ship dynamics in waves_rostok

8/12/2019 Ship Dynamics in Waves_rostok

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Universitt Rostock

Fakultt fr Maschinenbau und Schiffstechnik

Ship dynamics in waves

Prof. Dr.- Ing. Nikolai Kornev

Rostock2011


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1. Ship motion in regular sea waves.

1.1 Coupling of d ifferent ship oscillations.

The ship has generally six degrees of freedom which are called as surge , sway ,heave , heel (or roll) , yaw and pitch (see Fig. 1.1 for explanation of eachoscillation motion). In this chapter we consider first the case of the ship with zeroforward speed. Generally, different ship oscillations are strongly coupled. There arethree sorts of coupling: hydrostatic coupling, hydrodynamic coupling, gyroscopics coupling.

The hydrostatic coupling Is illustrated in Fig.1.2. If the ship draught is changed, the

centre of effort of vertical hydrostatic (floating) force is moving usually towards theship stern because the frames in the stern are more full than those in the bow region.The displacement of the centre of effort towards the stern causes the negative pitchangle. Therefore, the heave oscillations cause the pitch oscillations and vice versa.With the other words, the heave and pitch oscillations are coupled.

Hydrodynamic coupling can be illustrated when the ship is moving with accelerationin transverse direction (sway motion). Since the ship is asymmetric with respect tothe midship, such a motion is conducted with appearance of the yaw moment.Therefore, the sway and yaw oscillations are hydrodynamically coupled.

According to gyroscopic effect, rotation on one axis of the turning around the secondaxis wheel produced rotation of the third axis. This rule can be applied to the ship.For instance, if the ship performs rolling motion and the transverse force is acting onthe ship, it starts to perform the pitch oscillations. The gyroscopic effects are presentin the equation system (1.13). They are represented in i-th force equation by

products j i m j iV and by products j i m j i in the i-th moment equation.

In this chapter we consider the ship oscillations with small amplitude. For suchoscillations the coupling mentioned above can be neglected.


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Fig. 1.1: Ship motion with 6 degree of freedom (from [12]).

Fig. 1.2: Displacement of the centre of effort due to change of the ship draught(from [1]).

1.2 Classification of forces

According to the tradition used in ship hydrodynamics since almost a hundred years,the forces acting upon the ship are subdivided into hydrostatic forces, radiation anddiffraction forces. This subdivision can be derived formally utilizing the potentialtheory. The potential theory is still remaining the theoretical basis for the

determination of wave induced forces, since the most contribution to these forces iscaused by processes properly described by inviscid flow models.

Let us consider the plane progressive waves of amplitude A and direction w are

incident upon a ship, which moves in response to these waves. The ship oscillationcaused by waves can be written in the form

0

j jsin t, j 1,2, ...,6. (1.1)

The corresponding speeds of ship oscillations ju , j 1,2, ...,6. are:

j 0

j j

dU cos t, j 1,2,...,6.

dt

(1.2)

and accelerations:


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j 2 0

j j

dUa sin t, j 1,2,...,6.

dt (1.3)

Here 0j are small ship oscillations amplitudes and is the frequency. Within the

linear theory the ship oscillation frequency is equal to the incident wave frequency.

In what follows we use the linear theory and assume that the both waves and shipmotion are small. The total potential can be written, using the superpositionprinciple, in the form:

6

j j A

j 1

60

j j A

j 1

(x, y, z, t) U (x, y, z) A (x, y, z) cos t

(x,y,z) A (x,y,z) cos t

, (1.4)

where j(x,y,z) is the velocity potential of the ship oscillation in j-th motion with the

unit amplitude 0j 1 in the absence of incident waves, A (x,y,z) is the potential taking the incident waves and their interaction with

the ship into account.The first potentials

j(x,y,z) describes the radiation problem, whereas the second

one the wave diffraction problem. The potentials j(x,y,z) and A are

independent only in the framework of the linear theory assuming the waves and ship

motions are small. Within this theory A is calculated for the ship fixed in position.

The potentials must satisfy the Laplace equation j A0, 0 and appropriate

boundary conditions. The boundary conditions to be imposed on the ship surface arethe no penetration conditions (see also formulae (3.18) in the chapter 3 [2]): for radiation potentials

31 2

4

5

6

cos( , ); cos( , ); cos( , );

cos( , ) cos( , ) ;

cos( , ) cos( , ) ;

cos( , ) cos( , ) .

n x n y n zn n n

y n z z n yn

z n x x n zn

x n y y n xn

(1.5)

for wave diffraction potentials

0

A

n (1.6)

where n is the normal vector to the ship surface, directed into the body, (x,y,z) are thecoordinates of a point on the ship surface. The r.h.s. of the conditions (1.5) is thenormal components of the ship local velocities caused by particular oscillatingmotions.

The diffraction potential A is decomposed in two parts


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A p (1.7)

which is the potential of incident waves not perturbed by the ship presence andp is the perturbation potential describing the interaction between the incident waves

and the ship. The potential of regular waves is known (see Chapter 6 in [2]). Theboundary condition for p on the ship surface is

p

n n (1.8)

Away from the ship the radiation potentials j and the diffraction perturbation

potential p decay, i.e. 0, 0 p jr r .On the free surface the linearized mixed boundary condition (see formula (6.17) in[2]) reads

2

2 0

g

zt on z= 0. (1.9)

Substituting (1.4) in (1.9) yields forj(x,y,z) and A (x,y,z) :

2j

j 0g z

on z= 0. (1.10)

2

AA 0

g z

on z= 0.

Additionally in the wave theory the radiation condition is imposed stating that thewaves on the free surface caused by the potentials are radiated away from the ship.The potentials introduced above can be found using panel methods.

The force and the moment on the ship are determined by integrating the pressureover the wetted ship surface. The pressure can be found from the Bernoulli equationwritten in the general form:

2up gz C(t)

2 t

(1.11)

Here the potential is the potential of the perturbed motion. The constant C(t) which isthe same for the whole flow domain is calculated from the condition that the pressureon the free surface far from the ship is constant and equal to the atmosphericpressure:

ap C(t) (1.12)Substituting (1.12) in (1.11) gives:

2

a

up p gz

2 t

(1.12a)

Remembering that the ship speed is zero and perturbation velocities as well as thevelocities caused by incident waves are small we neglect the first term in (1.12a):

ap p gz

t

(1.13)

Together with (1.4) it gives


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60

a j j A

j 1

p p (x, y,z) A (x, y,z) sin t gz

(1.14)

The forces and the moment are then calculated by integration of ap p over thewetted ship area

S S

F pndS, M p(r n)dS.

(1.15)

The vertical ordinate z of any point on the wetted area can be represented as thedifference between the submergence under unperturbed free surface and free

surface elevation 0 . Substituting (1.14) in (1.15) one obtains

0

S S

60 2

j j

j 1 S

p

S

F g n dS g n dS

sin t n dS

A sin t n( )dS

(1.16)

Four integrals in (1.16) represent four different contributions to the total force: the hydrostatic component (the first term) acting on the ship oscillating on the

unperturbed free surface (in calm water), the hydrostatic component arising due to waves (the second term), the damping and the added mass component (the third term) and the hydrodynamic wave exciting force (the fourth term).

The moment is expressed through similar components.

The third term describes the force acting on the ship oscillating in calm water. The

last term arises due to incident waves acting on the ship. Within the linear theorykeeping only the terms proportional to the amplitude A and neglecting small terms ofhigher orders proportional to , 1nA n one can show that the integration in the lastterm can be done over the wetted area corresponding to the equilibrium state. Thus,the last term describes the force induced by waves on the ship at rest.

