Analytical calculation of uncoupled heave and roll, A parametric study of a barge.pdf

7/21/2019 Analytical calculation of uncoupled heave and roll, A parametric study of a barge.pdf

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Analytical calculation of

uncoupledheaveandrollAparametricstudyofabarge

TorEdvardSfteland

OFF600MarineOperations

Fall2012


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ABSTRACT

The theory of how vessels behave and their additional response at sea is called seakeeping

and is of great importance in all kinds of marine operations. The response is normally

obtained by different software using a finite element approach, but to understand and change

the properties of a vessel, one should have a general understanding of how the responserelates to different factors. The response of a vessel is divided into six degrees of freedom,

three linear and three rotational. In this report, one linear (heave) and one rotational (roll)

response will be in focus where the other four degrees of freedom can be calculated by similar

approaches as what is shown in this report. After analyzing the heave and roll motion of a

vessel with certain assumptions, the formulas for the motion of a barge is derived and the

additional assumptions is discussed thoroughly. The resulting motion can be calculated by

few easy obtainable parameters which are found in ship certification documents and weather

statistics. The roll response where calculated to be 30% larger than the response obtained by a

wave tank test of the same vessel due to necessary approximations described in this report.

The obtained formulas can be used to show the impact of change in vessel or sea parametersand should be used to compare different vessel motions.


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Table of content

1.INTRODUCTION.............................................................................................................. 82.STATEOFART.............................................................................................................. 103.METHODS.................................................................................................................... 12

3.1VESSELDEGREESOFFREEDOM..............................................................................................12

3.2UNCOUPLINGOFHEAVEANDROLL........................................................................................13

3.2.1 Lineartheory....................................................................................................13

3.2.2 Coupledmotions..............................................................................................13

3.2.3 Symmetricandanti symmetricmotions.........................................................14

3.3EQUATIONOFMOTION........................................................................................................14

3.3.1 Solutionsofequationofmotion......................................................................15

4.3.2 Particularsolution...........................................................................................15

3.3.3 Dynamicamplificationfactor..........................................................................16

3.3.4 Homogeneoussolution....................................................................................17

3.3.5

Phaseangle......................................................................................................

18

3.4HARMONICRESPONSEOFAVESSEL........................................................................................19

3.4.1 Dynamicamplificationofavessel...................................................................19

3.4.2 Addedmassforce............................................................................................20

3.4.3 Dampingforce.................................................................................................21

3.4.4 Restoringforce.................................................................................................22

3.4.5 Excitingforce...................................................................................................22

3.4.6 Striptheory......................................................................................................24

3.5HEAVEMOTION.................................................................................................................25

3.5.1 Equationofmotion..........................................................................................25

3.5.2 Addedmassinheave.......................................................................................25

3.5.3

Dampingin

heave

............................................................................................

27

3.5.4 Restoringforceinheave..................................................................................28

3.5.5 Excitingforceinheave.....................................................................................29

3.5.6 Naturalperiodinheave...................................................................................34

3.5.7 Solution............................................................................................................34

3.6ROLLMOTION....................................................................................................................38

3.6.1 Equationofmotion..........................................................................................38

3.6.2 Massmomentofinertiaandaddedmassmomentofinertiainroll...............38

3.6.3 Dampinginroll................................................................................................42

3.6.4 Restoringforceinroll.......................................................................................42

3.6.5 Excitingforceinroll.........................................................................................45

3.6.6

Naturalperiod

in

roll

.......................................................................................

46

3.6.7 Solution............................................................................................................47


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List of symbols

az Heave, added mass [kg]D2

za Heave, added mass for each strip [kg/m]

A Roll, added mass moment of inertia [kgm2]

D2A Roll, added mass moment of inertia for each strip [kgm2/m]

Aw Waterline area [m2]

bz Heave, damping coefficient [kg/s]D2

zb Heave, damping coefficient for each strip [kg/sm]

bzC Heave, critical damping [kg/s]

B Roll, damping coefficient [kgm2/s]

D2

B Roll, damping coefficient for each strip [kgm2

/sm]B Breadth of vessel [m]

BM Metacenter radius [m]

cz Heave, restoring force coefficient (stiffness) [N/m]

C Roll, restoring force coefficient [Nm2/m]

CA Added mass coefficient [-]

CB Block coefficient [-]

CW Waterline coefficient [-]

D Draft of vessel [m]DAFz Heave, dynamic amplification factor [-]

DC Characteristic dimension [m]

FA Approaching wave force [N]

FD Diffraction wave force [N]

Fz Heave, exciting force [N]

F0 Heave, exciting force amplitude [N]

g Gravity acceleration [m/s2]

GM Metacenter height [m]

I Mass moment of inertia [kgm2]

k Wave number [1/m]

KB Center of bouyancy [m]

KG Center of gracity [m]

L Length of vessel [m]

LW Wave length [m]

M Mass [kg]

Meq Equivalent mass [kg]

M Roll, exciting force [Nm]


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AM Approaching wave force [Nm]

M0 Roll, exciting force amplitude [Nm]

pD Dynamic water pressure [N/mm2]

rz Heave, frequency ratio [-]

r Roll, frequency ratio [-]

T Wave period [s]

Tnz Heave, natural period [s]

nT Roll, natural period [s]

U Vessel forward speed [m/s]

zw Vertical particle acceleration [m3]

z Heave, vessel motion [m]

z Heave, vessel velocity [m/s]

z Heave, vessel acceleration [m/s2]

Z Heave, amplitude of particular solution [m]

Z0 Heave, amplitude of homogeneous solution [m]

zp Heave, particular solution [m]

zh Heave, homogeneous solution [m]

Roll, vessel motion [-]

Roll, vessel velocity [1/s]

Roll, vessel acceleration [1/s2]

Submerged volume [m3]Density of seawater [kg/m3]

Vessel angle (deviation from x-axis) [rad]

i Roll added mass moment of inertia coefficient factor [-]

z Heave, phase amplitude particular solution [-]

0z Heave, phase amplitude homogeneous solution [-]

a Amplitude of radiating waves [m]

0 Linear wave amplitude [m]z Heave, damping frequency [-]

Roll, damping frequency [-]

Wave frequency [1/s]

e Encounter frequency [1/s]

dz Heave, damped (reduced) natural frequency [1/s]

nz Heave, natural frequency (eigen frequency) [1/s]

n Roll, natural frequency (eigen frequency) [1/s]


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1. Introduction

When doing operations at sea, it is important to understand and predict the huge

environmental forces which are present. The forces are difficult to anticipate, and its even

harder to calculate the response of an operating vessel. The vessel can be considered as an

oscillating system, with six degrees of freedom. By using certain assumptions, the

information needed to perform the calculations is narrowed to a few easy obtainable factors.

Those factors are found in ship certification documents and weather statistics.

The response of vessels at sea is described by three linear and three rotational degrees of

freedom. In this report one linear (heave) and one rotational (roll) response will be in focus.

The combination of heave and roll is important when calculating loads on cargo and

additional seafastening when cargo is transported at sea.

When operating at sea, there are three different areas that need to be taken into consideration.(TMC Marine consultants (2008))

Estimation of environmental conditions encountered by the vessel. This is based onhindcast or predicted weather data regarding wind, wind waves, and swell waves.

Determination of the vessels response characteristics. Operation criteria based on allowable accelerations for cargo, people, deck wetness,

etc.

In this report, focus is drawn towards the vessels response which is determined by choosing

some wave parameters based on North Sea environment.

Through a structured setup and well defined formulas, the analysis part of the report clearly

identifies the similarity between the rotational and linear motion. When assuming that heave

and roll is unaffected by each other, the dynamic response of heave and roll can be described

with the equations of motion with similar terms in both motions.

In addition to the analysis, the results are presented with numerical simulations and

parametric studies. The values are discussed and compared with relevant data obtained from a

wave tank test of a Standard North Sea Barge.


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2. State of art

Dynamic vessel response is normally calculated using finite element software program like

Maxsurf, VERES, Seaway or MOSES. These computer programs use the finite element

approach called strip theory. The strip theory is an approach where the vessel is divided intocertain amount of two-dimensional strips along the longitudinal axis, where the

hydrodynamic loadings on each strip are considered and summarized over the length of the

vessel. (J ourne (2001) p.5) These programs have proven to give accurate results for how

vessels behave at sea.

