Ship Theory I

7/27/2019 Ship Theory I

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Facu ty of Mec

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rof. Dr.-

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Chapter 1. Differential Equation of Ship Motion

1.1 Ship motion equations in the inertial reference systemThe ship is assumed to be a rigid body with a constant mass m. The differentialequations of the ship motion in the most general form are derived from the

momentum theorem: The rate of change of the momentum of a body is proportionalto the resultant force acting on the body and is in the direction of that force. Mathematically this theorem applied both for linear momentum and angularmomentum can be expressed as

dP F

dtd

D Mdt

, (1.1)

whereasd

dtis the substantial time derivative, P

and D

are respectively linear and

angular momentums of the ship, F and M are respectively total hydrodynamic forceand total hydrodynamic torque acting on the ship. The equations (1.1) are written inthe inertial system which is at the rest relatively to the earth (further referred as to theearth-fixed system). The forces acting on the ship comprise hydrostatic (buoyancy) forces, gravity forces, forces (thrust and transverse force) and moments supplied by the propulsion

system, ship resistance including wave resistance and drag caused by viscosity, (1.2) additional forces and moments caused by waves (wave-induced forces),

control forces and moments exerted by rudders or other steering devices, transverse force, lift and corresponding moments caused by the viscosity, forces and moments caused by wind, forces and moments caused by currents, forces and moments arising from acceleration through the water (added

mass).The linear and angular momentums can be expressed through the kinetic energy ofthe rigid body by differentiation on velocity components:

k k k

x y z

k k k

x y z

E E EP i j k ,

V V V

E E ED i j k .

(1.3)

where x y zV iV jV kV

and x y zi j k

are respectively linear and angular

velocity of the origin. The kinetic energy of the body is obtained by the integration of

the squared local velocity at each body point r ix jy kz

multiplied with the

elementary local mass dm:2

2 2k

m m m

2E (V r) dm mV 2V ( r)dm ( r) dm

(1.4)

Substituting the vector product

y z z x x yr i ( z y) j( x z) k( y x)

(1.5)into (1.4) one obtains


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2 2 2

k x y z

x y x z

m m

y z y x

m m

z x z y

m m

2 2 2 2y y z z

m m m

2 2 2 2z z x x

m m m

2 2 2 2x x y y

m m m

2E (V V V )m

2[V zdm V ydm

V xdm V zdm

V ydm V xdm]

z dm 2 yzdm y dm

x dm 2 xzdm z dm

y dm 2 xydm x dm

(1.6)

The coefficients 2 2 2 2xx yym m

I (y z )dm, I (x z )dm and2 2

zz

m

I (x y )dm are called

as inertia moments, xy xzm m

I xydm, I xzdm and yzm

I yzdm are deviation moments

or products of inertia, x ym m

S xdm,S ydm and zm

S zdm are static moments. With

these designations the formula for the kinetic energy of the body kE takes the form:2 2 2

k x y z

x y z x z y y z x y x z z x y z y x

2 2 2x xx y yy z zz

x y xy z x xz y z yz

2E (V V V )m

2[V S V S V S V S V S V S ]

I I I2 I 2 I 2 I

(1.7)

Substituting (1.7) into (1.3) and (1.1) one obtains the six coupled ordinary differentialequations

y yx z zz y y z x

y z x x zx z z x y

y yz x xy x x y z

y yx z zx x y z x y x z

y x yx x zx z y y

d d Sd V d d Sm S S F ,

d t d t d t d t d td V d d d S d S

m S S F ,d t d t d t d t d t

d d Sd V d d Sm S S F ,

d t d t d t d t d td V dd d V d

I S S I Id t d t d t d t d t

d S d Id I d SV V

d t d t d t

x zz x

y x z x zy y z x x y y z

y y x y y zz xy x z x z y

y yz x xz z x y x z y z

y y zz z x x zz y x x y z

d IM ,

d t d td d V d V d d


d I d I d Id S d SV V M ,

d t d t d t d t d td V dd d V d


d S d Id I d S d IV V M .d t d t d t d t d t

(1.8)


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(1.9)


5/102

Fig.2

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6/102

The ship mass distribution is symmetrical with respect to the plane xz. Therefore, the

products of inertia xyI and yzI as well as the static moment yS are zero in the ship

fixed system. This is the second advantage of the ship-fixed coordinates. Also, the

third productxz

I is often assumed to be zero. With these simplifications the vector

components are:

x x y z y y z x x z z z y x

x x xx y z z xz y y yy x z z x z z zz y x x xz

P mV S ,P mV S S ,P mV S ,

D I V S I ,D I V S V S ,D I V S I .

(1.12)Substituting (1.12) into (1.11) results in the general system describing the six degreeof freedom (6DOF) motion of the ship in the ship-fixed reference system:

yxz y z y x z y z x x z x

yz xx z z x y z x z y x y

yzx x y z x x z y x y z z

yx zxx z xz y y x z z x x z

y

ddVm S (mV S ) (mV S S ) F ,

dt dtdV d d

m S S (mV S ) (mV S ) F ,dt dt dtddV

m S (mV S S ) (mV S ) F ,dt dt

dVd dI S I V S V ( S S )

dt dt dt

(

z zz y x x xz z y yy x z z x x

y x zyy z x z y z x y x

z x xx y z z xz x z zz y x x xz y

yz xzz x xz x z x x z y y z

x y yy x z z x y x xx

I V S I ) ( I V S V S ) M ,

d dV dVI S S V S V S

dt dt dt

( I V S I ) ( I V S I ) M ,dVd d

I S I V ( S S ) V Sdt dt dt

( I V S V S ) ( I

y z z xz zV S I ) M .

(1.13)

This system is integrated numerically using modern numerical 6DOF solvers (CFX,STAR CCM+, OpenFoam). In this case the hydrodynamic forces are calculated bydirect integration of normal and shear stresses over the ship surface without thesubdivision according to physical nature of forces (1.2).

1.3 Ship motion equations in the ship-fixed coordinates with principle axes

The principle axes coordinate system is chosen from the condition that all off-diagonal elements of the inertia matrix (products of inertia)

xx xy xz

xy yy yz

xz yz zz

I I I

I I I

I I I

and the static moments are zero, i.e.,

xy xz yzI I I 0 (1.14)

x y zS S S 0 . (1.15)


7/102

The conditions (1.14) and (1.15) can be satisfied by a special choice of the locationof the origin and a special direction of the coordinate system axes. A sample of sucha system for the case of manoeuvring will be shown later.

In the principle axes system the ship motion equations take the form:

xz y y z x

yx z z x y

zy x x y z

xxx y z zz yy x

yyy x z xx zz y

zzz x y yy xx z

dVm( V V ) F ,dt

dVm( V V ) F ,

dtdV

m( V V ) F ,dt

dI (I I ) M ,

dtd

I (I I ) M ,dt

dI (I I ) M .

dt

(1.16)

The forces F

and moments M

have to be determined in the moving principle axescoordinate system.

1.4 Forces and moments arising from acceleration through the water

The physical nature of the forces and moments arising from acceleration through thewater is the inertia of the medium which the body is moving in. Traditionally these

forces are determined using the irrotational inviscid fluid model. This model isdescribed in details in [2], Chapters 1, 2 and 3. For students who did not attend in thelecture course Grundlagen der Schiffstheorie we give overview of basic principles ofthe theory of irrotational flows in the Appendix I.