1.3 Radiation force components

Let us consider the second term of the force6 6

0 s

2 j j j jj 1 j 1S SF sin t n dS U (t t ) n dS2

(1.17)Each component of this force is expressed as

6 6j0

2i j j ji

j 1 j 1S

dUF sin t n dS c

dt

(1.18)

The hydrodynamic coefficient jic is represented as the sum of two coefficients:

1ji ji jic

The force is then6 6 6

j s

2i ji ji ji j ji jj 1 j 1 j 1

dU1

F ( ) a U (t t )dt 2

(1.19)


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As seen from (1.19) the first component of the force is proportional to theacceleration ja whereas the second one is proportional to the velocity jU . The first

component is called the added mass component, whereas the second one- thedamping component.

Using the Greens theorem, Haskind derived the following symmetry conditions forthe case zero forward speed:

ji ij

ji ij

ji ij

c c

1.3.1 Hydrodynamic damping.

There two reasons of the hydrodynamic damping of the ship oscillations on thefree surface. First reason is the viscous damping which is proportional to the

square of the ship velocity

2

jDj UC S

2 . Within the linear theory this term

proportional to the amplitude ( 0j )2 is neglected. The main contribution to the

damping is done by the damping caused by radiated waves. When oscillatingon the free surface the ship generates waves which have the mechanicpotential and kinetic energy. This wave energy is extracted from the kineticenergy of the ship. Ship transfers its energy to waves which carry it away fromthe ship. With the time the whole kinetic energy is radiated away and the shiposcillations decay.

Similarly to the added mass one can introduce the damping coefficients. Thefull mechanic energy in the progressive wave with the amplitude A is (seechapter 6.4 in [2])

21E gA2

(1.20)

per wave length.

Fig.1.3: Illustration to derivation of damping coefficient.

The energy transported by waves through sides 1 and 2 (see Fig. 1.3) per time unit is


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21E 2 gA U2

(1.21)

where U is the wave group velocity. The damping coefficient is defined as2

ij jE U (1.22)

where ij is the coefficient of damping in i-th direction when the ship oscillates in j-thmotion. 2jU is the time averaged square of the ship oscillations speed. Obviously,

0 2

j2

j

( )U

2

and

0 2

j

ij

( )E

2

(1.23)

The group velocity (see formulae (6.39) and (6.40) in [2]):

c 1 g gU

2 2 k 2 (1.24)

since 2kg (see formula (6.21) in [2]. Equating (1.21) and (1.23) one obtainswith account for (1.24)

0 2

j2

ij

20 2

j2

ij ij 3 0

j

( )gA U

2

( )g g AgA

2 2

(1.25)

The damping coefficientij depends on the square of the ratio of the wave amplitude

to the ship oscillation amplitude causing the wave. The damping coefficient dependsalso on the frequency as 3 .

The damping coefficient of slender body can be found by integration of dampingcoefficients of ship frames along the ship length

L/2 L/2 L/2

22 22 33 33 44 44

L/2 L/2 L/2

L/2 L/2

2 2

55 33 66 22

L/2 L/2

B dx, B dx, B dx,

B x dx,B x dx

(1.26)

The damping coefficients of different frames are shown in Fig. 1.4, 1.5,1.6, 1.7 and1.8 taken from [1]. Solid lines show results obtained from the potential theory.Generally, the results show the applicability of the potential theory for calculation ofdamping coefficients. The accuracy of prediction is not satisfactory for the box B/T=8in heave and B/T=2 in sway because of the flow separation at corners which has asufficient impact on hydrodynamics in these two cases. The agreement for 44 is not

satisfactory (see Fig.1.8) because of dominating role of the viscosity for this type ofdamping. For the semi circle frame the damping coefficient in roll is zero 44 0 withinthe inviscid theory. One hundred per cent of the roll damping is due to viscosity.Usually

44

are determined using viscous flow models. It is remarkable, that the

damping coefficients depend on the frequency and amplitude (see Fig.1.8).


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Fig. 1.4: Added mass and damping coefficient of the semi circle frame at heaveoscillations.

Fig. 1.5: Added mass and damping coefficient of the box frame at heaveoscillations.

Fig. 1.6: Added mass and damping coefficient of the semi circle frame at swayoscillations.


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Fig. 1.7: Added mass and damping coefficient of the box frame at swayoscillations.

Fig. 1.8: Added mass and damping coefficient of the box frame at roll (heel)oscillations.

1.3.2 Added mass component.

When the ship oscillates, the force acting on the ship contains the componentassociated with the added mass like in every case of accelerated body motion. Thedifference with the case of the motion in unlimited space is the presence of the freesurface. The added mass ij have to be calculated with account for the free surface

effect. For their determination the panel methods can be used. The problem is

sufficiently simplified in two limiting cases 0 and . The boundary condition(1.10) can be written in the form:j

j

0 for 0,z

0 for .

on z= 0. (1.27)

The conventional mirroring method can be used for the case 0 (Fig.1.9). Themirroring frame is moving in the same direction for surge, sway and yaw. For theheave, roll and pitch the fictitious frame is moving in the opposite direction. At thefree surface, these tricks make the normal components of the total velocity induced

by the actual and the fictitious frames zero, i.e. j 0z

on z=0.


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In the case the tangential component of the total velocity should be zero,

sincex

j j

j j0 dx 0 0x x

on z=0. The modified mirroring method is

implemented for the case (Fig.1.10). The fictitious frame is moving in the

opposite direction for surge, sway and yaw. For the heave, roll and pitch the fictitiousframe is moving in the same direction as that of the original frame. These tricks makethe normal components of the total velocity induced by the actual and the fictitious

frames zero, i.e. j 0z

at the free surface on z=0.

Fig.1.9: Mirroring for the case 0

Fig.1.10: Mirroring for the case

Using mirroring method the added mass can be found using the panel without explicitconsideration of the free surface since it is taken into account by fictitious frames.

The added mass of slender body can be found by integration of added mass of shipframes along the ship length

L/2 L/2 L/2

22 22 33 33 44 44

L/2 L/2 L/2

L/2 L/2

2 2

55 33 66 22

L/2 L/2

A dx,A dx,A dx,

A x dx,A x dx

(1.28)

The added mass of different frames are shown in Fig. 1.4, 1.5, 1.6,1.7 and 1.8 takenfrom [1]. Like in case of damping coefficients the results of the potential theory are

not acceptable for roll added mass because of dominating role of the viscosity. Asseen from Fig.1.4-1.8 the added mass depend on the frequency .


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1.4. Hydrostatic component

Let us the ship is in the equilibrium state. The ship weight is counterbalanced by thehydrostatic lift. Due to small heave motion the equilibrium is violated and anadditional hydrostatic force appears. The vertical component of this additional

hydrostatic force can be calculated analytically from (1.16) for the case of smallheave motion

WP

S ST T

F g cos(nz)zdS g cos(nz)zdS gA

(1.29)

where is the increment of the ship draught, WLA is the waterplane area and T is the

ship draught in the equilibrium state. The roll and pitch hydrostatic moments for smallchange of the roll and pitch angles are

0M g GM , (1.30)

0 LM g GM , (1.31)

where and are the roll and pitch angle respectively, GM is the transversemetacentric height, LGM is the longitudinal metacentric height and 0 is the shipdisplacement.