When using finite element software to calculate allowable vessel response in terms of

operation criteria, certain data is required to get desired results. (TMC Marine consultants

(2008))

Principal particular of vessel (dimensions, displacement, draught, trim) Appendages (rudder, bilge keels, stabilizers) Loading conditions (GM) Centre of Gravity (KG, LCG) Radii of Gyration (for rotational degrees of freedom) Vessel speed through water Wave spectrum type Wave parameters (height, period, regular/irregular) Longitudinal weight distribution (for wave induced shear, moment and torsion)

By inserting the needed information into the relevant software, one is able to find the relevant

vessel response and vessel accelerations. Limit criteria for different operations is normally

given as maximum acceleration limits. The acceleration limits is chosen based on what kind

of operation that is considered and what kind of work that are conducted on the vessel (light

manual work, heavy manual work, etc.).

Of the possible software outputs, the response amplitude operator (RAO) is considered the

most important output of a normal vessel analysis. The RAO is also called the transfer

function and describes how the response of the vessel varies with frequency. By using the

given RAO for a vessel one is able to find out to what extent external forces with different

frequencies affect the response of the vessel. (Formsys (2012)) The RAO peek tells us which

frequencies that result in potential resonance of the system.

The different computer programs are developed by using analytical approaches combined

with model testing. The combination of surge, sway, heave, roll, pitch and yaw are affecting

each other, and some of the motions cannot be considered with a linear theory approach.

When assuming linear theory, it is possible to summarize all terms that is relevant for the

motion to get the final response. Roll motion is best described as a non-linear motion with

half empirical formulas as a result of model testing. (Journe J.M.J . (2001) p.5)


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When a vessel is moving at sea, the formulation of the resultant vessel motion of each of the

degrees of freedom is considered as simple harmonic functions with an amplitude, a phase

shift and a frequency. (Journe J.M.J . (2001) p.8)

Surge: )tsin(Xx xep Sway: )tsin(Yy yep Heave: )tsin(Zz zep Roll: )tsin( ep Pitch: )tsin( ep Yaw: )tsin( ep

Where the frequency term, e is the encounter frequency which describe how fast the waves

are approaching a moving vessel.

When obtaining the six harmonic functions, one can easily find the velocity and acceleration

functions by differentiating the particular solution one and two times respectively.


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3. Methods

When doing operations at sea, it is of great importance to understand how vessels are affected

by environmental forces at sea. How the ship is affected is described by vessels response in

six degrees of freedom system, where the response of each of these motions must becalculated to find the resultant motions of a vessel. The theory of how a ship is behaving is

called seakeeping.

The environmental loading affecting a ship is hard to analyze and predict, and it is even

harder to correctly calculate the influence on a given vessel. There are many methods for

calculating vessels motions, and the analytical results that come close to the real behavior of a

given vessel are all numerical methods based on finite element methods.

This chapter describes how vessel motions is calculated in general and presents analytical

formulas for how to calculate the motions of a barge based on proper assumptions which are

thoroughly described.

The goal is to calculate vessels motions with use of easy obtainable data. By using certain

formulas, one can quickly understand how the vessel is behaving without needing time

consuming finite element approaches or programming.

One rotational and one linear motion will be thoroughly discussed in this report.

3.1 Vessel degrees of freedom

Vessels motions are explained by its six degrees of freedom which is indicated infigure 3.01.

Sway, surge and heave are linear degrees of freedom and roll, pitch and yaw are rotationaldegrees of freedom. In this report, the result of heave and roll motions is highlighted and

considered.

Figure 3.01. Degrees of freedom with coordinate system

Source: Mitra et al. (2012)

It is important to be consequent when using different coefficient for the different motions.The coefficient for the different motions will be used according tofigure 3.01.


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3.2 Uncoupling of heave and roll

Vessels motions in sea water are divided into six degrees of freedom. The six degrees of

freedom are sway, surge, heave, roll, pitch and yaw respectively as shown in figure (3.01). A

vessel will behave differently regarding each degree of freedom where the response in each

direction will affect each other through coupling.

The resulting forces on a vessel due to heave and roll can be calculated by assuming that roll

and heave is uncoupled. This means that heave and roll is calculated by obtaining the

resulting motions when roll and heave is considered as two single degree of freedom systems.

If one where to consider heave and pitch, the motions will be strongly coupled. Both forces

are working in the longitudinal direction of the ship. By considering either the front or the

back of the ship, we see that the vertical motions is the sum of the vertical displacement in

heave and pitch and the motions will be highly influenced by each other.

3.2.1 Linear theory

Linear theory is a necessary assumption for general finite element calculations which is

normally used when calculating vessel motions. To use linear theory, the non-linear

contributions in any motion must be assumed small and neglected.

Linear theory let us calculate resultant forces by adding all contributions to the considered

force together. This concept is important when calculating a vessels response using the

equation of motions which is discussed later insection 4.3.1. All contribution related to mass,

added mass, damping, stiffness and exciting forces is obtained by linear theory where all

contributions related to the considered motion is added together.

3.2.2 Coupled motions

At sea there is a dynamic pressure field affecting a general vessel from all sides. When a ship

is oscillating in either direction, the vessel will influence and change this pressure field. This

means that the dynamic pressure field is changed when the vessel is moving in one of the six

degrees of freedom so that the other degrees of freedom will be affected due to that

movement.

The coupled motions in either direction can be found by assuming linear theory and harmonic

motions by summarizing properties for mass, added mass, damping, stiffness and exciting

forces in matrixes shown in eq. (3.01) where all terms consist of six times six matrixes. Incomparison the uncoupled single degree of freedom equation of motions is shown in eq.

(3.02).

6...1j,FCBAM6

1kjkjkkjkkjkjk

(3.01)

jjjjjjjj FCB)AM( (3.02)

Where Mjk are mass components, Ajk and Bjk are added mass and damping coefficients, Cjk

are hydrostatic restoring coefficients, and Fj are the exciting force and moment components.


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(Salvesen et al. (1970)) The numbers from one to six represents respectively surge, sway,

heave, pitch, sway, roll and yaw.

3.2.3 Symmetric and anti- symmetric motions

All degrees of freedom will affect each other to some extent, but some of the degrees of

freedom will be less coupled. When a ship has a lateral symmetry, the degrees of freedom can

be divided into symmetric and anti- symmetric motions. Surge, heave and pitch are the

symmetric motions, since the motion response to either of these motions will be the same on

either side of the ship. (Sway, roll and yaw is then anti- symmetric motions) The symmetric

and anti- symmetric motions are unaffected by each other when assuming linear theory.

(J ourne (2001) p.26)

By assuming linear theory and harmonic motions in case of long slender ships; roll and heave

will be uncoupled since they are symmetric and anti-symmetric motions. The motion on a

particular point on a vessel can then be found by adding the contributions due to heave and

roll.

3.3 Equation of motion

The equation of motion is a product of Newtons second law of motion which states that the

product of an objects mass and acceleration equals the sum of forces acting on the object. In

case of a vessel, the applied forces are the wave forces, radiation forces and hydrostatic forces

as shown in eq. (3.03). The general equation of motion for heave is given in eq. (3.04) and

represents the formulation of all six degrees of freedom.

F Fw FR FS (3.03)

The force F is the mass times acceleration, the wave-induced forces )FFF( DAw is the

approaching wave forces plus the diffraction forces due to a vessels disturbance of the

pressure field. The radiation forces due to harmonic motion )zBzAF( R are thehydrodynamic forces related to added mass and damping which points in the opposite

direction of the external forces. The hydrostatic restoring forces )CzF( S are also called

the stiffness which is the buoyancy force that restores the vessel to its initial position.

(Faltinsen (1990) p.41) The forces affecting a vessel can be written as the general equation of

motion aseq. (3.04).

)()( tFzczbzaM zzzz (3.04)

The form of the equation of motion will be the same for the motions in all six degrees of

freedom for a vessel moving in water. The only difference will be that the mass terms are

changed with mass moment of inertia, and the damping and stiffness term will relate to

rotation and the induced wave force must be converted into momentum force.


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3.3.1 Solutions of equation of motion

The equation of motion is applied for all six degrees of freedom and is treated the same for all

cases. The solution for uncoupled heave is shown, but it the equation is treated the same in

case of uncoupled roll motion.