1.4.1 Kinetic energy of the fluid surrounding the body. If the flow is

incompressible, inviscid and irrotional ( V 0

) the kinetic energy of the fluidsurrounding the moving body is

6 6

Fl1 1

1

2 i k ik i kE V V m (1.17)

where 1 x 2 y 3 z 4 x 5 y 6 zV V ,V V ,V V ,V , V ,V are components of linear and

angular velocities, whereas ikm are added mass. Generally, the body has 36 addedmass


8/102

11 12 13 14 15 16

21 22 23 24 25 26

31 32 33 34 35 36

41 42 43 44 45 46

51 52 53 54 55 56

61 62 63 64 65 66

m m m m m m

m m m m m m

m m m m m m

m m m m m m

m m m m m m

m m m m m m

(1.18)

Due to symmetry condition ik kim m the number of unknown mass is 21. Theadded mass are determined from the formulae (see [2]):

dSn

m k

S

iik

(1.19)

where S is the wetted ship area, is the density, i are potentials of the flow when

the ship is moved in i-th direction with unit speed. The potentialsi

satisfy the

Laplace equation2 2 2

2 2 20

i i i

x y z(1.20)

the boundary condition of the decay of perturbations far from the moving body

i r0 (1.21)

and no penetration boundary condition at each point (x,y,z) on the ship surface

31 2

4

5

6

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

cos( , ) cos( , );

cos( , ) cos( , );

cos( , ) cos( , ).

n x n y n z n n n

y n z z n yn

z n x x n zn

x n y y n xn

(1.22)

Here n

is the normal vector to the ship surface at the point (x,y,z),

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

n x ni n y nj n z nk . When the ship moves arbitrarily the potential of

the flow is the sum of particular potentials multiplied with corresponding components

of linear and angular velocities:6

1

k kk

V(1.23)

1.4.2 Momentum of the fluid surrounding the body. Let usconsider the amount offluid between the surfaces S (wetted ship surface) and which is located far fromthe ship. The momentum of this fluid is

Fl

U U

K VdU grad dU

(1.24)

According to the Gauss theorem


9/102

/Fl h h

U U S

K VdU grad dU ndS ndS F F

(1.25)

where hF

and /hF

are respectively the forces acting on the surface S and :

h

S

F pndS

(1.26)

/hF pndS

(1.27)

Since the shear stresses are zero in the inviscid fluid, only normal stresses arepresent in formulae (1.26) and (1.27).From the momentum theorem follows:

/h h Fl(F F )dt dK

(1.28)

The temporal change of the momentum reads:

Fl

S

dK d ndS d ndS V(Vn)dSdt

(1.29)

The last term considers the fact that a part of the momentum V(Vn)dSdt is

transported from the fluid volume U through the surface by the mass (Vn)dSdt

.

From (1.29) follows:

/ Flh h

S

dK d dF F ndS ndS V(Vn)dS

dt dt dt

(1.30)

Since the surface is motionless the integral and differentiation are commutativeoperators:

dndS ndS

dt t

(1.31)

The pressure in inviscid irrotational fluid is determined from the general Bernoulliequation:

2

0

Vp p

t 2

(1.32)

Substitution of (1.32) into (1.27) brings:2 2

/h 0

V VF pndS (p )ndS ( )ndS

t 2 t 2

(1.33)

With consideration of (1.31) and (1.33) the force acting on the surface S can beexpressed from (1.30) in the following form

/h h

S

2

S

2

S

d dF ndS ndS V(Vn)dS Fdt dt

d d VndS ndS V(Vn)dS ( )ndS

dt dt t 2

d VndS ( n V(Vn))dS

dt 2

(1.34)

We choose the surface located very far from the body. All perturbations decayaccording to the condition (1.21) so quickly that the last integral in (1.34) is zero.Therefore, we have

Flh

S

dP dF ndSdt dt

, (1.35)


10/102

where FlS

P ndS

is the linear momentum of the fluid. The components of the

force are (see formulae (1.22) and (1.23))6

1hx x k k

k 1S S S

6 61

k 1k k Flxk 1 k 1S

d d dF n dS cos(nx)dS V dS

dt dt dt x

d d ddS m V P

dt x dt dt

(1.36)

6

hy y 2k k Flyk 1S S

d d d dF n dS cos(ny)dS m V P

dt dt dt dt

6

hz z 3k k Flzk 1S S

d d d dF n dS cos(nz)dS m V P

dt dt dt dt

Similarly, the moment arising from acceleration through the water can be expressedthrough the angular momentum derivative:

Flh

S

dD dM (r n)dS

dt dt

(1.37)

where FlS

D (r n)dS

is the angular momentum of the fluid. The components of

moments are (see formulae (1.22) and (1.23))6

hx x 4k k Flxk 1S S

6

hy y 5k k Fly

k 1S S

hz z

S S

d d d dM (r n) dS (ycos(nz) z cos(ny))dS m V D ,

dt dt dt dt

d d d dM (r n) dS (z cos(nx) x cos(nz))dS m V D ,

dt dt dt dtd d

M (r n) dS (x cos(ny) ycos(nx))dSdt dt

6

6k k Flzk 1

d dm V D .

dt dt

(1.38)

The relation between the linear and angular momentums of the fluid and the kineticenergy can be found from formulae (1.36), (1.38) and (1.17)

Fl Fl FlFl

x y z

Fl Fl FlFl

x y z

E E EP i j k ,

V V V

E E ED i j k .

(1.39)

This relation has exactly the same form as the relation between linear and angularmomentums and kinetic energy of solid body (1.3)

1.4.3 Ship motion equations in the inertial reference system. The ship motionequations in the earth-fixed system (1.1) are rewritten in the form

Fl

Fl

d(P P ) F

dtd

(D D ) Mdt

(1.40)

Where, in contrast to (1.1), the forces F

and moments M

dont account for forces

and moments arising from acceleration through the water.


11/102

1.4.4 Ship motion equations in the ship-fixed reference system.

Fl Fl

Fl Fl Fl

d(P P ) (P P ) F

dtd

(D D ) V (P P ) (D D ) Mdt

(1.41)

where the forces and moments dont account for forces and moments arising fromacceleration through the water, since they are explicitly considered on the left hand

side of the equation by terms with FlP

and FlD

. Substitution of (1.36) and (1.38) into

(1.41) results in the following change of equations (1.13)6 6 6

yxz y z y x z y z x x z 1k k y 3k k z 2k k x

k 1 k 1 k 1

6 6 6y z x

x z z x y z x z y x 2k k z 1k k x 3k k yk 1 k 1 k 1

yzx x

ddV dm S (mV S ) (mV S S ) m V m V m V F ,

dt dt dt

dV d d dm S S (mV S ) (mV S ) m V m V m V F ,

dt dt dt dt

ddVm S (m

dt dt

6 6 6

y z x x z y x y z 3k k x 2k k y 1k k zk 1 k 1 k 1

yx zxx z xz y y x z z x x z y z zz y x x xz z y yy x z z x

6 6 6

4k k y 3k k z 2k k y

k 1 k 1 k 1

dV S S ) (mV S ) m V m V m V F ,

dt

dVd dI S I V S V ( S S ) ( I V S I ) ( I V S V S )

dt dt dt

dm V V m V V m V

dt

6 6

6k k z 5k k x

k 1 k 1

y x zyy z x z y z x y x z x xx y z z xz x z zz y x x xz

6 6 6 6 6

5k k z 1k k x 3k k z 4k k x 6k k yk 1 k 1 k 1 k 1 k 1

yz xzz x xz

m V m V M ,

d dV dVI S S V S V S ( I V S I ) ( I V S I )

dt dt dt

dm V V m V V m V m V m V M ,

dt

dVd dI S I

dt dt

x z x x z y y z x y yy x z z x y x xx y z z xz

6 6 6 6 6

6k k x 2k k y 1k k x 5k k y 4k k z

k 1 k 1 k 1 k 1 k 1

V ( S S ) V S ( I V S V S ) ( I V S I )dt

dm V V m V V m V m V m V M .