1.5 Waveexciting force

The wave exciting force per pS

F Asin t n( )dS

contains two components. The

first component determined by the integration of the incident potential

SA sin t n dS

is referred to as the hydrodynamic part of the Froude- Krylow

force. This force called as Smith effect is calculated by the integration of waveinduced pressure as if the ship is fully transparent for incident waves. The fullFroude- Krylow force contains additionally the hydrostatic force arising due to changeof the submerged part of the ship due to waves (the second term in (1.16)). The

second component PS

A sin t n dS

takes the diffraction effect (the contribution of

scattering potential p to pressure distribution) into account. As shown by Peters

and Stokes the Froude Krylov force is a dominating part of the wave induced forcesfor oscilations of slender ships in directions j=1 (surge), 3 (heave) and 5 (pitch).


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1.6 Motion equations

The linearized decoupled motion equations of the ship oscillations are written in the

form added damping hydrostatic wave excitingmass forces forces forcesforce

11 11 ,per

22 22 ,per

33 33 WP ,per

xx 44 44 0 ,per

yy 55 55 0 L ,per

zz 66 66 ,pe

m A B F (t),

m A B F (t),

m A B gA F (t),

I A B g GM M (t),

I A B g GM M (t),

I A B M

r(t).

(1.32)

The weight is not present in the second equation of the system (1.32) because it iscounterbalanced by the hydrostatic force at rest. The additional hydrostatic force

WPgA is the difference between the weight and the full hydrostatic force. Thesystem (1.32) is written in the principle axes coordinate system [3].

1.7 Haskinds relation .

One of the most outstanding results in the ship oscillations theory is the relationderived by Max Haskind who developed in 1948 the famous linear hydrodynamictheory of ship oscillations. Haskind shown how to calculate the wave inducedhydrodynamic force utilizing the radiation potentials j and the potential of incident

waves . The determination of the diffraction potential p what is quite difficult can

be avoided using this relation which is valid for waves of arbitrary lengths.

The Greens formula for two functions and satisfying the Laplace equation is

[ ] 0wS

dSn n

, (1.33)

where wS is the flow boundary (wetted ship surface plus the area away from the ship,

see the sample in Chapter 3.2). Particularly, the relation (1.33) can be applied toradiation potentials j . Since the potential p satisfies the Laplace equation and the

same boundary conditions as the radiation potentialsj , the Greens formula (1.33)

can also be applied toj and p

[ ] 0

w

j p

p j

S

dSn n

(1.34)

The last term in (1.16) is the wave induced force


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,per pS

F A sin t n( )dS XAsin t

where pS

X n( )dS

. Taking (1.5), (3.34) and (1.8) into account we get

w

j

j p

S

X ( ) dSn

(1.35)

( )

w w w

j p j p

p j j j

S S S

dS dS X dS n n n n

(1.36)

p

n n

090

( )

w

j

j j

S

X dS

n n

(1.37)

The formula (1.37) is the Haskinds relation. As seen the wave induced force can becalculated through the radiation and free wave potentials avoiding the determinationof the diffraction potential p .

The calculation of the integral (1.37) is a complicated problem because the incidentwaves dont decay away from the ship and the integral (1.37) should be calculatedover both the surface far from the ship and the ship wetted surface. Note that thepotential does not decay away from the ship. The method of the stationary phase

[16] allows one to come to the following force expression using the Haskinds relation

(1.37): 22

,

0

, ( )8 ( / 2)

i per ii i ii i

kF B where B X d

g c

Here c is the phase wave velocity (celerity) and is the course angle.

Let us consider the slender ship ( , ) 0B x z in a beam wave ( 090 ). The wettedarea is approximately equal to the projection on the symmetry plane y=0,

[0, ] [0, ]wettedS L T .

( )

w w

j j

j j

S S

X dS dS

n n n

33cos( , ) 2

wS

B Bn z X dS

n z z

The coefficient 2 arises due to the integration over two boards ( , )y B x z and( , )y B x z . Using the potential of an Airy wave (see formule (6.18) in [2]) estimated

at y=0 one can find the potential :

sin /kz kzAg

e ky t ge

For the case of a vertical cylinder for which the vertical force does not depend on the

wave course angle the damping coefficient 33B takes a very simple form [16]:


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20

33

2

( / 2)

kz

T

g BB e dz

c z

(1.38)

2. Free oscillations with small amplitudes.

Let us consider a ship in the equilibrium position at calm water condition. The shiphas zero forward speed. If a perturbation acts on the ship, it performs oscillatingmotions in three directions: heave, roll (heel), pitch.

Yaw, surge and sway motions did not arise at calm water conditions. The reason isthe presence of restoring hydrostatic forces in heave, roll and pitch directions.

The motion equations of the free oscillation read:

33 33 WP

xx 44 44 0

yy 55 55 0 L

(m A ) B gA 0,

(I A ) B g GM 0,

(I A ) B g GM 0.

(2.1)

In ship theory the equations (2.1) are written in the normalized form:2

2

2

2 0,

2 0,

2 0,

(2.2)

where

33 5544

33 xx 44 yy 55

B BB, ,

2(m A ) 2(I A ) 2(I A )

(2.3)

are damping coefficients and

0WP 0 L

33 xx 44 yy 55

g GMgA g GM, ,

m A I A I A

(2.4)

are the eugen frequencies of non damped oscillations.

The equations (2.2) are fully independent of each other. The solutions of theequations (2.2) written in the general form:

22 0 (2.5)is given as:

ptCe (2.6)Substitution of (2.6) into (2.5) yields the algebraic equation

2 2p 2 p 0 (2.7)which solution is

2 2

1,2p (2.8)

If the system has no damping the solution isi t

1,2p i Ce C(cos t i sin t) (2.9)


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The system oscillates with the constant amplitude and frequency . That is why thefrequency is referred to as the eugen frequency.

For real ships the damping coefficient is smaller than the eugen frequency andthe equation (2.8) has two solutions:

2 21

2 2

2

p i i ,

p i i ,

(2.10)

In turn, the solution of the differential equation ist i tCe e Ce (cos t i sin t) (2.11)

It describes damped oscillations with decaying amplitude which decrease is governedby the factor t t

te ,e 0 . The rate of the decay is characterized by the damping

coefficient . The frequency of damped oscillations is2 2 (2.12)

Due to damping the frequency of the oscillations is shorter whereas the periodis longer:

2 2

2 2T

(2.13)

Since

2 2

2 2 2T

(2.14)

From (2.12) and (2.14) we obtain the frequencies for different types of oscillationswhich are listed in the table below:

Table 10.1: Frequencies and periods of different oscillation typesOscillation Eugen frequency Frequency of

dampedoscialltions

Period ofoscillations

HeaveWP

33

gA

m A

2 2

33WP

m AT 2

gA

Rolling0

xx 44

g GM

I A

2 2

xx 440

I AT 2

g GM

Pitch0 L

yy 55

g GM

I A

2 2

yy 55

0 L

I A

T 2 g GM

The damping is characterized by the logarithmic decrement which is thelogarithm of the ratio of the oscillation amplitude at the time instant t to that atthe time instant t+T, i.e.

tT

( t T )

(t) ee

(t T) e

(2.15)

The logarithm of the ratio (2.15) ist

T

( t T )

(t) e 2

ln ln ln e T(t T) e

(2.16)


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The ratio

is called as the referred damping factor . The decay of the oscillation

amplitude is equal to this factor multiplied by 2. Referred damping factors fordifferent types of oscillation can be found from this definition. The results obtainedunder assumption are listed in the table 10.2.