When solving a differential equation, two solution must be considered; the homogeneous

solution zh(t), and the particular solution zp(t). The total solution is z(t)=zh(t)+zp(t). (Rao

(2011) p.261)

The particular solution zp(t) and homogeneous solution zh(t) is the solution of respectively eq.

(3.05) andeq.(3.06).

)()( tFzczbzaM zzzz (3.05)

0)( zczbzaM zzz (3.06)

4.3.2 Particular solutionLet us consider eq. (3.05) as a general differential equation, where the equation equals a

harmonic force with a frequency e so that )tsin(F)t(F e0z . The systems mass are

azz M)aM( , the damping is bz, and the stiffness is cz.

The equation can be solved as response of a damped system under harmonic force. The

following calculations are also illustrated in Rao. (RAO (2011) p.271)

)tsin(Fzc

dt

dzb

dt

zd)M( e0z

p

z2

p2

az (3.07)

Since the force is harmonic, the solution ofeq. (3.05), zp(t) is also assumed harmonic with a

phase difference between the exciting force and the motion which is z .

)tsin(Zd

zd),tcos(Z

d

dz),tsin(Zz ze

2e2

p2

zee

p

zep (3.08)

By substitutingeq. (3.08) intoeq. (3.05), eq. (3.09) is obtained:

)tsin(F)tcos(b)tsin(McZ e0zeezze2eazz (3.09)

By using the trigonometric relations:

sintcoscostsin)tsin(

sintsincostcos)tcos(

Equation (3.10) and (3.11) is obtained by equating coefficients of sinuses and cosines

respectively.

)tsin(F)tsin()sin(b)cos(McZ e0ezezz2eazz (3.10)


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0)tcos()cos(b)sin(McZ ezezz2eazz (3.11)

The amplitude Z is obtained from eq. (3.10) and the mass terms are inserted.

( azz M)aM( ) The phase angle, z is obtained from eq. (3.11) and is given in the

following equations.

2ez22

ezz

0

b)aM(c

FZ

(3.12)

z2ez

ez1z

caM

btan (3.13)

3.3.3 Dynamic amplification factor

To understand which frequencies that give the largest response of the vessel,eq. (3.12) can beshowed in a different form giving a relation between the relative response and frequency rate.

Let us divide the numerator and denominator ineq. (3.12) with the stiffness coefficient, cz.

2

z

ez

2

2e

z

z

z

z

z

0

c

b

c

)aM(

c

c

c

F

Z

(3.14)

The following relations for natural frequency, nz (the coefficient number z is used todistinguish the natural frequency of heave and roll) damping ratio, and frequency ratio, rare given ineq. (3.15) and substituted intoeq. (3.14) to obtaineq. (3.16).

nze

nzz

z

z

znz r,

)aM(2

b,

aM

c

(3.15)

2220z

r2r1

1

F

ZcDAF

(3.16)

This relation will be the same for all single degree of freedom systems which is oscillating in

harmonic motions. The dynamic amplification factor in eq. (3.16) is plotted for different

damping frequencies infigure 3.02


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Figure 3.02. Dynamic amplification factor

Figure 3.02shows how much a general force will influence the response motion of a system

related to the frequency rate. When the force frequency rate equals zero, the response is the

same as the constant force divided by stiffness. And in case of relative low force frequencies,

the motion is highly dependent on the restoring force cz.

A case where force frequencies are close to the natural frequency is in general undesirable.

This is where resonance occurs. This is called the damping controlled region since the value

of damping is controlling how high the possible amplitude motion can be. In case of a

damping frequency of 0.1, the response is five times larger than the external force divided by

the stiffness of the system.

Cases of large force frequencies related to natural frequency, is called mass dominated

motions. This is because the response in general will decrease depending on the value of the

mass system.

How the dynamic amplification factor relates to ship motions will also be discussed in section

3.4.1.

3.3.4 Homogeneous solution

The homogeneous solution is also called the transient motion. This motion is the result of aninitial force where the system is left alone to oscillate. This motion will in all realistic cases

die out do to damping since the solution is the result of a motion with external force equals

zero.

When the system is undergoing a change of motion, like in lifting operations or sudden wind

increase, there will be a change in the system that must be accounted for. The motion will die

out after a while, but if the vessel is not designed for this initial response, the ship or cargo

will be damaged.

The homogeneous solution, zh(t) is obtained by solvingeq. (3.06) with given initial conditions

for motion and velocity at time zero.


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0zczbz)M( zzaz (3.17)

In most realistic cases of motion, the damping ratio, would be between zero and one whichis called an underdamped case. The solution of the homogeneous solution would be in the

form shown ineq. (3.18). (Rao (2011) p.160, p.270)

)tsin(eZz 0zdt

0hnz (3.18)

nz2

dz 1 (3.19)

Where the amplitude, Z0 is multiplied by an exponential term which goes to zero in a slope

that depends on the damping ratio, . The damping frequency dz is the reduced natural

frequency which also depends on the damping ratio, shown in eq. (3.19). The amplitudeand phase angle of the homogeneous solution is given in respectively eq. (3.20) and eq.

(3.21).

2

1

2

zeznz00nz2dz

2

z00 cosZsinZzz1

sinZzZ

(3.20)

)sinZz(

cosZsinZzztan

2 z0d

zeznz00nz10z

(3.21)

Where Z is the amplitude of the particular solution, z is the phase angle of the particular

solutions, 0z and 0z

is the initial movement and speed when the time is zero.

3.3.5 Phase angle

When considering the particular solution of equation of motion, the phase angle is defined as

the phase difference between the external force and the motion of a single degree of freedom

system. In case of ship motions, the exciting force will have a frequency equal the encounter

frequency e .

In case of a vessel, each of the six single degree of freedom systems will have its own phase

angle, telling when each motion is occurring with respect to the force. How the phase angle

relates to damping ratio, and frequency ratio, r is derived by dividing the nominator anddenominator by the stiffness ineq. (3.13)

2

1z

r1

r2tan (3.22)

The phase angle is dependent on the damping where a small damping results a smaller phase

angle between the motion and the force frequency.


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3.4 Harmonic response of a vessel

As shown in section 3.3, the equation of motion can describe the behavior of vessels. The

accuracy of the solution depends on how the terms for radiation, restoring, mass and external

forces are obtained. In this section the relation between these terms and a vessels behavior is

discussed. In section 3.5 and section 3.6 different ways to calculate those terms in case of

heave and roll is considered.

3.4.1 Dynamic amplification of a vessel

The dynamic amplification factor for the equation of motions is calculated and described in

section 3.3.3. Figure 3.03shows which terms in the equation of motions that have most effect

on the response of the system related to different frequencies. Where r= nze / .

Figure 3.03. Dynamic amplification factor

Figure 3.03describes three different behaviors of vessel motions. (Journe and Massie (2001)

p. 6-24)

1. In case of low relative force frequencies, ne .The motion of a vessel isdominated by the restoring forces of the system, the hydrostatic term. When this

occurs, the vessel motions will follow the wave in a very stiff motion. In case of

small vessels this occurs since the vessel length is relatively small compared to the

wave length. The vessel will then follow the waves and feel all waves as stiff

movements. Seefigure 3.04.

2. In case of force frequencies close to the eigenfrequency, ne , resonance occur.When there is little damping in the system, the motions in these force frequencies is

critical. The damping dominated area should in cases of vessel motion be narrowed as

much as necessary to be avoided.


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3.4.3 Damping force

The damping in an oscillating or vibrating system tells us how much energy that is dissipating

while the system moves. By looking at a system induced with an initial force where no other

external forces is working on the system, the response and displacement of the system will

gradually decrease due to the reduction in energy. The damping term is telling us how muchenergy that is dissipated in the process. (Rao (2011) p.160)

In case of a vessel moving in fluid, the damping of the system is the total of all contributions

reducing the energy in the system. The real damping of a system is affected by viscosity, eddy

currents and the energy in the waves that the vessel is able to create when moving. The

contributions in roll damping are summarized to get the equivalent damping and are shown in

eq. (3.23).