dt

(1.42)

1.4.5 Ship motion equations in the ship-fixed reference system along the x-axis. The system (1.42) takes the simplest form for the case of the straight ship

motion along the x-axis ( y z x y zV V 0 ):

x11 x

dV(m m ) F

dt (1.43)

As seen the fluid inertia results in the increase of the real mass m by the additional

virtual mass 11m . The total mass is becoming larger due to inertia of the fluid. That is

why the mass 11m is called as the additional mass. The effect of the fluid inertiamakes the ship motion milder. i.e.


12/102

If the

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Chap

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board side directiose of the

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(1.44)

e ship

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,,course

acles,


13/102

Important questions for the specification of ship manoeuvrability may include [3]: Does the ship keep a reasonably straight course (in autopilot or manual

mode), under what conditions (current, wind) can the ship berth without tug

assistance?

Up to what ratio of wind speed to ship speed can the ship still be kept on allcourses?

Can the ship lay rudder in acceptable time from one side to the other?The characteristics usually used to regulate the manoeuvrability are discussed in thenext sections.

2.3 Main assumptions of the theory

The ship manoeuvring theory is based on the following assumptions: the ship motion is occurred only in the horizontal plane xy. Heave velocity,

rolling and pitching are neglected ( z x yV 0, 0 ).

The Froude number is small and the free surface deformation is neglected.The mirror principle is used to model the free surface effect.Hydrodynamically the ship is considered as a doubled body.

The doubled body has two symmetry planes that is why the ship has only eight

added mass: 11 22 33 44 55 66 26 35m ,m ,m ,m ,m ,m ,m ,m . The static moment of the doubled

body and the product of inertia are zero, i.e. zS 0 and xzI 0 . The system (1.42) isreduced to:

2x11 22 y z z 26 x x

y z22 11 x z 26 x y

yzzz 66 x y 22 11 26 x x z z

dV(m m ) (m m )V (m S ) F ,

dtdV d

(m m ) (m m )V (m S ) F ,dt dt

dVd(I m ) V V (m m ) (m S )( V ) M .

dt dt

(2.1)

2.4 Equations in the ship-fixed coordinates with principle axes

The principle axes coordinate system was chosen in Section 1.3 from the conditionthat all off-diagonal elements of the products of inertia and the static moments of

body are zero. It simplifies the equation system. However, many terms proportional tooff-diagonal elements of the added mass matrix remain. For example, the system

(2.1) contains terms with 26m . The motion equations have the simplest form if the

axes are principle axes of the coupled system body+fluid. The system with principleaxes can easily be found for the doubled body moving in the horizontal plane fromthe following conditions: the x axis is along the longitudinal axis of the doubled body, the xy and xz are symmetry planes, the position of the origin is found from the formula

26 xm S 0 (2.2)

Remember that the origin in the equation (1.67) was chosen from the condition thatonly body static moment is zero xS 0


14/102

Let us consider the sum 26 x 2S m

m S (x cos(n, y) ycos(n, x))dS xdm . The ship

can be considered as a slender body. The normal vector to the slender body has thefollowing asymptotic estimations which are valid on the most part of the ship length:

2

/ ( ),

cos( , ) ( ),

cos( , ) (1),

/ (1),

(1).

y L O

n x O

n y O

x L O

O

(2.3)

Therefore, the asymptotic estimation for the sum 26 xm S for the slender body reads

26 x 2 2

S m S m

m S (x cos(n, y) ycos(n, x))dS xdm x cos(n, y)dS xdm (2.4)

The condition (2.2) can be satisfied by shifting the origin by gx :

2

2 2

2

2

( ) cos( , ) ( ) 0

cos( , ) cos( , ) 0

cos( , )

cos( , )

g g

S m

g

S m S m

m Sg

S

x x n y dS x x dm

x n y dS xdm x n y dS dm

xdm x n y dS

xm n y dS

Using the middle value rule// //

2 2 22cos( , ) ( cos( , ) ) S S

n y dS x n y dS x m and

/m

xdm x m the last formula is rewritten in the form

/ //22

22g

m x mx

m m

(2.5)

Here/x is the ship gravity center and

//x is the hydrodynamic center. If the origin

lies at the point gx the system (2.1) takes the simplest form

x11 22 y z x

y22 11 x z y

zzz 66 x y 22 11 z

dV(m m ) (m m )V F ,

dtdV

(m m ) (m m )V F ,dt

d(I m ) V V (m m ) M .

dt

(2.6)

The aim of the ship trajectory calculation is also determination of the ship position inthe earth- connected coordinates system 0 0x y . Two following equations are used for

this purpose (see Fig.5):


15/102

0 0dx dyVcos( ), Vsin( ).dt dt

(2.7)

Here is the course angle calculated from the equation:

z

d

dt

(2.8)

Combining (2.6), (2.7) and (2.8) we obtain the full system of ship equation in thehorizontal plane:

x11 22 y z x

y22 11 x z y

zzz 66 z x y 22 11

t

0 0

0

t

0 0

0

t

z

0

dV(m m ) (m m )V F ,

dtdV

(m m ) (m m )V F ,dt

d(I m ) M V V (m m ),

dt

x (t) x (0) V cos( )dt,

y ( t) y (0) Vsin( )dt,

(t) (0) dt.

(2.9)

2.5 Munk moment

The second term x y 22 11V V (m m ) on the r.h.s in the moment equation isreferred as to the moment of Munk who investigated this moment forZeppelins.

The Munk moment appears in the full form only in the inviscid fluid. In theinviscid potential fluid the flow around the ship hull is shown in Fig.6. In thebow area on the lower side we have the deceleration of the flow and increaseof the pressure. On the upper side the flow is accelerated and the pressuredecreases. As a result a lift force appears in the bow region. An opposed flowprocess takes place in the stern area in the inviscid flow. Here the deceleration

arises on the upper side whereas the flow acceleration appears on the lowerone. The negative down force counterbalances the lift and the total force iszero according to the DAlambert paradox. However, these two forces producethe moment which is exactly the Munk moment,

This moment is called also as the unstable moment. It can be explained at

small drift angles . The velocity components are expressed through the ship speedand the drift angle:

x yV Vcos , V Vsin (2.10)

Since 11m is much less than 22m , the Munk moment is

2 2

22 11 22sin 2 sin 22 2MunkV V

M m m m . This moment is the moment which


16/102

causemomethe pr

of the

In thetakendoesn

Munk

measmomethe rig

zM V

2.6 EqClassiin termstudy t

the instnt increasence of

drift angl

real viscinto acct appear

moment

rementsnt automht hand

y 22V (m

(

(m

Fig.6. Illu

uations in

al form os of thehe ship ya

bility. Wies this asmall dri

:

us fluidunt in tand the

(Fig.6).

in realtically. Tide of th

11) as a t

11

22

m )(V

m )( V

tration of

terms of

the manrift angle

stability.

x

y

dV

dtdV

dt

th the othngle. Int angle

z dMd

he flow ie wing tunstable

ery ofte

iscous flat is wh

e momen

otal yaw

zz

os V

sin V

(I

the Munk

the drift

euvrabilitand trajecThe time

Vcos

t

dVsin

dt

er wordseed, the

0 is po

2unk m

the steheory bymoment

n the y

uids andit is co

t equatio

oment, i

z66

in ) (m

cos ) (

d)

dt

oment. a

ngle and

equationtory curvderivative

dsin

dt

dVcos

dt

, if a smaadditionitive, i.e.