If the meatcentric heights GM and LGM are getting larger, the periods

xx 44

0

I AT 2

g GM

and yy 55

0 L

I AT 2

g GM

as well as the damping factors

44

xx 44 0

B

2 (I A ) g GM

and 55

yy 55 0 L

B

2 (I A ) g GM

decrease. Therefore,

the smaller are the metacentric heights the larger are the oscillations periods and thefaster these oscillations decay.

Table 10.2: Referred damping factors for different oscillation typesOscillation Referred damping factorHeave

33

33 WP

B

2 (m A ) gA

Rolling44

xx 44 0

B

2 (I A ) g GM

Pitch55

yy 55 0 L

B

2 (I A ) g GM

3. Ship oscil lations in small t ransverse waves (beam see).

The formalism developed in this chapter is based on the following assumptions:

waves are regular, waves amplitudes related to the wave lengths are small. Wave slope is small. wave length is much larger than the ship width, The ship has zero forward speed.

From the first two assumptions it follows, that the collective action of waves on shipcan be considered through the superposition principle. Therefore, the theory can bedeveloped for the interaction of the ship with a single wave with given length andamplitude. The effects of different waves are then summed. For the case of smallwaves the oscillations are decoupled. The hydrodynamic, hydrostatic and gyroscopiccoupling effects are neglected.

The perturbation forces (see the last column in the equation system (1.32)) arise dueto wave induced change of the hydrostatic forces and due to hydrodynamic effectscaused by orbital motion in waves. The orbital motion causes the hydrodynamic

pressure change which results in the wave induced hydrodynamic forces.


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In each frame, the pressure gradient induced by waves is assumed to be constantalong the frame contour and equal to the pressure gradient at the centre A on thefree surface. When considering the roll and pitch oscillations in transverse waves it isadditionally assumed that the ship draught change and the ship slope relatively tothe free surface are constant along the ship.

The wave ordinate is given by the formula derived for the progressive wave (seechapter 6 in [2])

0 A sin( t k ) (3.1)

where A is the amplitude, is the wave propagation direction and is the frequency.

In this section the incident waves are perpendicular to the ship (see Fig.3.1). Thewave propagation direction is in direction, i.e. =. The waves induce roll andheave oscillations. The curvature of the free surface is neglected, the free surface isconsidered as the plane performing angular oscillations and translational oscillations

in vertical direction.

Fig.3.1:

3.1. Hydrostatic forces and moments.

The hydrostatic forces during the heave oscillations are calculated neglecting thewave surface slope. The hydrostatic force acting on the ship with draft increment in

the wavewith the ordinate 0 (see Fig.3.2) ishydr

WP 0 WPF gA ( ) gA ( Asin t) (3.2)

where 0 at the point A . Since the wave slope is neglected, the dependence of 0 on is not considered. The first part WPgA is the restoring hydrostatic force

which is already present in (1.32). The second part WPgA A sin t is the wave

induced hydrostatic force.


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Fig.3.2: Illustration of hydrostatic force

The additional hydrostatic moment is determined from the analysis of the Figure 3.3.

Fig.3.3: Illustration of hydrostatic moment

The relative slope of the ship to the free surface is , where is the wavesurface slope

2

0A

0

dAk cos t Acos t cos t

d g

(3.3)

A is the amplitude of the angular water plane oscillations. The hydrostatic pressure

increases linearly in direction perpendicular to the free surface plane. Therefore, therestoring moment is the same as in the case if the free surface is horizontal and theship is inclined at the angle . The restoring moment is known from the shiphydrostatics

hydr

0 0 AM g GM ( ) g GM ( cos t) (3.4)

The first part 0g GM is the restoring hydrostatic moment, whereas the second

part 0 Ag GM cos t is the wave induced hydrostatic moment.

3.2 Hydrodynamic Krylov- Froude force.

Hydrodynamic forces arise due to wave induced hydrodynamic pressures. From theBernoulli equation the pressure is (see (1.13))


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2

a

up gz p

2 t

(3.5)

The constant pressure ap does not need to be considered since being integrated

over the ship wetted surface results in zero force and moment. The first term in (3.5)

is neglected within the linear theory under consideration. The second term results inforce and moment considered above in the subsection 3.1. The remaining term

unstpt

is responsible for hydrodynamic effects caused by waves. If the

interaction between the ship and incident waves is neglected (Krylov- Froudeformalism) the potential can be written as the potential of uniform unsteady parallelflow:

u (t) (3.6)

where u (t) is unsteady velocity of the flow in the wave

0du (t)dt

(3.7)

The gradient of the hydrodynamic pressure in vertical direction reads:

2unst0

u (t)pAsin t

t t t

(3.8)

The force caused by unstp on each frame is calculated by the integration of the

pressure over the frame wetted area0 unst

dyn unst

0

2 2

pdF p cos(n )dC dz cos(n )dC

z

Asin tdz cos(n )dC Asin t cos(n )dC

(3.9)

Note that is positive in this derivation. Since the integral cos(n )dC is equal tothe frame area taken with the opposite sign, i.e. fA , the hydrodynamic force causedby waves takes the form:

dyn 2

fdF Asin tA (3.10)

Being integrated along the ship length this force gives the force acting on the wholeship length

L L L

dyn dyn 2 2 2

f f 0

0 0 0F dF d Asin tA d Asin t A d m Asin t m

(3.11)

The hydrodynamic moment acting on the ship framedyn unst 2dM p ( cos(n ) cos(n ))dC Asin t ( cos(n ) cos(n ))dC

(3.12)Within the linear theory considering small ship slopes the last integral in (3.12) iszero, i.e.

L

dyn dyn dyn

0

dM 0 M dM 0 (3.13)

3.3 Full Krylov- Froude force and moment.


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The full Froude Krylov force takes the form:1 2

WP WP 0F gA Asin t m Asin t gA Asin t m (3.14)

The first term is caused by hydrostatic effect, whereas the second one byhydrodynamic effects. The second term is referred in the literature to as the Smitheffect.The full Froude Krylov moment contains only the wave induced hydrostaticcomponent:

1

0 AM g GM cos t (3.15)

3.4 Force and moment acting on the ship frame in accelerated flow.

These forces are determined using the concept of the relative motion. Let us j is a

ship displacement in j-th direction. As it has been explained in previous chapters, theforce acting on the body moving with the acceleration j in a liquid at rest is equal to

the product of added mass with the acceleration taken with opposite sign, i.e. jj jA .If the liquid moves with the acceleration jL relative to motionless body, the force

acting on the body is towards the acceleration direction, i.e. jj jLA . If both body and

liquid move with accelerations the total force is jj j jLA ( ) . Similarly, the damping

force can introduced being proportional to the relative velocity jj j jLB ( ) . The first

components of both forces jj jA and jj jB are already represented by the first

and the second columns in the motion equations (1.32). The second components

jj jLA and jj jLB represent the hydrodynamic forces due to interaction between the

incident waves and floating body. Remembering that jL Acos t and2

jL Asin t we obtain the lift force caused by the interaction between the ship

and incident wave:2 2

33 33F A Asin t B A cos t (3.16)

In roll oscillations the ship moves with the angular velocity and angularacceleration . The free surface oscillates with the angular velocity andacceleration . Taking from (3.3) we obtain the roll moment caused by theinteraction between the ship and incident wave:

4 32 2

44 A 44 A 44 44M A cos t B sin t A A cos t B Asin tg g

(3.17)

3.5 Full wave induced force and moment.

In the section 1.5 we divided the wave induced forces into the Froude Krylov part andthe interaction force. Commonly, the Froude Krylov force is the dominating part of thewave induced forces.