BKLwefeq BBBBBB (3.23)

Where Beq is the equivalent damping and is discussed for roll in section 3.6.3. The terms inthe equation must be defined by semi empirical formulas. Bf is damping due to skin friction

which is dependent on viscosity, Be is damping due to eddy making and vortexes created in

the movement, BW is the radiation damping, BL is damping due to lift forces and BBK is the

damping due to a bilge keel which has a significant effect on a systems damping. This

damping is not in phase with the velocity of the system and can be multiplied by the

amplitude to make it in phase with the velocity. The damping in case of roll is therefore given

as ineq. (3.24). (Chakrabarti (2000))

)B(B eq (3.24)

In case of a vessel, the ship will create waves and thus dissipate energy in each motion. When

considering a linear system like surge, sway and heave, the damping will be proportional to

the velocity. This is because the total damping is approximately equal to the dissipated energy

in the radiation waves which is in phase with the speed of the system. (Journe and Massie

(2001) p. 6-11) The concept of radiation damping is illustrated in figure 3.05.

Figure 3.05. Radiating waves

Source: Journe and Massie (2001) p. 6-14


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Figure3.05 illustrates the concept of radiation waves in case of a vessel moving up and down

in heave. When the vessel is moving it will create additional waves with amplitude a . The

energy dissipated in these waves is equal to the damping of the system. (Faltinsen (1990)

p.46)

The damping term and added mass terms are called radiation forces affected by a change infrequency of the system. They are also affected by viscosity which makes them impossible to

calculate by purely analytical formulas (without more than few approximations). In most

cases these terms are calculated by strip theory and a numerical approach described insection

3.4.6.

3.4.4 Restoring force

The restoring force of a oscillating is defined as the systems ability to restore itself to its

original position. In case of a vessel it is dependent on hydrostatics and stability properties of

the vessel. The relation for a spring in general is that an applied force equals the stiffness of

the spring multiplied by the displacement of the spring as shown in eq. (3.25). The same

relation follows here, and the stiffness relates to the restoring forces given from geometrical

properties and buoyancy.

kjkj CF (3.25)

Where Fj is applied force, k is the related displacement and Cjk is the equivalent stiffness ofthe system. The stiffness is found by considering a static system affected by a given force

which induces a displacement. The term is considered negative with respect to the force since

the restoring force is countering the force. Every term in the equation is known except the

stiffness.

3.4.5 Exciting force

To understand how vessels behave at sea it is important to understand the exciting forces

applied on a certain vessel. At sea there are three kind of environmental loads present at all

times. Those are wave, current and wind forces.

Forces from currant would be constant and must be taken into consideration when discussing

forces on a drifting structure, but is in general neglected when considering moving vessels.

Wind forces are also important in case of drifting vessels, and should be taken into

consideration in case of roll motions of vessels with significant height. (Journe and Massie(2001))

The wave exciting forces are divided into two kinds of forces, diffraction forces and

approaching wave forces. It is also important to understand the meaning of the encounter

frequency, e which is discussed in this section.

Approaching wave forces

Approaching wave forces are found by considering wave forces on a rigid body at sea,

restrained from oscillation. These forces can be found by using linear wave theory where the

dynamic wave pressure is given as Dp ineq. (3.26) (Faltinsen (1990) table 2.1) The total wave


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force is found by integrating this pressure over the area it is working on, and this force is

called the Froude-Kriloff force. (Faltinsen (1990) p.59)

)kxtsin(egp ekz

0D (3.26)

Where and g is the density of seawater and the gravity constant, 0 is the wave amplitude,

e is the encounter frequency, t is the time, k the wave number, and x is the position relative

to the center of gravity of the vessel. (Positive direction against the front of the ship)

Diffraction forces

The considered approaching wave forces form an undisturbed pressure field around the

vessel, but since the vessel in truth is moving, the vessel will disturb this pressure field and

this disturbance must be taken into account. In case of bodies with small breadth (diameter in

case of a cylinder) compared to the wave length, the Morrison equation is applied. The force

will then be related to the added mass and the acceleration of the vessel. (Faltinsen (1990)p.61)

Encounter frequency

When calculating forces on a vessel there will be two relevant wave frequencies and it is

important not to mix them. Those frequencies are the frequency of the waves and the

encounter frequency which tells us the frequency of the encountered waves depending on the

speed and direction of the vessel.

Lets consider a point on a vessel with a constant forward velocity U [m/s] with an angle .

The point will encounter two waves in a time Te. The phase velocity of the waves is c=g/ and the wave length is Lw as shown in figure 3.06.

Figure 3.06. Encountering periods, TeSource: Faltinsen (1990) figure 3.12

wee LcTT)cos(U (3.27)

e

e

2T

and2w

g2L

by assuming deepwater. When substituted into eq. (3.27), eq.

(3.28) is obtained.


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2e

g2g)cos(U

2

(3.28)

By solvingeq. (3.28) with respect to e , eq. (3.29) is obtained.

cosg

U2

e (3.29)

Eq. 3.29 is also given by DNV (DNV-RP-C205 (2010)).

3.4.6 Strip theory

To use the equation of motion and calculate the response of a vessel, a numerical approach is

used for calculating the forces on a vessel. The strip theory is a numerical theory where the

cross section of the ship at a distance x from the center of gravity, is considered as one strip

with infinite length. This means that each strip is unaffected by other strips. A ship is dividedinto as many strips as necessary where the hydrodynamic properties are added together by

integrating the two-dimensional strips over the length. (Journe (2001) p.5) Seefigure 3.07.

Figure 3.07. Representation of strip theory

Source: Journe (2001) figure 2.7

A few assumptions is necessary for using the strip theory on vessels in the computer code forSeaway which is used to calculate wave induced loads and motions on ships. (Journe

(2001) p.5)

- The vessel is a rigid body floating in an ideal fluid- The motions of the vessel is linear or can be linearized

That a vessel is floating in ideal fluid, means that all assumptions for linear wave theory is

applicable so that the sea water is assumed homogeneous, incompressible, free of surface

tension, irrotational and without viscosity.


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To assume that the vessel motion is linear, one must neglect the effect of what is happening

above water. Waves on deck is then neglected. Other than that, the strip theory has been

proved effective for predicting ship motions with length divided by breadth larger than three.

3.5 Heave motionThe heave motion of a vessel is the vertical response of the vessel. When assuming uncoupled

motions which are described in section 3.2, the heave motion of the ship is considered as a

single degree of freedom where all terms related to the equation of motion must be calculated

for this specific motion.

When finding the relevant equation of motion in heave, all necessary assumptions will be

given and described in its respective section.

3.5.1 Equation of motion

The equation of motion is defined in section 3.3 and eq. (3.04) is improved by considering asinusoidal force with amplitude F0 moving in phase with the encounter frequency defined in

section 3.4.5.

)tsin(Fzczbz)aM( e0zzz (3.30)

3.5.2 Added mass in heave

When a vessel is moving in a fluid, additional water will move with the vessel. As described

earlier the fluid will oscillate with the moving body and the added mass is in reality a mass

equivalent to the hydrodynamic force related to the movement of the fluid. (Faltinsen (1990)

p.42)

The added mass is normally calculated by using strip theory. Where the vessel is divided into

many elements, (depending on the shape of the vessel) the hydrodynamic properties for each

strip are considered and integrated over the length (x), as shown ineq. (3.31). In section 3.2.3

the concept of added mass was discussed and concluded that the added mass goes to zero

when the frequency goes to zero.

dx)x(aaL

D2zz (3.31)

Added mass equals half a circle

The added mass of each strip can be considered as half a circle with diameter equal to the

breadth of the vessel as shown in figure 3.08 thus making the added mass independent of the

wave frequency. By using this approximation one assume that the added mass is independent

of the waves. (Sarkar and Gudmestad (2010))


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Figure 3.08, Added mass equals half a circle

The formula for area of half a circle is8

2dA

, where d is the diameter of the circle. By

using the strip theory, we find that the added mass is the sum off all strips that will have thefinal form of half a cone in case of a triangular shape or half a cylinder in case of a

rectangular shape. We find the added mass by integrating the 2D sections over the length.

8

Ba

2D2

z (3.32)

)(8

)(2

2

22

L

LL

Dzz dxxBdxxaa

(3.33)

8

LBa

2

eargb,z

(3.34)

In case of a rectangular barge the added mass is the same as half a cylinder with diameter, B

and length, L as seen ineq. (3.34).

Added mass, DNV

DNV uses the same equation as obtained in eq. (3.34) but advice that the added mass is

multiplied by a factor, CA. (DNV-RP-H103 (2011) Table A-1) Figure 3.06 is the relevantinformation extracted from the relevant table where the added mass per unit length is given.