20 cos2V

n area isKutta c

s approx

w mom

capturemon to c

n and to

.e.

22 z

11 z

z

m )V

m )V

.

)-inviscid f

trajector

s are writtture. Thisof speed

Vcos

Vsin

ll drift anl momenit causes

0

changedndition.mately o

nt zM i

the Muarry theconsider

x

y

sin F ,

cos F ,

luid, b) vis

curvatur

en in nonform is vomponen

sin ,

V cos

le appets arisingfurther in

. This chhe dow

nly a hal

determi

ks partunk mo

the com

cous fluid.

e

dimensioery convets (2.10) a

,

rs, thisdue tocrease

nge isforce

of the

ned in

of theent to

ination

(2.11)

al formnient tore:

(2.12)


17/102

where point over the quantity means the time derivative of this quantity, i.e.

dV dV , .

dt dt

To derive the non-dimensional equation form one uses the typical force

representations:2 2 2

x x L y y L z z L

V V VF C A ,F C A , M m A L

2 2 2

(2.13)

Introducing the non-dimensional time , non-dimensional angular velocity andinstantaneous trajectory radius R

z

V LtV / L, L / V L/ R,

R V (2.14)

the dimensional time derivatives V and are expressed through the non-

dimensional ones/V

V,

/ and / by:

2 2 /

/

2 2 2 //z

z

dV V dV V 1 dV V VV ,

dt L d L V d L V

d V d V,

dt L d L

d V d V V d V 1 dV V V.

dt L d L L d L V d L V

(2.15)

Here/ / /dV d dV , ,

d d d

. From the second formula in (2.14) follows that

dimensionless angular velocity is the dimensionless trajectory curvature.

Using dimensionless mass and inertia moments

zz 6611 22x y

3L L L

I mm m m m, , .

A L A L A L2 2 2

(2.16)

and substituting (2.12), (2.13), (2.14) and (2.15) into (2.11) one obtains:/

/x x y x

/

/y y x y

//

z

Vcos sin sin C ,

V

V sin cos cos C ,V

Vm .

V

(2.17)

Chapter 3. Determination of added mass.

3.1 General solution

The basis for an exact determination of added mass is the formula


18/102

dSn

m k

S

iik

(3.1)

where i are potentials of the flow when the ship is moved in i-th direction with unit

speed. These potentials can be found from the solution of the integral equation (3.2)

which was derived in [2] from the no penetration condition

MN ii i 2

MNS

1 cos(n, R ) qV q dS 0

4 R 2 (3.2)

Here the component of the inflow velocity is calculated depending on i:

1 2 3

4 5

6

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

cos( , ) cos( , ), cos( , ) os( , ),

cos( , ) cos( , )

V n x V n y V n z

V y n z z n y V z n x xc n z

V x n y y n x(3.3)

Once the source intensity is found from (3.3), the potential i is calculated according

to the definitioni

i 2 2 2S

1 q ( , , )(x, y, z) dS

4 (x ) (y ) (z )

(3.4)

Substituting (3.4) in (3.1) one calculates all added mass. Nowadays thenumerical solution of the equation (3.2) presents no serious difficulties and canbe performed by any code using panel methods.

For some simple bodies there are analytical solutions. For instance, for anelliptical cylinder the following analytic formulae are valid

2

2 2 2 21 1 2 2 6 6; ; 8

m b m a m a b (3.5)

where a and b are semi axis of the ellipse (a>b).

The analytic solution which is the most interesting for shipbuilding is thesolution for rotational ellipsoid. Unfortunately, this solution is cumbersome andcontains non elementary functions. The results of calculation using thissolution are presented in Fig.7 for added mass coefficients.


19/102

In thebetwe

3.2 AdAnothe

assum

slende

where

this co

since t

written

Expres

Fig.

implest an the widt

ded mass

r way of

ption. Let

body esti

C is the s

ntour. Th

e contou

for the ad

6m

sion for

7 Dimen

pproach, th B and th

of the sl

determin

us cons

mations (

22m

hip frame

formula

added m

ed mass

6(S

can be fo

ionless a

he largeste draught

nder bod

tion of

der the

.3) the for

2 cos( ,S

n

contour a

3.6) is ea

ss is calc

66m :

cos( , )n y

und from t

dded ma

axis 2a iT.

y.

dded ma

dded m

mula for t

0

)L

y dS

nd

/

22m

sier than

ulated fro

cos( ))y nx

he followi

s of rotat

the ship l

ss is the

ss 22 m

is mass c

2 cos( ,C

n

2cos( ,

C

n

he origin

2D theo

0

L

C

S

g asympt

ional ellip

ength, 2b

use of t

2 cos( S

n

n be writt

0

)L

dCdL

)y dCis t

l one 22m

ry. Simila

6 cos( , )x n y

tic analys

soid

is a middl

he slend

, )y dS . Us

en as foll

/22dL

e added

2 co S

r formulae

CdL

is:

e value

r body

ing the

ws:

(3.6)

ass of

s( , )n y dS

can be

(3.7)


20/102

26 2 2

S S

62 26 6 2

S S

6 2

m (x cos(n, y) ycos(n, x))dS x cos(n, y)dS,

m m cos(n, y)dS x cos(n, y)dS

x

(3.8)

Substituting the last result in (3.7) gives

2 2 2 /66 2 2 22

0 0 0

cos( , ) cos( , )L L L

C C

m x n y dCdL x n y dC dL x m dL

(3.9)L

/26 2 2 22

S S 0

m (x cos(n, y) y cos(n, x))dS x cos(n, y)dS xm dx (3.10)

Similarly, added mass 33 35,m m and 55m can be found. Unfortunately, the slender body

theory is not capable of simplifying the formulae for mass like 1km

, since the effect ofthe motion in x direction is assumed to be neglected. The mass

/22m can be found

using 2D panel method which is much easier than 3D version of this method.