To calculate the full wave induced force we have to note that the Smith effect is

already represented in the force jj jL jj jLA B . All hydrodynamic effects are taken


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into account. Only the hydrostatic part of the Froude Krylov force should be added

to jj jL jj jLA B to get the full wave induced force:2

,per WP 33 33F gA A Asin t B Acos t (3.18)

The full moment is the sum of (3.15) and (3.17):2 3

2

,per 0 44 44M ( g GM A ) A cos t B Asin tg g

(3.19)

3.6 Equations of ship heave and rol l osci llations

Substitution of all forces derive above into the original differential equations results intwo following decoupled ordinary differential equations

33 0 33 0 WP 0 0(m A )( ) B ( ) gA ( ) m (3.20)

xx 44 44 0 xx(I A )( ) B ( ) g GM ( ) I

(3.21)The solution of both equations can be represented as the sum

inh free

inh free , where free and ree are free heave oscillations:t

free Ce (cos t i sin t)

, tfree Ce (cos t i sin t)

satisfying the

homogeneous equations:

33 33 WP(m A ) B gA 0

xx 44 44 0(I A ) B g GM 0

When the free oscillations decay , 0free free t , the solutions of the equation

(3.20) and (3.21) tend to the solutions of inhomogeneous equations:

33 33 WP 33 0 33 0 WP 0(m A ) B gA A B gA (3.22)

xx 44 44 0 44 44 0(I A ) B g GM A B g GM

The inhomogeneous equation (3.21) is written in terms of relative roll angle

( )r in the normalized form:2

( ) ( ) 2 ( )2 cos1

r r r

A t

k

, (3.23)

where 44/ xxk A I . The solution of (3.23) is seeking in the form

( ) ( ) cos( )r rA t (3.24)

Substituting (3.24) into (3.23) and separating terms proportional to cos t andsin t gives two equations:

2( ) 2 2( )cos 2 sin

1

r

A Ak

(3.25)

( ) 2 22 cos ( )sin 0rA (3.26)

It follows from (3.25) and (3.26)

42( ) 2 2 2 2 2 2 2 2 2 2

2( ) cos 4 sin 4 sin ( )cos

1

r

A A

k


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2

( ) 2 2 2 2 2 2 2 2 2( ) sin 4 cos 4 cos ( )sin 0rA

The sum of two last equations

42

( ) 2 2 2 2 2 2 2

2( ) 4 cos

1

r

A A

k

allows one to find the ratio ( ) /rA A

2( )

2 2 2 2

/ (1 )

(1 ) 4

r

A

A

k

, (3.27)

where

and

. Eigen frequency and damping coefficient are

given by formulae (2.3) and (2.4). The phase of the response relative to that of theinput (phase displacement) is found from (3.26):

2

2

1arctg

(3.28)

Similar solutions are obtained for the heave oscillations:

2( )

2 2 2 2

/ (1 )

(1 ) 4

r

A k

A

(3.29)

2

2

1arctg

(3.30)

with33/k A m ,

and

.

3.7 Analysis of the formula (3.27)

The formula (3.27) can be rewritten as follows:2( )

2 2 2 2

/ (1 )

(1 ) 4

r

A A A

A A

k

or 2

2 2 2 2

/ (1 )1

(1 ) 4

A

A

k

(3.31)

The physical meaning of terms in (3.31) is obvious from the following expression

1amplitude of ship roll oscillations

enhancementamplitude of wave angle oscillations

(3.32)

Since the function2

2 2 2 2

/ (1 )

(1 ) 4

k

is positive

2

2 2 2 2

/ (1 )0

(1 ) 4

k

the ship

roll amplitude is larger than the the amplitude of the angular water plane oscillations

A , i.e. 1A

A

.


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Fig.3.4: Ship as linear system

The ship can be considered as a system with the waves as input and the resultingmotion as the output (Fig.3.4). As seen from (3.31) this system is linear for smallamplitude oscillations. In terms of linear system theory the formula (3.31) reads

1output

enhancementinput

(3.33)

The linear system is time invariant. The output produced by a given input isindependent of the time at which the input is applied. The function 1 enhancement

which characterizes the system response in the frequency domain is called thefrequency response function.

The enhancement function2

2 2 2 2

/ (1 )

(1 ) 4

k

goes to zero if referred frequency

becomes zero. At very large frequencies ,2

2 2 2 2

/ (1 )1 / (1 )

(1 ) 4

kk

.

The enhancement is maximum in the resonance case1

1

1 2 1 2

. Strictly speaking the resonance frequency

1 2

is not equal to the eugen frequency

, i.e. . Since is small,

this discrepancy can be neglected . Typical dependence of the ratio

( )r

A

A

on

the referred frequency is presented in Fig. 3.5.


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Fig. 3.5: Response funct ion versus referred frequency

Typical dependence of the phase displacement on the referred frequency ispresented in Fig. 3.6.

Fig. 3.6: Phase displacement versus referred frequency

The phase displacement is equal to / 2 in the resonance case 1 for every

damping. For 0

the phase displacement disappears. For

the

phase displacement tends to . The largest relative roll angle occurs in the

resonance case either at wave crests or at wave troughs (Fig.3.7). Indeed, themagnitude of the relative roll angle in the resonance case


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( ) ( ) ( )cos( ) sin2

r r r

A At t

attains the maximum ( )rA at sin 1t . It

corresponds to wave crest 0 A sin t A and wave trough 0 A .

Fig. 3.7: Ship oscil lations in resonance case.

Fig. 3.8: Oscillation of a raft with a big metacentric height

At very large metacentric height GM the eugen frequency is also getting large

0

xx 44

g GM

I A

. The referred frequency for a limited wave frequency

tends to zero 0

. The relative roll angle amplitude and phase displacement

are zero. The floating body moves together with the free surface as shown in Fig.3.8like a raft.

Similar results are obtained from analysis of the heave oscillations formulae (3.29)

and (3.30).

3.8 Sway ship osci llations in beam sea.

The equation describing the sway oscillations is (see the second equation in thesystem (1.32)):

22 22 ,per m A B F (t)

(3.34)

The wave exciting force ,perF (t) consists of two components of hydrostatic and

hydrodynamics nature. As seen from Fig. 3.9 the hydrostatic force ishyd 2 2

0 0F g Acos t m Acos t (3.35)


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Fig. 3.9: Illustration of the frame in beam waves.

The horizontal oscillations of the wave surface can be presented in harmonic form(see formula (6.23) in [2]):

0

0 0

20

( , ) cos ,

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

cos

kzy z A e ky t

A t A t

A t

(3.36)

The hydrodynamic component of the wave induced force is written in the similar formas (3.16):

2

22 220 0F A B

(3.37)

Substitution of (3.37) and (3.35) into (3.34) gives:

22 22 22 220 00m A B m A B

(3.38)or

22 220 0(m A )( ) B ( ) 0

(3.39)The solution of the equation is written in the form:

0( ) tCe

(3.40)which substitution into (3.39) allows one to find

22

22

B

m A

(3.41)

The parameter is positive. Therefore, 0 0( ) 0t

t tCe

. As soonthe transitional process is finished, the ship oscillates together with the wave

0 sinA t (3.42)

3.9 Ship oscillations at fin ite beam to wave length ratio /B L and draught tolength ratio.

The analysis presented above was carried out for the case of a very long wave, i.e.both the beam to length ratio /B L and the draught to length ratio /T L are small.The results for roll oscillation obtained for the case / 0, / 0B L T L are extended tothe case / (1), / (1)B L O T L O using reduction coefficients. According to thistraditional in shipbuilding approach the wave amplitude is multiplied with thereduction coefficient , i.e.