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Table 3.01. Added mass coefficients from DNVSource: DNV-RP-H103 (2011) Table A-1

The total added mass of a rectangular shape oscillating under water is considered in table

3.01. In this case, the only interest is the equivalent added mass under the vessel, so the added

mass is half the value given in the table. The added mass is then given as ineq. (3.35) andeq.

(3.36). The values of 2a and 2b from thetable 3.01 is respectively the breadth and draft of ourvessel

2

ACa RA

D2z (3.35)

8

LBCL

2

ACa

2

AR

Aeargb,z

(3.36)

Where CA is the added mass coefficient. The added mass coefficient is found by interpolating

the relation between breadth and draft with respect to values intable 3.01.

3.5.3 Damping in heave

When the frequency of external forces working on the ship is the same as the natural

frequency, the motion amplitude will go to infinity in case of no damping. As discussed in

section 3.4.1, these motions are called damping controlled motions.

The contributions of damping are shown in eq. (3.23). All contributions will affect the total

damping of the system, but the radiation damping will be very large compared to the other

contribution in case of heave. The radiation damping can be calculated without empirical

formulas and is the only damping that gives satisfactory results for the single degree of

freedom systems except roll. (Chakrabarti (2000))

When a vessel is moving vertically in water it will create its own waves. This process is

called radiation, and vessels (especially barges) moving vertically in heave motion will

displace a lot of water thus losing a lot of energy, making radiation damping the largest

contribution to damping in heave as shown in figure 3.07. The total damping in heave is

therefore assumed equal to the radiation damping.

Wz Bb (3.37)


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There will be other contributions such as damping due to eddy vortexes and viscosity but they

will be very small compared to the radiation damping. So saying that radiation damping is the

only damping term present in case of heave is a very good approximation for the motion.

The radiation damping can be found using an energy relation where the total energy, E in a

fluid volume, is the sum of kinetic and potential energy where is the displaced volumeof the system as shown in eq. (3.38). (Newman (1977) p.266)

dgz5,0)t(E 2 (3.38)

Faltinsen has made a formula for heave damping using Newmans energy relation based on

how much water a vessel will displace. (Faltinsen (1990) p.47) The formula is related to

radiation wave amplitude a , vessel movement z, and frequency , of the waves as shown

ineq. (3.39).

3

22

aD2z

g

zb

(3.39)

The radiation wave amplitude a , illustrated in figure 3.07, must be found by a forced motion

test related to each force frequency. Eq. 3.39 takes the energy of each created wave into

consideration making this a close to truth relation for calculating the damping of the system.

The total damping is found by integrating over the length.

3.5.4 Restoring force in heave

The restoring force of a body oscillating vertically in water is related to a vessels ability to

restore itself to its original position. The original position is at z=0 at the waterline area in

stillwater. The restoring force is based on the geometry of the vessel which defines a vessels

restoring force.

To find the restoring force we consider a static system with a static force, based on buoyancy

and the additional displacement.

zcF z (3.40)

The force is equivalent to the mass of displaced water multiplied by the gravity constant,which equals the stiffness, zc times the depth of the displaced water, Ddisplace.

displacedzdisplacedwdisplaced DcgDAgF (3.41)

This gives that the stiffness, cz is proportional to the waterline area in heave motion of any

vessel if the waterline area is constant during the vertical motion.

gBLgAc wz (3.42)


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3.5.5 Exciting force in heave

When considering the uncoupled heave of a vessel the vertical motions is considered alone

and the only relevant external forces are the vertical wave forces. The external heave force,

Fz(t) can be divided into approaching wave forces and diffraction forces as discussed in

Section 3.2.6.

)()()( tFtFtF DAz (3.43)

These forces are calculated differently depending on the size of the structure relevant to the

wave length. In case of large structures both diffraction and inertia forces must be accounted

for. The relevant force regimes are presented infigure 3.09where, 0 is the wave amplitude,

DC is the characteristic dimension and LW is the wave length found by the deepwater relation

presented insection 3.2.5.

Figure 3.09. Different wave force regimes

Source: DNV-RP-H103 (2011) figure 2-3

In case of heave, the characteristic dimension will be either the length or the breadth of the

barge. These dimensions are large compared to the wave length making it necessary to takediffraction forces into account.

Two general assumptions is made for the calculations:

- Linear wave theory is applicable- Deepwater relations is applicable

Formulas used in linear wave theory are found intable 3.02.


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Table 3.02.Llinear wave theory

Source: Faltinsen (1990) table 2.1

Approaching wave forces

The approaching wave force corresponds to the undisturbed pressure field and is called the

Froude-Kriloff force. This force is applicable for both large and small structures and is

calculated by integrating the pressure over the surface of the vessel. The approaching wave

force can be calculated as in eq. (3.44) where pD is the dynamic pressure found from table

3.02. (Faltinsen (1990) p.61)

S Dz,A dsp)t(F(3.44)

)sin(0 kxtegpkz

D (3.45)

By substituting eq. (3.45) into (3.44) a general formula for the Froude-Kriloff force is

obtained.

S

kz0z,A ds)kxtsin(egF (3.46)


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To obtain the resultant force in heave, eq. (3.46) must be integrated over its bottom surface

and the same assumptions as for strip theory is applied. In case of a barge, the resulting force

is calculated by eq. (3.47), where z is the draft of the barge.

2/B

2/B

2/L

2/L

kD

0eargb,Az dxdy)kxtsin(egF (3.47)

Diffraction forces

The diffraction forces can be calculated using the Morison equation. This formula can only be

applied for waves with wave lengths, LW>5DC where DC is the characteristic dimension as

shown in figure 3.09.The Morison equation can be modified in terms of added and the

relevant wave acceleration, zw found from table 3.02 as a3 in the table. (Faltinsen (1990)

formula 3.36)

The usability of the formula is confirmed by DNV with the following statement: In caseinertia dominated, large volume structures where the diffraction loads are much larger than

drag induced global loads, a Morison load model with predefined added mass coefficients can

be added to the radiation/diffraction model. (DNV-RP-H103 (2011) 2.3.4.5) (The radiation

model refers to added mass and damping forces)

zzz,D waF (3.48)

)sin(0 kxtekgwkz

z (3.49)

To obtain the resultant force in heave, eq. (3.48) must be integrated over its length the sameassumptions as for strip theory is applied by taking possible change in the added mass into

consideration and integrate the 2D- added mass over the length.

L

kz0

D2zz,D dx)kxtsin(ekgaF (3.50)

The resulting diffraction wave force is calculated byeq. (3.51), where z is half the draft of the

barge. (Since the resultant force would be applied at a depth of the center of buoyancy where

z=D/2)

2/L

2/L

2

kD

0D2

zz,D dx)kxtsin(ekgaF (3.51)

Minimum characteristic dimension, DC

As mentioned earlier, the formula for the diffraction forces gives reliable answers in cases

where the wave length is five times larger than the characteristic dimension. In case of a

barge where the characteristic dimension of heave is the breadth, the formula is used for large

wave periods for best results. Since wave length is related to wave period in deepwater as in

eq. (3.52).


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2W T

2

gL

(3.52)

Requirement for use of Morison equation:

5LD WC (3.53)

Figure 3.10 shows which wave periods the formula for diffraction forces give best results

relating to the characteristic dimension, DC.

Figure 3.10. Minimum characteristic dimension, where DC,min=LW/5

Total wave forces

By substituting eq. (3.51) and (3.47) into eq. (3.43). The total force for a barge can be

calculated ineq. (3.54).

dx)kxtsin(ekgadxdy)kxtsin(eg)t(F 2kD

0

2/L

2/L

D2z

2/B

2/B

2/L

2/L

kD0z

(3.54)

The following trigonometric relation is used.

)tsin(2

kLsin

k

2dx)kxtsin(

2/L

2/L

(3.55)

By deriving the integral in eq. (3.54) and using the trigonometric relation in eq. (3.55) the

harmonic wave force of a barge can be calculated byeq. (3.56) (Faltinsen (1990) p.89) Wave

frequency is exchanged with encounter wave frequency, since we are interested in the motion

response on the barge.


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)tsin(*F)tsin(2

kLsin

k

2ekgaegB)t(F e0e

2

kD

0D2

zkD

0z

(3.56)

In case of a vessel moving in water the frequency of the waves, will be equal the effective

frequency e which is derived insection 3.4.5. Ineq. (4.54) B is the breadth of the barge, L isthe length of the barge, is the density of sea water, g is the gravity constant o is the wave

amplitude, k is the wave number, Dza2 is the added mass of a strip (half circle with breadth B)

and D is the draft.