3.3 Added mass of the slender body at small Fn numbers.

In what follows we use the concept of doubled body assuming the Froude number issmall and water surface deformation effects can be neglected. An effective way to

get/22m is the use of the Lewis theory which became a classical way to determine

the added mass in naval architecture. Lewis used theory of conformal mapping1

which is applicable only for two dimensional flows. According to this theory (see alsochapter 5.8.1 in [2]) the physical plane z x iy is mapped into an auxiliary plane

i . The skill is to find such a mapping function z( ) and inversion mappingfunction (z) so that the flow around the contour is mapped into the flow around acylinder. Lewis succeeded in mapping of a special class of doubled ship frames,called further as Lewis frames, into cylinders. The Lewis frames have the form typicalfor ship frames in the middle ship area. In the bow and stern regions Lewis framesare deviated significantly from the typical frames. Lewis inversion mapping function iswritten in general form

3

a bz

z z

, (3.11)where a and b are real coefficients. Changing a and b one gets a family ofLewis frames. Lewis performed a serial calculation for various frames andpresented his results in a form of a resulting diagram shown in Fig. 8. Heintroduced the coefficient (referred as to the Lewis coefficient) which is theratio of the added mass of the frame to that of the cylinder with radius T

/22

2

mC

T (3.12)

Therefore, C=1 for the cylinder. C for different Lewis frames are presented in

Fig.8 depending on H 2T / B and spA /(BT) , where spA is the frame area.

1 see, for instance, en.wikipedia.org/wiki/Conformal_map


21/102

Fig. 8

Lewisform isrewritt

The slmass iflow isThe slareas ton addMunk

Here

largestdraugh

directiMunk

Lewis co

ata are ustill unknn in the fo

22m

nder bods obtaineconsidernder bodhe flow ised massorrection

2 ( , )a b an

axis 2a ist T. The

n are neroposed t

efficients

is

seful espwn. Usinllowing fo

/22

0

L

m dL

theory oby integ

d as a ty theoryessentiallan also bactors. T

222

3

( ,

( ,

m

R a b

R a b

3 ( , )R a b a

the ship ladded m

lected wifind 11m i

dependi

the fram

cially in tLewis co

m:

2

0

( )L

C x T

/26 2

0

L

m xm

added mation of fo dimensorks well

y three ditaking in

e idea of

2 2222 _

66 _

( , )

elli

elli

a b m

m

m

e the Mu

ength, 2bss 11m ca

hin the sln a simila

11m

g on H

area (ta

e prelimiefficient t

( )x dL , 66m

0

L

dL x

ass is a same masional onein the miensional

to accounMunk bec

_22

66

,

/

/

slender

psoid

psoid

m

m

m

nks corr

is a middn not be d

ender boway like

1( , )R a b m

2T/B an

en from [

ary shipe formula

2 /22

0

L

x m dL

2( ) ( )x T x d

rip theorys along thcorrespo

ddle ship. The effet within thmes obvi

66 3_ _

_

( ,

slender ell

slender ell

R a

ction fact

le value betermined

y theory.

22 but wit

22_slender.

spA /

1])

esign whe (3.6), (3

2

0

( L

x C

. It meanse ship leding crosarea. In tt of threeslenderus from f

66 _)

,

.

slen

ipsoid

ipsoid

b m

rs (see

etween thsince the

To overch different

BT) , whe

en the ex.9) and (3

2) ( )T x dL ,

that the rgth. Ever

s sectionse bow adimensioody theo

ollowing f

,der

ig.9). Ag

e width Bperturbati

me thiscorrectio

re spA

ct ship.10) are

(3.13)

esultingframe

(strip).d sternal flow

y usingrmulae

(3.14)

in, the

and thens in x

roblemfactor:

(3.15)


22/102

The ad

the facLewis

As se

body tdensitvolum

This re

about f

With th

The fathat of

ded mass

t into accooefficient

n from (3

eory is e. If the ellimultiplie

Therefore,

theory is e

sult can b

ive, eight

e Munk c

tor is ithe doubl

22 _ _ slenderm

unt, that tis one.

.16) the h

actly equpsoid is inwith thethe hydr

qual to th

used for

er cent,

Fi

rrections t

troducedd body.

ellipsoid and

e cross s

22 _

66 _

slender

slender

ydrodyna

al to thethe equiliensity (Ardynamic

the ship

rough esti

hereas th

g. 9 Mun

he added

11 1

22 2

66 3

1R

2

1R

2

1R

2

into (3.17)

66 _slenderm

ction of t

_

_

ellipsoid

ellipsoid

ic mass

olume ofrium stat

chimedesass 22 _m

ass.

mation of

mass 6m

s correc

mass areL

2

0

L2

0

L2

0

C(x)T

C(x)T

C(x)T

because

ellipsoid are

e rotation

2

3

4,

34

15

ab

ab

22 _slenderm

the ellipsin the w

law).

lender obtai

the mass

6 is about

ion facto

calculated

2

x)dx,

(x)dx,

(x)x dx.

the added

obtained

al ellipsoi

obtained

id multipliter the m

ned using

22m . The

zz1.4I [1].

s.

from form

mass of t

from (3.13

is cylind

using the

ed with thss is equ

the slend

dded ma

ulae

he hull is

) taking

r which

(3.16)

slender

e waterl to the

er body

s 11m is

(3.17)

half of


23/102

Chapter 4. Steady manoeuvring forces

The forces acting on the ship can be subdivided into steady manoeuvring forces,propulsion forces, forces arising on control elements, wave induced and forcescaused by wind and current. In this chapter we consider the steady force component.

The steady manoeuvring forces arise on the body moving with steady linear andangular velocities due to viscosity influence. The physical reason of the inception ofthe steady manoeuvring forces is illustrated in Fig. 6. If the ship moves with a steadydrift angle in an inviscid flow, the lift force arises in the bow region whereas the downforce acts on the stern area (Fig.6a). The resulting force is in accordance with the DAlambert paradoxon zero. In the viscous fluid the flow in the stern area is changeddue to influence of the boundary layer developing along the ship surface beginningfrom the bow. As a result the down force disappears at the stern part and theresulting force is not zero. This component is referred to as the steady manoeuvringforce caused by the drift angle. The forces and moments appear also if the shipmoves with any steady linear and angular velocity. In the manoeuvrability theory the

steady forces arising due to drift angle and yaw angular velocity are of importance.

4.1 Representation of forces

Using the Reynolds averaged Navier Stokes equations (RANSE) technique, thesteady forces can be calculated by direct integration of normal and shear stressesover the wetted ship area. This way requires huge computer resources, is timeconsuming and the prediction accuracy is often not satisfactory. The experiment isstill remaining a main source of the force data used for prediction of manoeuvrability.

The experimental methodology is based on the representation of forces in form ofdifferent approximations. For instance, one uses a multivariate Taylor series

expansion about the equilibrium condition x y z x y zV V,V V 0 :

x y z x y z

j

n x y z x y z k nj 0 k V V,V V 0

1F (V ,V ,V , , , ) V F

j! x

, (4.1)

where n x y z x y zF (V ,V , V , , , ) is the force component2, n=1,2,,6, ,.,

4 x y z x y z x x y z x y zF (V ,V ,V , , , ) M (V ,V ,V , , , ) ,., 1 x 2 y 5 yV V V,V V ,..., V ,... .

As a rule the force coefficient are calculated through the coefficients

x y z x y zC ,C ,C ,M ,M ,M 2 2

x,y,z x,y,z L x,y,z x,y,z L

V VF C A , M m A L,

2 2

which are represented in the form of Taylor series. The coefficients

x y z x y zC ,C ,C ,M ,M ,M are the function of kinematic parameters and similarity criteria

such as the Froude and Reynolds numbers. The derivatives

x y z x y zk V V,V V 0

x

are determined about the equilibrium condition x y z x y zV V, V V 0 .

