). http://www.eagle.org/NEWS/press/parametricroll.html.

Voytkunsky YI, editor. 1985. Ship static: Ship theory handbook.(Russian) Vol. 2. Ship Dynamics. Leningrad (USSR): Su-dostroenie.

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