Wave number

The wave number and frequency are related through the dispersion relations shown intable

3.02where k=2/g in case of deepwater. The wave frequency must not be mixed with theeffective frequency e shown ineq. (3.29) which tells us how fast the vessel is encountering

the waves. The encounter frequency is already a function of wave number k, where k

describes the ratio of waves per meter in the equivalent wave direction. In other words, the

wave frequency is used to calculate the wave number when considering the force on the

barge.

Wave force when wave frequency goes to zero

Since the resulting wave forces in heave is not applicable for all wave periods it interesting to

look at the worst case values of the function. The largest value of force amplitude, F0 in eq.

(3.56) is found when the wave frequency goes to zero. The dispersion relation is shown ineq.

(3.57).

gk

2 (3.57)

Remember that )sin( in case of small angles. When eq. (3.57) is substituted into eq.(3.56), eq. (3.58) is obtained.

LeDaegBF g2D

02D2

zg

D

0o

22

(3.58)

If the frequency is close to zero, the diffraction term ineq. (3.58) will be much smaller than

the approaching wave, and the exponential term will be close to one so that the equation is

simplified to a relation between the stiffness of the system, zc and wave amplitude, 0 .

0z0o cgBLF (3.59)

By calculating F0 by eq. (3.59) the largest force for the relevant wave height is obtained. This

is also the basic relation for a spring in a static system where the force equals the stiffness

multiplied by the displacement. (P=kx)


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3.5.6 Natural period in heave

The natural period is considered one of the most important parameters when analyzing

motions of a vessel. The natural period is also called the resonance period, since induced

vibrations with period equal to the natural periods results in resonance.

If a system, after an initial disturbance is left to vibrate on its own, the period of the

oscillations without any external forces is known as the natural period. A vibratory system

having a number of degrees of freedom will have as many distinct natural periods of

vibration. (Rao (2011) p.62)

To calculate the natural period the motion is assumed uncoupled from the other degrees of

freedom.

General formula of uncoupled natural period of heave, Tz is given provided by DNV, where

the mass equals the added mass plus the mass (DNV-RP-H103 (2011) 2.3.3.4).

z

z

nz

nzc

aM2

2T

(3.60)

By substituting values for the mass (eq. (3.61)) and the stiffness (eq. (3.42)) for a rectangular

barge intoeq. (3.60) the relation for natural beriod is given ineq. (3.62).

BLDM (3.61)

Ma1gD2gBL M

a1BLD

2T z

z

eargb,nz (3.62)

Where D is the static draft of the barge, g is gravity acceleration constant, az is added mass

given insection 3.3.3and M is the total mass of the barge.

3.5.7 Solution

The equation of motion for heave motion is given ineq. (3.63):

)tsin(Fzczbz)aM( eozzz (3.63)

Where M is the total mass of the barge, az is the added mass, bz is the damping, cz the

stiffness, F0 the force amplitude, e the equivalent wave frequency, t is time and z is the

vertical motion of the barge.

Table 3.03shows examples of which formulas that can be used in the equation of motion and

which assumptions that is needed. When for example strip theory is assumed, all relevant

assumptions mentioned insection 3.4.6 is used.


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Table 3.03. Terms in equation of motion

Particular solution

The equation of motion has the same form as the equation of motion presented in section

3.3.1, and the solutions of the equation will be in the same form as proven earlier.

The equation of motions is derived insection 3.3.2, and the same solutions are applicable for

roll motion. To understand the vessels behavior, the response with respect to displacement,

velocity and acceleration are of interest and is presented ineq. (3.64) to eq. (3.70). Mark that

the frequency is the encounter force frequency, edefined insection 3.4.5.

)sin()( zep tZtz (3.64)

)cos()( zeep tZtz (3.65)

)tsin(Z)t(z ze2ep (3.66)

Where the motion amplitude, Z and phase amplitude , is given ineq. (3.67) andeq. (3.68).

Term

Assumptions used

for generalformulas

General formulas in

heave for a vessel

Additional

assumptions forbarge

Barge formulas

za Strip theory )(2

L

Dz dxxa

Added mass isunaffected by wave

frequency

No viscosity

Deepwater

8

LBC

2

A

zb

Only dampingbecause of radiation

Energy relation influid from Newman

Linear wave theory

L3

22

a dxg

z L

g

z 3

22

a

zc Constant waterlinein range of motion

gAw Rectangular

waterline in range ofmotion

gBL

)(tFz

Strip theory

Morison equation

Linear wave theory

Long relative wavelength

No other forcecontributions

zz

S

zD wadsnp Sinusoidal waves

B


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z0z

2

zz

22z

zo

c

FDAF

r2r1

c/FZ

(3.67)

2

1

)(tan ezz

ez

z aMc

b

(3.68)

Where DAFz is the dynamic amplification factor, cz is the stiffness, F0 is the force amplitude,

r is the frequency ratio and , is the damping ratio.

2zz22

z

z

r2r1

1DAF

(3.69)

nz

ezr

(3.70)

zz

z

zC

zz

c)aM(2

b

b

b

(3.70)

Homogeneous solution

When considering vessel movement in a given seastate, the homogeneous solution is

normally neglected since the homogeneous solution is damped out. But if the vessel has

undergone a change like a sudden wind increase or an impact, the homogeneous solution is of

interest.

The homogeneous solution is shown insection 3.3.4and is in case of heave represented in eq.

(3.71) toeq. (3.74).

)tsin(eZ)t(z 0zdt

0hnz (3.71)

t0zdnz0zdd0h nze)tsin()tcos(Z)t(z (3.72)

t

0zddnz0zd

2

d

0zdnz0zddnz

0hnze

)tsin()tsin(

)tsin()tcos(Z)t(z

(3.73)

nz2

dz 1 (3.74)

Where the amplitude, Z0 is multiplied by an exponential term which goes to zero in a slope

that depends on the damping ratio, . The damping frequency dz is the reduced natural

frequency which also depends on the damping ratio . (eq. (3.70)). The amplitude and phaseangle of the homogeneous solution is given in respectivelyeq. (3.75) andeq. (3.76).


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2

1

2

zeznz00nz2dz

2

z00 cosZsinZzz1

sinZzZ

(3.75)

)sinZz(

cosZsinZzz

tan2 z0d

zeznz00nz1

0z

(3.76)

The initial responses in terms of motion and velocity, 0z and 0z must be given or assumed to

obtain the homogeneous solution.

Total solution

If the vessel is influenced by a sudden change in external forces, the homogeneous solution

must be taken into account, so that the total response of the system is the sum of

homogeneous solution and particular solution. To obtain eq. (3.64) linear theory is assumed

so that the total response is the sum of two responses.

)t(z)t(z)t(z hp (3.77)

If the motion has been uniformly increasing until the considered response is given we can

neglect the transient, homogeneous solution and the total response of the system will equal

the particular solution.

)t(z)t(z p (3.78)


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3.6 Roll motion

The roll motion of a vessel is the rotational response around the x-axis of the vessel, when the

x-axis is pointing toward the front of the vessel from the center of gravity. When assuming

uncoupled motions, (see section 3.2) the roll motion of the ship is considered as a single

degree of freedom where all terms related to the equation of motion must be calculated for

this specific motion.

The equation of motion and additional solutions is in the same form as in case of heave, only

the motions are rotational instead of linear.

Necessary assumptions will be given and described in the related section.

3.6.1 Equation of motion

The equation of motion defined insection 3.3and applied for heave insection 3.5can be used

in the same way in case of roll with some modifications.

- Mass [kg] IM mass moment of inertia [kgm2]- Added mass [kg] Aaz added mass moment of inertia [kgm2]- Damping [kg/s] Bbz (rotational) damping [kgm2/s]- Stiffness [N/m] Ccz (rotational) stiffness [Nm2/m]- Exciting force [N] 00 MF exciting moment [Nm]- Linear motion [m] z rotational motion [rad]

Beside the difference in coefficients and units, the equation of motion is treated the same for

linear and rotational motions. The frequency of motion is the same but a face lag between the

two motions must be taken into account. The equation of motion for roll is shown in eq.