2 For the sake of brevity both force and moment are meant here and further under the term force


24/102

4.1.1 Hypothesis of quasi steady motion.

Application of the Taylor series implies the hypothesis of quasi steady motion. Thelatter means that the forces are fully determined by instantaneous values of kinematic

parameters neglecting the unsteady effects. The motion history influence isneglected. Strictly speaking the ship hydrodynamics depends on the ship states inprevious times, because the wave surface, boundary layer and wake depend on theship trajectory. However, the unsteady effects can be neglected if the characteristictime scales of the hydrodynamic processes are much smaller than the characteristictimes of the ship motion. With the other words the ship motion is much slower thanthe change of the hydrodynamics characteristics. In this case the hydrodynamics isfully determined by instantaneous ship kinematic characteristics. With the other

words, it is assumed that the hydrodynamic coefficients x y z x y zC ,C ,C ,M ,M ,M are

frequency independent. This assumption is not necessary if the motion is modelled

using coupled 6DoF simulation (see chapter 10).

4.1.2 Truncated forms.

In the shipbuilding the maximum order of the derivatives in the representation (4.1) isthree. General forms of (4.1) for different bodies are given in [4]. The representation(4.1) contains high-order derivatives which are hardly to determine. There are noreliable theoretical or empirical means to calculate many of the second and third-order terms [4]. That is why the expansion (4.1) is used in a very truncated form,which can be derived by further analysis showing that only a part of the derivativeshas an essential impact on the ship dynamics. Additionally, the expansions (4.1) are

significantly simplified if the ship symmetry is taken into account. In this casex y x yF (0, V , 0, 0, 0, 0) F (0, V , 0, 0, 0, 0) ,

y y y yF (0, V , 0, 0, 0, 0) F (0, V , 0, 0, 0, 0) , (4.2)

z y z yM (0, V ,0, 0, 0,0) M (0, V , 0, 0,0, 0) .

Some of derivatives in (4.1) are zero. For instance, due to symmetry of the drag with

respect to the velocity component yV and z , the derivatives of the drag on yV and

on z at y z x y zV V 0 are zero:

x y z x y zx y z x y z

x x

y z V ,V V 0V ,V V 0

F F0, 0

V

(4.3)

These facts are used to truncate the expansions (4.1).

4.1.3 Cross f low drag principle.

The Taylor series expansion was also revisited using the so-called cross flow dragprinciple taken from the wing theory. Let us consider the steady ship motion with

velocity components xV and yV . The dependence of the transverse force arising on

the ship is shown in Fig. 10 depending on the drift angle:


25/102

Fig. 1nonlin

The de

of two

Here

discusdepenextremessenti

where

middlFormaconsid

The ncalcul

cross

causeprove

The re

sign o

althou

term

. Typicalear part o

pendence

componen

y is th

ed in thes on theely low aial already

the line

point.

l applicaeration gi

onlinearte the n

low Vsin

by theto be pr

sult (4.7)

the nonli

h intuitive

2 is rewrit

dependef the forc

of the tra

ts:

linear c

ing theowing aspspect ratiat small d

yC

r mome

tion ofves:

componnlinear

. The a

cross flportional

as confi

near forc

ly it is cle

en in the

ce of the.

sverse fo

yC

mponent

y (see [2],ct ratio. TAR 2T

rift angles

2

/

Y d

V TL

t compo

he Tayl

y

dC ( )

d

nt seemomponen

ditional t

w. Accto the dri

C

med in m

both at

r that C

orm .

yC

transver

rce coeffi

y( ) C

whereas

chapter 5he ship uL . For sand overc

,2 zm

ent zdM

r series

3y d1

6 d

to bet yC , B

ransvers

rding toft angle s

2y yC

easureme

positive

( )

Therefore

y) C

se force

ient on th

yC

yC is th

.3) the ratider the

uch a wiome the li

2

/

zdM

V T

/ d is ca

expansi

y 3

3...

proportiotz consi

force

Betz thequared b

nts. The0 and

y ( ) . To

,

yC

n the dri

drift ang

nonline

o betweerift angleg the noear com

2 .4

lculated

on for t

al to

ered the

y is inter

nonlinet not cub

roblem n

egative

avoid thi

ft angle,

le yC ( )

r compon

two comis a wingnlinear ponent

round th

he case

3 .In owing un

reted as

r compoed as in (

w is the0 drift

contradic

is the

onsists

(4.4)

ent. As

onentswith anrt q is

(4.5)

e wing

under

(4.6)

der toer the

a drag

nent is4.6):

(4.7)

positive

angles,

tion the

(4.8)


26/102

The representation with the second order terms yC is used by Norrbin [13],

SNAME [4], Sobolev and Fedjaevsky [14]. On the contrary Abkowitz [15] uses theterms of the third order to represent the nonlinear components of forces.

4.2 Representation of forces in the manoeuvrability theory

With considerations of facts discussed above the force representationproposed by SNAME [4] for manoeuvrability theory reads

2 2 22 2x x x

x x y z x x y z y z2 2y z y z

F F F1F (V ,V ,0,0,0, ) F (V ,0,0,0,0,0) V V

2 V V

(4.9)

y y

y x y z y z

y z

2 2 2 3y y y y 2

y z y y z z y z2 2 2y z y z y z

F FF (V , V ,0,0,0, ) V

V

F F F F1 1V V V V

2 V V 6 V

(4.10)

z zz x y z y z

y z

2 2 2 32z z z z

y z y y z z y z2 2 2y z y z y z

M MM (V ,V ,0,0,0, ) V

V

M M M M1 1V V V V

2 V V 6 V

(4.11)

Usually, the series expansions (4.9) - (4.11) are applied for force and moment

coefficientsx y

C ,C andz

m .

In the simplest case the series expansion for the transverse force used in the linearmanoeuvrability theory contains a restricted number of terms:

2

y y L

VF C A

2

y y 0 R y 0 0 y 0C C ( ,0, ) C ( ,0,0)( ) C ( ,0,0) (4.12)

where y xa tan V / V , zL / V ,2 2x yV V V , R is the rudder deflection. The

expansion is valid in the vicinity of any operation point 0 R,0, . As seen in (4.12) it is

common to represent the forces through the force coefficient which is approximatedin the form of the Taylor series expansion on the drift angle y xa tan V / V and the

non-dimensional angular velocity zL / V .

Abkowitz [13] proposed force representation using terms up to the third orders.


27/102

The mtakespropos

4.3 Ex

Despita maimetho

Fig. 1

Corpo

st generaonlinearit

ed by the

erimenta

of rapidsource

s of force

1 Model

ation [5].

l form of finto acc

Krylov Shi

l determi

developmf manoedetermin

test wit

rce repreount. A spbuilding

ation of

nt of numvring forction.

h PMM

sentation iample ofesearch I

teady m

erical mete data.

in ice p

s the polysuch a renstitute (s

noeuvrin

hods theHere we

erformed

omial reppresentatie section

g forces

xperimendiscuss t

by Oce

(4.13resentatioon is the4.3.3 bel

t is still reree expe

anic Con

)n whichmethodw).

ainingimental

sulting


28/102

4.3.1 The planar motion mechanism (PMM)

The PMM is used in manoeuvring studies conducted in open water and ice (seeFig.11). This technique has been pioneered in the USA by Gertler (1959) andGoodman (1960). The PMM allows a model to move in exact, preprogramming

patterns while forces, moments and motion around the model are recorded. Themodel is towed in a testing tank and oscillates harmonically around a steadyreference motion. The amplitude of oscillations and the frequency are prescribed bythe PMM. For instance the PMM installed at the Oceanic Consulting Corporation, St.Johns, Canada [5] produces the sway oscillations with the amplitude of 4 meters, thesway velocity amplitude of 0.7 m/s and yaw rates up to 60 degree per second in thetowing tank with the length of 200 m and the width of 12 m.