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redA A (3.43)The ship oscillations at / (1), / (1)B L O T L O are smaller than these at

/ 0, / 0B L T L due to two reasons Hydrostatic force is smaller because the submerged volume is smaller due to

wave surface curvature, Hydrodynamic force is smaller because the velocities caused by the orbital

motion are not constant as assumed above. They decay with the increasingsubmergence as exp( )kz .

The first reduction factor is mainly due to the finite beam to length ratio / (1)B L O .

Let us consider first the reduction coefficient for the heave oscillations. The factor

B considers the reduction of the hydrostatic force due to the finite beam to length

ratio. To estimateB the fixed ship is considered at the time instant t / 2 when

the wave crest is in the symmetry plane (Fig.3.10).

Fig.3.10

The free surface ordinate

0 A sin( k ) A cos k 2

The hydrostatic force obtained in the previous analysis is

0 wpR gAA (3.44)whereas the actual one is calculated by the integral:

( ) /2/2 /2

/2 0 /2

2 ( )cos 2 cos sin

2wp

BL L

true

A L L

gA kBR gA k d d gA k d d d

k

(3.45)

Using the Taylor expansion for sin ( )kB 3( ) ( )

sin ...2 2 48

kB kB kB

the final formula fortrueR takes the form:


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/2 /2 3

/2 /2

/22 23

/2

2 ( ) 2 ( )sin ( / 2 )

2 48

1 ,24 2

L L

true

L L

L

wp wp

wpL

gA kB gA kBR d kB d

k k

gAk k I gAA B d gAA

A

(3.46)

where/2 /2

3

/2 /2

1, ,

12

L L

wp

L L

A Bd I B d

The reduction of the hydrostatic force can be taken by the following coefficient intoaccount:

2

21

21

2

wp

wp

B

wp wp

k IgAA

A k I

gAA A

(3.47)

The second reduction factor is mainly due to the finite draught to length ratio/ (1)T L O . The factor T considers the reduction of the hydrodynamic force due to

the finite draught to length ratio. The reduction coefficient is given here withoutderivation:

2 3

1 2 2 22(2 ) 6(3 2 )

T

T T T

L L L

(3.48)

where is the coefficient of the lateral area / ( )LAA LT .

The total reduction coefficient is calculated as the product of B and

T neglecting their mutual influence:

B T (3.49)

The formula (3.49) is valid at 4, 8L L

B T . For heave calculations one can use the

formula (3.29) with A instead of A .

Reduction coefficient of the roll oscillations can be calculated from the expressiongained from regression of experimental data:

1/2

2exp 4.2( ) ,2

rBT

GMR R

g

(3.50)

Here r is the metacentric radius. Amplitude of roll oscillations is found from (3.31)

withA instead of A . A sample of the reduction coefficient for a real ship is

presented in Fig. 3.11.


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Fig. 3.11. Reduction coefficient of the heave oscillations.

4. Ship oscillations in small head waves.

4.1 Ship oscil lations

Let us consider the ship oscillations in small head waves coming from the stern

0wave , where is the wave course angle. The wave ordinate, wave orbitalmotion velocity and acceleration are:

)sin(0 ktA (4.1)

0 cos( )A t k (4.2)2

0 sin( )A t k (4.3)

Fig. 4.1: Illustration of the ship in head waves.

The perturbation force acting on the section AB (Fig.4.1) can be represented as thesum of

the hydrostatic Froude Krylov force 0gB ,

the hydrodynamic Froude Krylov force 0( )f ,

the interaction force 33 0 33 0( )


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, 33 0 33 0 0( ) ( )per

f

dFgB

d

(4.4)

Integrating ,perdF

d

over the ship length we obtain the whole wave induced force

,perF . If the ship is symmetric with respect to the midship = , f = f , 33 = 33 , 33 = 33 the formula for ,perF

is simplified to:

2

, 33

33

cos sin

cos cos cos sin

per f

L

L L

F A A k d t

A k d t A gB k d t

2

33 33

, ,

cos sin cos cos

sin

f

L L

per per

A gB A k d t A k d t

F t

(4.5)

where

2 2

2 2

, 33 33

33

, 2

33

cos cos ,

cos

arctan ;cos

per f

L L

Lper

f

L

F A gB A k d k d

k d

gB A k d

(4.6)

The wave exciting moment is calculated by multiplication of,perdF

d

with the arm :

,

,

2

33 33

, ,

sin cos sin sin

sin

per

L

f

L L

per per

dFM d

d

A gB A k t A k t

t

(4.7)

where

2 2

2 2

, 33 33

2

33

,

33

sin sin ,

sin

arctan ;sin

per f

L L

f

Lper

L

M A gB A k d k d

gB A k d

k d

(4.8)

Substitution (4.5) and (4.7) in the third and sixth equations of the system (1.32) gives:

33 33 WP ,per ,per m A B gA F sin t

(4.9)

yy 55 55 0 L ,per ,per I A B g GM sin t

(4.10)


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33 33 WP ,per ,per (m A ) B gA F sin t

(4.11)

yy 55 55 0 L ,per ,per (I A ) B g GM sin t

(4.12)

Dividing both equations by the coefficient of the first term one obtains:

2

,

2

,

2 sin

2 sin

per

per

f t

f t

(4.13)

where

, ,

33 55

,per per

yy

F Mf f

m A A

, 33 55

33 yy 55

B B,

2(m A ) 2(I A )

,

WP 0 L

33 yy 55

gA g GM,

m A I A

.

Solution of (4.13) is seeking in the form ,sin perA pert , ,sin perA pert (4.14)After some simple manipulations the amplitudes of the heave and pitch oscillationsas well as the phase displacements are obtained from (4.13) and (4.14):

22 2 2 24

A

f

,2 2

2arctanper

(4.15)

2

2 2 2 24A

f

,2 2

2arctanper

(4.16)

In the resonance case the phase displacement is equal to / 2 , i.e./ 2per in case and / 2

per

in case .

4.2 Estimations of slamming and deck flooding

Results of ship oscillations obtained in the previous section can be used forpractically useful estimations. For instance, we can estimate the slamming and deckflooding. Using the relations derived above

)sin(0 ktA ,

,sin

per

A pert ,

,sin

per

A pert (4.17)

one can display the ship positions in head waves as shown in Fig. 4.2

Fig. 4.2. Position of ship at different time instants in a head wave.


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Let us represent the formulae (4.17) in the form:

0 sin cos cos sinA t k A t k (4.18)

, ,sin cos cos sinper perA per A pert t

, ,sin cos cos sinper per

A per A pert t

The local change of the draft is:0 1 2( ) ( ) cos ( )sinz x x f x t f x t (4.19)

where

1 , ,

2 , ,

( ) cos sin sin

( ) sin cos cos

per per

A per A per

per per

A per A per

f x A k x

f x A k x

(4.20)

Fig. 4.3. Curves max ( )y z x and ( )y z x .

A sample of the curve ( )y z x is shown in Fig. 4.3. The maximum draft is then2 2max 1 2( ) ( ) ( )z x f x f x (4.21)

The curvemax ( )y z x shows the contour of maximum wave elevations along the

ship board in the symmetry plane whereas the curvemax

( )y z x the minimum waveelevations. Both curves are symmetric with respect to the equilibrium water plane.

Deck flooding takes place if max ( )z x H , where H is the board height.