(3.79) and will be the same in the same form as other rotational single degree of freedom

systems. Keep in mind that there will be an important phase displacement between the

different responses.

)tsin(MCB)AI( e0 (3.79)

3.6.2 Mass moment of inertia and added mass moment of inertia in roll

The mass terms in the equation of motion in case of roll, must be transferred into rotational

mass called mass moment of inertia.

Mass moment of inertia

The mass moment of inertia is a measure of an objects resistance to changes in rotation.

Mass moment of inertia has the same relationship to angular acceleration as the relation

between mass and linear acceleration, and is therefore applicable in the equation of motion.

(http://www.engineeringtoolbox.com/moment-inertia-torque-d_913.html)

Mass moment of inertia is the integral of the distance to the rotated mass squared and

integrated over the total mass as shown ineq. (3.80).


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dVrdmrI body 22 (3.80)

As the coordinate system in figure 3.01, the applied mass moment of inertia is the rotation

around the x-axis. To simplify the equation, the barge with additional cargo has a shape as a

box with a uniform load distribution and a total height of KG2 . The integral in eq. (3.80) isthen transferred into Cartesian coordinates in eq. (3.81) and the rectangular barge is

considered as a box with length L (along x-axis), breath B (along y-axis) and height two

timesKG , which is calculated ineq. (3.99) (along z-axis).

KG

KG

B

B

L

LV

dxdydzzydVzydVrI2/

2/

2/

2/

22222 (3.81)

The integral is solved as shown in eq. (3.81) and the final solution is shown in eq. (3.82).

Note that; bodyKGLBM 2 , where body [kg/m3] is the evenly distributed weight for the

assumed volume of the ship.

KG

KGbody

KG

KG

B

Bbody dzz

BLBdydzzyLI 2

22/

2/

22

12 (3.82)

2222

21212

2

122 KGB

MKGBKGLBI body

(3.83)

Added mass moment of inertia

In this section, three ways of calculating the added mass will be considered and compared.

Added mass moment of inertia equals the resulting moment of half circles.

Similar to the added mass in heave the added mass in roll can be calculated by considering

the momentum of relevant half circles. The mass moment of each strip is then integrated over

the length, and the added mass is assumed unaffected by the wave frequency.

Based on figure 3.12 the added mass moment of inertia is presented in eq. (3.85) as the

integral of the small dys multiplied by the arm, y. (the formula is the result of triangular

relations) The half circle is considered as sinusoidal with amplitude B/4. To simplify theequation a formula for triangular shapes is used in case of the small half circles. (This would

be the smaller of the contributions making the assumption insignificant)

The mass moment of inertia is defined in this section as the distance to the rotated mass

squared and integrated over the total mass as shown in eq. (3.80). It will be the same for

added mass moment of inertia shown ineq. (3.84).

dVydmyA 22 (3.84)

The volume which must be integrated is shown infigure 3.12andeq. (3.85).


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dyy2DBydyB

2ysin

4

By2A

2/D2/B

2/B

22/B

0

2D2(3.85)

The integral ineq. (3.85) is solved using the website www.wolframalpha.com, which is proven a

reliable source when it comes to calculation of complicated equations.(http://www.wolframalpha.com)

96

DBD4B6D

32

B42A

222

4

42D2

(3.86)

For a general ship shape with inconstant values for breadth and draft they would be functions

of x and treated as such when the total added mass is obtained by integrating over the length.

In case of a barge, the added mass is obtained witheq. (3.87).

96DBD4B6D

32B4L2A

222

4

42

eargb, (3.87)

Added mass moment of inertia, DNV

DNV uses a similar approach for calculating added mass in cased of rectangular cross

sections. The relation between breadth and draft is taken into account by a factor, 1 or 2 asshown intable 3.01. (DNV-RP-H103 (2011) Table A-1).

The total added mass of a rectangular shape with a circular motion under water is considered

in table 3.01. As discussed in section 3.5.1 the values in table 3.01 must be divided by twosince the vessel is not totally submerged.

The formula for added mass it then given ineq. (3.88), where values for i is defined in table3.04, where a=B/2 and b=D/2.

4

iD2

2

B5.0A

(3.88)

Table 3.04. Added mass moment of inertia coefficients from DNV

Source: DNV-RP-H103 (2011) Table A-1

In case of a rectangular barge, the resulting added mass is obtained by multiplying by length

ineq. (3.89).


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3.6.3 Damping in roll

In case of heave damping, only the radiation damping is considered since the contribution due

to radiating waves is very large compared to the other damping effects. This is not the case

for roll. By considering a sircular sylinder in water, one can imagine the bodys influence on

the fluid in case of heave and roll. By moving the cylinder up and down in the water, we seethat the motion creates waves in the water, which are the radiating waves. In comparison, the

roll motion will create no significant waves making the energy lost to radiating waves very

small. This means that other contributions to damping which is discussed in section 3.4.3

suddenly are of greater importance.

The total roll damping is found by summarizing all contributions of roll and multypling by

the estimated roll velocity as shown ineq. 3.92andeq. (3.93). (Chakrabarti (2000))

eqBB (3.92)

BKLwefeq BBBBBB (3.93)

The radiation damping in roll can be calculated with eq. (3.39) where the amplitude of the

radiation waves must be found using a oscillating test and the linear motion z is the rotational

displacement, . The radiation damping accounts for between 5% and 30% of the total rolldamping for typical cargo ships, but can be larger for low draft barges. (ITTC-RP (2011))

3

22

aD2W

gB

(3.94)

3.6.4 Restoring force in roll

The restoring force of a body oscillating with an angle in fluid is related to a vessels ability to

restore itself to its original position. The original position is defined as the angle 0 , wherethe vessel is initially stable.

The restoring force which is also called the up-righting moment is calculated by considering

the buoyancy of the vessel, and the vessels ability to restore to its original position. We

calculate the restoring force by multiplying the total buoyancy of a vessel with the horizontal

distance between the center of gravity and the center of buoyancy after inclination, the

distance from G to Z infigure 3.14.


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Figure 3.14. Illustration of restoring moment

The restoring force, rM is calculated byeq. (3.95)

C)sin(GMgGZgM r (3.95)

C is the restoring force. By assuming small angles the restoring coefficient C for vessels is

defined as ineq. (3.96).

GMgC (3.96)

Stability

The static stability of floating structures is defined as the up-righting properties of the

structure when it is brought out of equilibrium of balance by a disturbance in the form of a

force or a moment. (Journe and Massie (2001) p. 2-6)

To find the distance from the initial metacenter, M to the center of gravity, G, formulas

obtained by stability calculations is applied.

A few distances must be calculated before obtaining the value of GM which is defined in

eq. (3.97)

KGBMKBGM (3.97)

Where all values represents distances which is shown in figure 3.14, where K is the center of

the keel, G is the center of gravit, B is the center of buoyancy and M is the initial metacenter.

The values are defined for a rectangular barge. A few assumptions is made for calculating the

stability of a barge and is shown below.

- Uniform weight distribution- Rectangular barge with flat bottom and vertical sides- The cargo is trimmed both longitudinal and transverse direction- Relative small angles of motions


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Center of buoyancy, KB is obtained by Archimedes law and is for a rectangular barge thestatic draft divided by two.

2

DKB eargb (3.98)

Center of gravity, KG is obtained by summarizing masses of the concerned vessel with thelength from the keel to the respective masses and divide the sum by the total mass. In case of

a barge with ballast and cargo, the distance from keel to center of gravity is obtained by eq.

(3.99).

oargcballasteargb

oargcoargcballastballasteargbeargb

MMM

COGMCOGMCOGMKG

(3.99)

Metacenter radius BM is found by considering the geometry of the vessel and findingequilibrium of rotation. The metacenter radius in case of small angles is calculated by eq.

(3.100). (Journe and Massie (2001) p. 2-13)

D

BIBM

12

2

(3.100)

Small angles of motion

When assuming small angles of motion, we simplify our equations so that )sin( . Thegraph in figure 3.15 illustrates the definition of small angles and how this approximation will

affect our calculations. The graph is a comparison of the angle and the deviation between)sin( and. In case of angles as large as 20 degrees, the deviation from the true value will

not be larger than 2%.

Figure 3.15. Sinus of small angles


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3.6.5 Exciting force in roll

When considering the uncoupled roll motion of a vessel, contributions from wind should also

be taken into account. By assuming linear theory the contributions can be summed together as

ineq. (3.101).