The idea of PMM in the simplest version can be easily illustrated using the Taylorseries expansion (4.9)-(4.11). Usually the expansions are used to find the forces onthe left-hand side of the formulae assuming that all derivatives on the right-hand side

are known. In the PMM methodology the forces are measured. The right hand sidesof the formulae x x y zF (V ,V ,0,0,0, ) , y x y zF (V ,V ,0,0,0, ) and z x y zM (V ,V ,0,0,0, ) are

known. The kinematic parameters x y zV , V , are prescribed by the PMM at every time

instant. Performing tests one obtains, say, M measurement points. The followingconditions are valid for each i-th measurement point:

2 2 22 2x x x

xi xi yi zi yi zi xi2 2y z y z

F F F1F (V ,0,0,0,0,0) V V F

2 V V

(4.13)

2 2 2 3y y y y yi y 2

yi zi yi zi yi yi zi zi yi zi yi2 2 2yi z y z y z y z

F F F F F F1 1V V V V V FV 2 V V 6 V

(4.14)2 2 2 3

2z z z z z zyi zi yi zi yi yi zi zi yi zi2 2 2

y z y z y z y z

zi

M M M M M M1 1V V V V V

V 2 V V 6 V

M

(4.15)

where I=1,M the measurement point number. Having 16 measurement points, one

can calculate 16 unknown derivatives in the system of linear equations (4.13)-(4.15).To increase the reliability of prediction, the number of experimental points is muchmore than the number of unknown derivatives. The resulting system is over defined(the number of equations is larger than the number of unknowns). In this case thederivatives are found from the condition that the optimal set of derivatives providesthe minimum of residuals of the equations (4.13)-(4.15).

The approach using derivatives imply the quasi steady motion. The influence ofunsteady effects, influence of frequencies in harmonic motions is not considered. Toovercome this disadvantage Bishop and Parkinson [7] proposed to represent forcesthrough the Fourier expansions based on the oscillatory derivatives following to the

experience from the airplane aerodynamics. The PMM equipped with the harmonicsanalysis device is capable of determining the oscillatory derivatives as well (see [7]).


29/102

For instance, if only the yaw oscillation motion 0 0d

sin t, cos tdt

is

studied, the representation of the transverse force looks like

1 2 1 0 2 0Y(t) a V a a V sin t a cos t (4.16)

Three remarkable points should be noted, considering the last formulae there is an explicit dependence of forces on time, additional term proportional to takes unsteady effects (delay of forces

change with respect to kinematic parameters change) into account,

coefficients 1 2a ,a depend not on the time rather than on frequencies .

If A and B the coefficients of the Fourier expansion for the force Y(t) provided frommeasurements:

Y(t) Acos t Bsin t 2 0 1 0A a ,B a V From this we obtain unknown coefficients in (4.16):

2 0 1 0a A / , a B / V

4.3.2 Rotating-arm basin

The rotating-arm basin is the traditional and well-tried facility to determine themanoeuvring forces. The rotating arm is installed in a round form basin withdiameters varying from 15 meters to 75 meters. For instance the rotating arm basinof the Krylov Shipbuilding Research Institute is 70 m with depth of 6.7 meters. Thesketch of the facility is presented in Fig.12. The model installed on the rotating armat arbitrary drift angle is free for heave and pitch motions. Changing the distance fromthe model to the basin centre allows one to control the model angular velocity. Thefrequency of rotation is changed in order to vary the linear speed of the ship motion.The drag, the transverse force and the yaw moment are measured usingdynamometers. The forces and moments obtained from measurements are

approximated as functions of and z . Numerical differentiation of these

approximations is then used to determine the derivatives. Unsteady effects are fullyneglected in the rotating- arm basin tests.

One of the difficulties in the rotating-arm tests is the determination of forces at z 0

since z 0 due to restriction on the arm length. This problem is easily solved, if the

rotating-arm tests are supplied by tests in towing tank at z 0 and 0 . Anotherway which doesnt require additional towing test measurements is the utilization ofsymmetrical conditions for forces and the moment. Let us consider the figure 13

showing the ship in two turning motions along a circle trajectory at z 0, 0 and

z 0, 0 .

The following conditions can be established just from the analysis of the fig.13:

z z

z z

z z z z

X( , ) X( , ),

Y( , ) Y( , ),

M ( , ) M ( , ).

(4.17)

The conditions (4.17) are applied to find the hydrodynamic characteristics at

z zmin using the measurements done at z zmin , 0 , where zmin is the


30/102

minim

at ( ,

Y( ,

zmin

zmin angula

m angula

z ) the

z ) is the

z zmin

s illustratr velocity

velocity

measure

multipli

can be f

d in Fig.nd both p

Fig. 12

hich can

ment is

d by (-

und from

4. This positive an

ketch of

e attaine

erformed

). The

the inter

rocedurenegative

the rotati

in the fa

at ( ,

orces an

olation of

requiresdrift angle

g-arm fa

ility. To o

z ) . The

d mome

forces b

easurems.

ility [5].

btain the

measure

t in the

tween

ents with

orce Yforce

range

zmin and

positive


31/102

Fig.

4.3.3 I

The mreal shangulamotion

Fig. 13

14. Gen

entificati

thod useips. Durinr velocitieequation

wo turni

ralization

on metho

the datathese teas well aystems c

g ship m

of rotati

d

obtainedts the kins the accen be writt

tions at

g-arm te

rom the tmatic parlerationsen in the f

z 0,

ts to the

sts with sameters ore measurm

0 and z

range

elf-propellthe ship

red depen

0, 0 .

min z

d modelsotion, lin

ding on ti

zmin .

or withear ande. The


32/102

2 2 22 2x x x

xi xi yi zi yi zi2 2y z y z

11 i i i i i 22 i zi i

2 2 2y y y y yi

yi zi yi zi yi yi zi zi2 2yi z y z y z

F F F1F (V ,0,0,0,0,0) V V

2 V V

(m m )(V cos V sin ) (m m )V sin ,

F F F F F1V V V V

V 2 V V

i

3y 2

yi zi2y z

22 i i i i i 11 i zi i

2 2 2 32z z z z z z

yi zi yi zi yi yi zi zi yi zi2 2 2y z y z y z y z

z

zz 66

F1V

6 V

(m m )(V sin V cos ) (m m )V cos ,

M M M M M M1 1V V V V V

V 2 V V 6 V

d(I m )

dt

(4.18)where i is the number of measurement. The system (4.18) can be considered as asystem of linear equations for determination of coefficients on the left hand side. Thecoefficients are assumed to be constant during the motion apart of the drag

xi xiF (V ,0,0,0,0,0) which can be found from any empirical method. Again, like in PMM

tests we have more experimental points and the resulting system (4.18) is overdefined (the number of equations is larger than the number of unknowns). In thiscase the derivatives are found from the condition that the optimal set of derivativesprovides the minimum of residuals of the equations (4.18). For that different methodsof optimization theory are used.