Slamming takes place if max ( )z x T

There are three zones limited by curvesmax ( )y z x can be distinguished along the

ship board (see Fig.4.3):

Allways dry area (white), Allways wetted area (red), Intermediate area (orange).

A sample of flooding curves for a real ship is given in Fig.4.4


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Fig. 4.4. Sample for a real ship

5. Seasickness caused by ship oscillations

Symptoms of the seasickness are giddiness (Schwindelgefhl), headache(Kopfschmerz), sickness (belkeit) and vomiting (Erbrechen). The seasickness isthe reason of work capacity reduction, memory decline (Rckgang derGedchtnisleistung), motion coordination (Bewegungskoordinierung), reduction ofmuscular strength, etc. Diagram of Sain Denice (fig. 5.1) shows the influence of thevertical acceleration on the seasickness depending on the oscillation period.


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Fig.5.1. Influence of the vertical acceleration on the seasickness depending on theoscillation period

Fig.5.2 Influence of the vertical acceleration on the seasickness depending on theoscillation period.

For the irregular sea state the similar diagram was proposed by Krappinger (Fig.5.2)who estimated the percentage of people suffering from the seasickness dependingon the root mean square deviation and frequency.


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According to standards developed in US Navy the oscillations have no significanteffect on the work capacity if the amplitude of the roll oscillations is under eightdegrees, the amplitude of pitch oscillation is below three degrees, the verticalaccelerations does not exceed 0.4 g whereas the transversal accelerations 0.2g.The upper limit of the roll angle for the deck works is 20 degrees which corresponds

to the reduction of the work capacity of about 50 percent.

At present the seasickness has insufficiently been studied in medical science. Asshown in the study by Vosser, the seasickness is developed at a certain level ofoverloads and then can remain even the ship oscillations decay. It is shown indiagram 5.3 presenting the number of passenger on a cruise liner /n n suffering

from the seasickness during eighty hours of the journey. At the journey beginning

the vertical acceleration /g was less than 0.1 and only 16 percent of passengerwere sick. As soon as the vertical acceleration attained 0.4g more than 80 percent ofpassengers were sick. In spite of the ship oscillation decay after 30 hours of the way

the number of sick passengers is not reduced. On the contrary this number is slightlyincreased during the next 24 hours. Only after 36 hours the seasickness retreated.The next diagram 5.4 illustrates the fact that the adaption to seasickness is relativelyweak.

Fig.5.3. Number of passengers suffering from seasickness on a cruise linersdepending on vertical accelerations.

Diathesis to seasickness depends on the individual properties of organisms. Thereare many people who had never had problems with seasickness. However there areexperienced seamen who suffers from this sickness the whole professional life.


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Fig.5.4 Adaption to seasickness.

6. Ship oscillations in i rregular waves.

6.1 Representation of irregular waves

The irregular waves can be both two dimensional and three dimensional (Fig.6.1).

Fig.6.1 Irregular seawaves, 1- two dimensional, 2 three dimensional. (Fig. from [14])


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Fig. 6.2 Profile of an irregular wave. (Fig. from [14])

A feature of the irregular waves distinguishing them from regular ones is the non-recurrence of their form in time (Fig. 6.2). The following relations between wavelengths L and wave heights h are recommended in practical calculations for swell:

3/40.17h L Zimmermann,1/20.607h L British Lloyd,

0.60.45h L det Norske Veritas.Within the linear theory the irregular waves can be represented as the superpositionof regular waves with different amplitudes, frequencies and course angles, as shownin Fig. 6.3

6.1.1. Wave ordinates as stochast ic quantit ies

The wave ordinate is the stochastic function with a certain probability density function(see Fig.6.1). The p.d.f. distribution of the real irregular wave ordinates is Gaussian.i.e,

2 20( ) /(2 )

1p.d.f . e

2 D

, (6.1)

where 0 is the mathematical expectation (in our case 0 0 ), is the standarddeviation:

2 20( ) D (6.2)

D is the dispersion. Probability 1 2P( ) of the event, that the ordinate lies in

the range between 1 and 2 is then

22 22 2 2

1 1 1

/ 2D

/(2 ) t

1 2

/ 2D

1 1P( ) p.d.f ( )d e d e dt

2 D

(6.3)


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Fig. 6.3 Representation of irregular wave through the superposition of regular waves.(Fig. from [14])

Fig.6.4: p.d.f. of the wave ordinate.

The last integral is known as the probability integral


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2x

t

0

2(x) e dt

(6.4)

satisfying the following properties( x) (x), ( ) 1, ( ) 1 (6.5)

Using the probability integral, the probability 1 2P( ) takes the form

2 11 2

1P( )

2 2D 2D

(6.6)

The probability 2 2P( ) P( ) is the probability of the event that does

not exceed 2 :

22

1P( ) 1

2 2D

(6.7)

The probability 1 1P( ) P( ) is the probability of the event that larger

than 1 :

11

1P( ) 1

2 2D

(6.8)

In the probability theory is shown that the p.d.f. of the amplitude of a stochasticquantity having the Gaussian p.d.f. distribution satisfies the Raleigh law:

2a / (2D )a

ap.d.f .( ) e

D

(6.9)

The probability that the amplitude is larger than * is

2 *2

a

*

/(2D ) /(2D )* aa aP( ) e d e

D

(6.10)

When evaluating the wave height an observer determines the middle height of onethird of the highest waves. This height is referred to as the significant wave heightand designated as 1/ 3h .

Dependence between the dispersion and the significant wave is2

1/ 3D 0.063h (6.11)

6.1.2. Wave spectra

Irregular waves are considered as the superposition of infinite number of regularwaves of different frequencies, amplitudes and course angles (Fig.6.3). According tothis concept the wave elevation ( , , )x y t is represented in form of Fourier Stieltjesintegral:

( , , ) ( , ) exp[ ( cos sin ) ]x y t Real dA ik x y i t (6.12)Here is the wave frequency, k is the wave number 2 /k g , is the wavecourse angle and ( , ) is the phase angle. The quantity ( , , )dA t is the function


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43/51

Since the equations are linear the responses of the ship to each regular wave can becalculated separately. In this case one can obtain the history of oscillations in time.

However, from point of view of practical applications only the statistical parameters ofoscillations are of importance. To determine them, the ship is considered as the

dynamic system. The seaway is the input which is transformed by the ship intooscillations considered as the output. In the statistical theory shown, that if the inputsignal has the Gaussian p.d.f. distribution the output signal has also the Gaussianp.d.f. distribution. With the other words, the ship oscillation parameters (roll angle,etc) obey the normal Gaussian law whereas the amplitudes of oscillation parameterssatisfy the Raleigh law. The only unknown value in these distributions laws is thedispersion D .

Let us consider the roll oscillations of a ship with the zero forward speed. As shown inthe previous lectures the ratio of the roll oscillations amplitude (output signal) to thewave slope amplitude (input signal) is given by the formula

2 2(r )A A A A

2 2 2 2 2 2 2 2A A A

/(1 k ) /(1 k )1 ( )

(1 ) 4 (1 ) 4

where2

2 2 2 2

/(1 k )( ) 1

(1 ) 4

(6.26)

is the so called response function. Since the wave spectral density is proportional tothe wave ordinates squared and taking the superposition principle into account, weobtain the following relation between the spectral density of the seaway and thespectral density of oscillations

2S ( ) ( )S ( ) (6.27)or

22

2 2 2 2

/(1 k )S ( ) S ( ) 1

(1 ) 4

(6.28)

Dispersion of the roll oscillations and the standard deviation are found from thedefinitions (6.2) and (6.18)

2

0 0

D S ( )d ( )S ( )d , D

(6.29)

Similar formulae can be obtained for angular roll velocity and acceleration2

0

D S ( )d , D

(6.30)

4

0

D S ( )d , D

(6.31)The dispersions obtained from (6.29), (6.30) and (6.31) determine fully the irregularship oscillations in heavy seaway. Using them the following further parameters canbe calculated

Most probable amplitude of oscillations corresponding to the maximum of theap.d.f .( ) distribution


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m (6.32)

The probability that the roll amplitude exceeds m is 60.6 %.