)t(M)t(M)t(M)t(M windDA (3.101)

If a standard north sea barge is considered and the cargo is has a low center of gravity (or is

not much affected by the wind) the total exciting force is assumed equal the approaching

wave force for simplicity.

)t(M)t(M A (3.102)

The approaching wave force can be calculated by using the Froude-Kriloff force which is

described insection 3.4.5and assuming strip theory. Rotational force is considered, making it

necessary to multiply by the length to the force, which is y as shown in figure 3.01.Theapproaching wave force is derived byeq. (3.103). (Journe and Massie (2001) p. 7-9))

S

DA dsnyp)t(M (3.103)

Where n represent the direction of the motion. By substituting the hydrodynamic pressure

fromeq. (3.45) into eq. (3.103), eq. (3.104) is obtained. In case of head sea, the wave forces

would be zero. By changing the wave direction to y direction in eq. (3.104), the waves are

perpendicular to the barge making the equation applicable for roll in beam sea.

2/B

2/B

2/L

2/L

kD0A dxdy)kytsin(egyMM (3.104)

The integral is solved in eq. (3.105) and eq. (3.106) using wolfram alpha which is

computational software. (http://www.wolframalpha.com)

2/B

2/B2

kD0

k

)kytcos(ky)kytsin(egL)t(M

(3.105)

2

tsin*M2

tsin2

BkcosBk

2

Bksin2

k

egL)t(M 02

kD0 (3.106)

Notice that the exciting force in roll has shifted with an angle /2 which seem logical sincethe maximum heave force is happening on top of a wave crest, and the maximum roll force is

happening in the steepest part of the wave.


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Wave force when wave frequency goes to zero

By substituting the formula for wave number in deep water from eq. (3.57) into eq. (3.106),

eq. (3.107) is obtained.

)0(gLM 40

3

0

(3.107)

This function doesnt tell us much since the formula is basically zero divided by zero, but the

boundary condition is discussed insection 4.4.2.

3.6.6 Natural period in roll

As discussed earlier the natural period an important parameter when analyzing motions of a

vessel. To calculate the natural period, the motion is assumed uncoupled from the other

degrees of freedom.

General formula of uncoupled natural period of roll T is given provided by DNV, where the

mass moment of inertia equals the added mass plus the mass moment of inertia. (DNV-RP-

H103 (2011) 2.3.3.5)

C

AI2

2T

n

n (3.108)

By substituting values for the mass moment of inertiaand stiffness in roll (respectively eq.

(3.83) andeq. (3.96)) for a rectangular barge intoeq. (3.108) the relation for natural period is

given ineq. (3.109).

BLDM (3.109)

I

AKGB

GMgGMgBLD

I

AKGBM

T ebn 112

212

112

2

2

22

22

arg,(3.110)

Where D is the static draft of the barge, B is the breadth of the barge, g is the gravity

acceleration constant, A and I are respectively the added mass moment and mass momentof inertia given insection 3.6.1.


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3.6.7 Solution

The equation of motion for roll motion is given ineq. (3.79):

2

tsinMCB)AI( e0 (3.111)

Where M is the total mass of the barge, A is the added mass moment of inertia, B is the

damping coefficient, C the stiffness coefficient, M0 the force amplitude, e the equivalent

wave frequency, t is time and is the rotational motion of the barge. The force frequency has

shifted a half circle due to the phase shift in the external loading proven insection 3.6.5.

Table 3.05shows examples of which formulas that can be used in the equation of motion and

which assumptions that is needed.

Table 3.05. Terms in equation of motion for roll

TermAssumptions used

for generalformulas

General formulas in rollfor a vessel

Additionalassumptions for

bargeBarge formulas

I dVrbody 2

Barge +cargo hasshape as a

rectangular box

Uniform weightdistribution

22 212

KGBM

A Strip theory

No water on deck )(2LDz dxxa

Added mass isunaffected by

wave frequencyNo viscosity

Deepwater

4

i2BL5.0

B

Energy relation influid fromNewman

Empirical dampingvalues fromChakrabarti

Linear wave theory

No water on deck

eqBB

BKLwefeq BBBBBB

eqBB

BKLwefeq BBBBBB

C

Correctly

calculation of

GM

No water on deck

GMg Rectangular bargewith flat bottom GMgBLD

)t(M

Strip theory

Morison equation

Linear wave theory

Long relative wavelength

No other forcecontributions thanapproaching wave

forces

S

D dsnyp

Sinusoidal waves

B


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Particular solution

The equation of motion has the same form as the equation of motion presented in section

3.3.1, and the solutions of the equation will be in the same form as proven earlier.

The equation of motions is derived insection 3.3.2, and the same solutions are applicable for

roll motion. To understand the vessels behavior, the response with respect to displacement,

velocity and acceleration are of interest and is presented ineq. (3.112) toeq. (3.70). Mark that

the frequency used is the encounter force frequency, e defined insection 3.4.5.

)tsin()t( ep (3.112)

)tcos()t( eep (3.113)

)tsin()t( e2ep (3.114)

Where the motion amplitude, and phase amplitude , is given in eq. (3.115) and eq.

(3.116).

C

MDAF

r2r1

C/M 0

222

o(3.115)

2)AI(C

Btan

2e

e1

(3.116)

Where DAF is the dynamic amplification factor, C is the stiffness, M0 is the force

amplitude, r is the frequency ratio and , is the damping ratio in roll. Its important to

notice the reduction in phase shift due to the phaseshift of the force amplitude as seen in eq.

(3.111) andeq. (3.106).

222 r2r11DAF

(3.117)

n

er (3.118)

C)AI(2

B

B

B

C

(3.119)


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Homogeneous solution

The homogeneous solution is shown insection 3.3.4and given for heave insection 3.5.7. The

homogeneous solution will be in the same form where the changes are defined in section

3.6.1.

Total solution

If the vessel is influenced by a sudden change in external forces, the homogeneous solution

must be taken into account, so that the total response of the system is the sum of

homogeneous solution and particular solution. To obtaineq. (3.120) linear theory is assumed

so that the total response is the sum of two responses.

)t()t()t( hp (3.120)

If the motion has been uniformly increasing until the considered response is given we can

neglect the transient, homogeneous solution and the total response of the system will equal

the particular solution.

)t()t( p (3.121)


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4. Results

The results are presented from numerical simulations and parametric studies. The calculations

is done in excel, and the relevant formulas needed for calculations is referred to by equation

number or table number and is found in chapter three.

Since most formulas used in the results are discussed thoroughly in the analysis part, the

result are presented as informative excel tables and charts where only key values are

discussed.

4.1 The case

Dynamic vessel response is normally calculated using different finite element software. To

calculate the hydrodynamic properties of vessels, the strip theory which is described in

section 3.4.6must be applied. This is because hydrodynamic properties like radiation (added

mass and damping) and diffraction forces are normally not constant over the length of the

ship due to wave properties and geometrical changes of the ship.

In case of a rectangular barge, the hydrodynamic properties can be considered constant over

the length of the barge, making it easy to use the strip theory, even when the calculations are

done by hand. In case of ships it would be possible to divide the ship into ten or twenty strips,

do similar calculations as for the barge and do the integrations by summarizing the terms for

all strips to get the total response.

4.1.1 Input

The considered vessel is a North Sea barge which is used for standard tow operations in theNorth Sea. The dimensions and weight of the barge is given intable 4.01.

Table 4.01. Standard North Sea Barge, input

Source: Greenway shipping, Greenbarge

Description Coefficient Value Unit

Totallength L 91,44 [m]

Totalbreadth B 27,43 [m]

TotalHeight H 6,10 [m]Lightshipweight,barge Mbarge 2361 [t]

Deadweight Meq,max 9025 [t]

COGabovekeel COGBarge 3,05 [m]

StandardNorthSeaBarge:

After the vessel is chosen, there are only two parameters which can be changed, (if no

appendages are added) the mass and the center of gravity. The natural frequency of a vessel is

dependent on mass, added mass and stiffness of the structure. Since the added mass is

assumed unaffected by frequency, (section 3.5.2 and section 3.6.2) the mass and stiffness

terms are based on the geometry and weight of the vessel, the natural frequency is only

dependent on the deadweight on the vessel (cargo, ballast, fuel, etc.).


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Table 4.02 shows a loading case which is considered.

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