Averaged amplitude of roll oscillations (mathematical expectation)

1.252

(6.33)

The probability that the roll amplitude exceeds is 45.6 %.

The probability that the roll amplitude exceeds the value * :* 20.5( / )*

Ap( ) e

(6.34)

Averaged frequency and averaged period of oscillations:

2

,T 2

(6.35)

Number of ship inclinations (semi periods) within the time interval t:

t

2tN

T (6.36)

Number of ship inclinations within the time interval t provided the roll angleamplitude is larger than * :

* 2

*

0.5( / )*

t A

2t

N N P( ) eT

(6.37)

The formulae (6.32)-(6.37) are derived under assumption that the oscillations obeythe Gaussian distribution law.

7. Strip theory

7.1 Assumptions Strip theory considers a ship to be made up of a finite number of transversetwo dimensional slices, which are rigidly connected to each other.

Each slice is treated hydrodynamically as if it is a segment of an infinitely longfloating cylinder;


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All waves which are produced by the oscillating ship (hydromechanic loads)and the diffracted waves (wave loads) are assumed to travel parallel to the (y,

z )-plane - of the ship. The fore and aft side of the body (such as a pontoon) does not produce waves

in the x -direction. For the zero forward speed case, interactions between the cross sections are

ignored.

Strip theory is valid for long and slender bodies only.

In spite of this restriction, experiments have shown that strip theory can beapplied successfully for floating bodies with a length to breadth ratio larger thanthree, 3/ BL , at least from a practical point of view.


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The appearance of two-dimensional surge forces seems strange here. Then, the2-D hydrodynamic sway coefficients of this equivalent cross section are translated to2-D hydrodynamic surge coefficients by an empirical method based on theoreticalresults from three-dimensional calculations and these coefficients are used todetermine 2-D loads.

7.2 Account for the ship velocityRelative to an oscillating ship moving forward in the undisturbed surface of the fluid,the displacements, hj , velocities,

hj

, and accelerations, hj , at forward ship speed

V in one of the 6 directions j of a water particle in a cross section are defined by:

Equation 2.5 3

Is a mathematical operator which transforms the potentials , defined

in the earth bounded (fixed) co-ordinate system, to the potentials ,defined in the ships steadily translating co-ordinate system with speed .

7.3 Force representation

Relative to a restrained ship, moving forward with speed in waves, the equivalent

j constant components of water particle displacements velocities

and accelerations in a cross section are defined in a similar way by:


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Ordinary strip theory by Korvin-Kroukovsky & Jacobs

In the equations above, and are the 2-D potential mass and damping

coefficients. is the two-dimensional quasi-static restoring spring term, as

generally present for have, roll and pitch only. is the two-dimensional Froude-Krilov force or moment which is calculated by integration of the directional pressuregradient in the undisturbed wave over the cross sectional area of the hull.

7.4 Mathematical prob lem formulationThe two-dimensional nature of the problem implies three degrees of freedom ofmotion:

Vertical or heave,

Horizontal or sway and Rotational about a horizontal axis or roll.

The following assumptions are made: the fluid is incompressible and inviscid, the effects of surface tension are negligible, the fluid is irrotational and the motion amplitudes and velocities are small enough that all but the linear

terms of the free-surface condition, the kinematic boundary condition on thecylinder and the Bernoulli equation may be neglected.

Given the above conditions and assumptions, the problem reduces to the followingboundary value problem of potential theory. The cylinder is forced into simple


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harmonic motion with a prescribed radian frequency of oscillation,, where the superscript may take on the values 2, 3 and 4, denoting swaying,

heaving and rolling motions, respectively.It is required to find a velocity potential:

Satisfying the following conditions:1. The Laplace equation:

Equation 3.4 - 2

in the fluid domain, i.e., for and outside the cylinder.2. The free surface condition:

Equation 3.4 3

on the free surface outside the cylinder, while g is the acceleration ofgravity.

3. The seabed boundary condition for deep water:

Equation 3.4 44. The condition of the normal velocity component of the fluid at the surface of

the oscillating cylinder being equal to the normal component of the forced

velocity of the cylinder. i.e., if is the component of the forced velocity of the

cylinder in the direction of the outgoing unit normal vector , then

Equation 3.4 5This is the kinematic boundary condition on the oscillating body surface, beingsatisfied at the mean (rest) position of the cylindrical surface.

5. The radiation condition that the disturbed surface of the fluid takes the form ofregular progressive outgoing gravity waves at large distances from thecylinder.

7.5 Franks method of pulsating sourcesAccording to Wehausen and Laitone [1960], the complex potential at zof a pulsating

point source of unit strength at the point in the lower half plane is:


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Equation 3.4 12 is applied at the midpoints of each of the N segments and it isassumed that over an individual segment the complex source strength Q(s)remainsconstant, although it varies from segment to segment. With these stipulations, the setof coupled integral equations (Equation 3.4 12) becomes a set of 2N linearalgebraic equations in the unknowns:

Thus, for i=1,2, , N:

where the superscript (m)denotes the mode of motion.The hydrodynamic pressure at (xi, y i)along the cylinder is obtained from the velocitypotential by means of the linearized Bernoulli equation:

as:

where and are the hydrodynamic pressure in-phase with the

displacement and in-phase with the velocity, respectively and denotes the densityof the fluid.

for the added mass and damping forces or moments, respectively.


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Literature

1. Kleinau D., Theorie des Schiffes, Manuskript, University of Rostock, 2001 (inGerman).

2. Kornev N., Schiffstheorie I, Shaker Verlag, 2009, 162 S (in German).3. Kornev N., Ship Theory I (ship manoeuvrability), Manuscript, 2011.4. Bertram V., Practical ship hydromechanics, Butterworth-Heinemann, 2000,

270 p.5. Lewandowski E., The dynamics of marine craft, World scientific, 2004, 411 p.6. Handbook of the ship theory. Editor Voitkunsky, 19857. www.oceaniccorp.com8. Bishop R.E., and Parkinson A.G., On the planar motion mechanism used in

ship model testing, Phil. Transactions of the Royal Society of London, SeriesA, Mathematical and Physical Sciences, Vol. 266, No. 1171, 35-61.

9. www.ksri.ru/eng1/

10.www.becker-marine-systems.com11.Schneekluth H., Hydromechanik zum Schiffsentwurf, Herford-Koehler,1988.

12.Brix J., Menoeuvring technical manual, 1993, Seehafen Verlag.13. Bronsart, R. Manuscript of lectures.14. Makov J., Ship oscillations, Kaliningrad, 2007.15. Price W.G. and Bishop R.E., Probabalistic theory of ship dynamics, Halsted,

London, 1974.16. Newman J., Marine hydrodynamics, MIT Press, 1984.

Task:Develop the theory of vertical oscillations of a very sharp cone with the draught T=10m and the diameter of 1 m in regular and irregular waves using the Haskinds relation(1.38). The added mass 33A can be neglected.

ship dynamics in waves_rostok

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