4.3.4 Approximations of steady manoeuvring forces

Variousseries of experimental measurements were performed and approximated bydifferent shipbuilding research organisations. Empirical methods of determination ofmanoeuvring forces are listed in the table 1.

Table 1.

Method ReferenceAbkowitz, M. A. (1964) Abkowitz, M. A. (1964). Lectures on Ship Hydrodynamics -

Steering and Manoeuvrability. Technical Report Hy-5. Hydro-and Aerodynamic Laboratory. Lyngby, Denmark

NORRBIN (1971) NORRBIN, N.H.Theory and Observations on the Use of aMathematical Model for Ship Manoeuvring in Deep andConfined Waters SSPA, Gothenburg, Sweden, PublicationNo. 68, 1971

CLARKE (1983) CLARKE, D. , GEDLING, P. , HINE, G., The Application ofManoeuvring Criteria in Hull Design Using Linear TheoryTransactions of the RINA, London, pp. 45-68, 1983

CLARKE/HORN (1997) CLARKE, D. , HORN, J.R.,Estimation of HydrodynamicDerivativesProceedings of the 11th Ship Control SystemsSymposium, Southampton, U. K.,Vol. 3, pp. 275-289, 1997

OLTMANN (2005) OLTMANN, P., Identification of Hydrodynamic Damping

Derivatives a Pragmatic Approach, InternationalConference on Marine Simulation and ShipManoeuvrability, Kanazawa, Japan, August 25th 28th,


33/102

SNAM

KSRI

Approapproxof forc

0 cause

The fir

where

coeffici

x0C is

approx

pp mL / T

block c

(1993)

imationimation pres and it

. The f

by the dri

t compon

LA is the

ents are a

the ship

imated d

( mT is th

oefficient

2

Ls

HL

proposedoposed bis valid f

rces are

ft angle,

nt is repr

lateral are

pproximat

drag at

pending

draught

f the late

03

WANDOientific, 2

andbookningrad,

by theKRSI is

r all drift

subdivide

hereas th

sented in

a, ppL is th

ed as follo

ero drift

on the

at the mi

al area

SKI E.,04, 411 p

n ship theudostroe

rylov Shadvantagangles in

d in two

second

the form:

e ship len

ws:

ngle. Th

roude nu

dship), po

. The angl

he dyna

ory, editornie, Vol. III

pbuildingous becathe wide

compone

ne arises

th betwe

coefficie

mber Fn

sition of t

e x is de

ics of ma

Prof. Voit., 1985

Researcse it takrange fro

ts. The fi

due to an

n two per

nts in for

pV / gL

e center

ermined f

ine craft,

kunski,

h Instituts full nonm zero to

rst comp

ular velo

pendicular

mulae (4.

, ratios

of gravity

rom Fig.1

orld

e. Thelinearity, i.e.

nent is

ity z .

(4.19)

s. The

(4.20)

20) are

ppL / B ,

gx and

.


34/102

The bl

for frabecomframe

choice

ck coeffic

es depice V-shapeumber w

of the are

Fig

Fig.1

ient of the

ed in Figd. If the sere the b

C is illu

Fig.17

.15 Deter

6 Shape

lateral are

.16 . Herip has U-

uttock alle

strated in

etermina

ination o

f frames i

a is calcul

i is theshaped frviates in t

Fig.18.

tion of i in

the angle

the stern

ated from

rame atmes alon

he symm

the formul

x

area

the formul

hich the

g the wholtry plane

a (4.21)

a

U-shapedle lengths,(see Fig.1

(4.21)framesi is the7). The


35/102

The fo

For shiright) t

Fig

Here

is perf

Table

mula (4.2

ps with cie followin

.19 Frame

1 is the tri

rmed acc

. Calculat

Fig.18

1) is valid

ar shapeg formula

s in the st

angle of

rding to t

ion of 2 .

Samples

for ships

stern (Fihave to

rn area: l

for fra

for fra

the ship a

e table 2.

g gx x /

f the choi

with conv

. 19 left)e used:

ft- cigar-

deadwood

es depict

es depict

the rest.

pp

e of the a

ntional st

nd well d

haped ste

.

ed in Fig.1

ed in Fig.1

Calculatio

rea CA .

ern shape

veloped

rn, right-

9 left

9 right

of the ru

shown in

eadwood

ell develo

ning ship

Fig.16.

(Fig.19

ped

trim 2


36/102

Fn

0.34

0.42

0.46

The lat

The co

n 0.4

n 0.4

n 0.5

eral area

efficients i

2

L , the b

n (4.20) a

lock coeffi

e calculat

gx

gx

gx

gx

0.0

gx

gx

0.0

gx

ient BC a

d from th

0.015

0.015

0.02

gx 0

0

0.04

gx 0.

0.01

re calculat

following

coeffici

0

ed from:

approxim

nts

ations:


37/102

Coeffic

Table

L/ B

4 L /

6 L /

8 L /

Where

ients are

. Calculat

6

8

10

3a and 3b

alculated

ion of 1a a

are calcul

according

d 1b .

0.93

0.95

0.97

0.93

0.95

0.97

0.93

0.95

0.97

ted from t

to the tabl

0.95

0.97

0.95

0.97

0.95

0.97

he table 4

e 3.

c

.

efficients


38/102

Table

mT / L

0.04

0.06

The p

middle

. Calculat

0.04

m / L 0.0

m / L 0.0

rameter

frame m

ion of 3a a

is the ra

.

d 3b .

tio of the block coe

fficient to the block coefficient of the


39/102


40/102

It is as

ship re

calcula

1

Chapbody

The foand aconce20). Fiship

yV

sumed th

sistance

ted from t

//

/2,

er 5. Ctheory

rmalism oymptotict of the arst, we cootion the

sin in

t the rota

x and to

e followin

zL

V

lculatio

the slenstimationtive crossnsider the

ship fra

direction.

ion with t

he side f

g approxi

of ste

er body ts (2.3). Tsection wsteady s

me in th

The mom

22P m

e angula

rce yC .

ations:

dy man

heory ishe goverith the thiip motion

e cross

entum of t

y C(

velocity

he contrib

oeuvrin

ased on ting equatkness xunder aection is

he flow in

2)T (x)V s

z does not

ution to th

forces

he potentions areat the abositive drmoved

he cross

n x

contribut

e yaw mo

using sl

ial inviscierived u

scissa x (ift angle.with the

ection re

to the

ment is

ender

theorying theee Fig.Due tovelocity

ds:

(5.1)


41/102

Accorddoubleearth fi

The ch

section

replac

Substit

Herewi

Since t

takes t

ing to thed body atxed refere

ange of th

due to

d by the d

ution of th

th the deri

he drift an

he form

ig.20 Act

momentux is calcunce syste

e flow mo

otion wit

erivative

e equation

Y

vative of t

dY

dx

gle is ass

ive cross

theoremlated thro

Y

entum is

h the vel

n x coordi

Y(5.1) into

d( P) dx

dx dt

e transve

Y

x

med to b

dYdx

section a

, the transgh the ti

d( P)

dt

caused b

ocity xV

nate dx

d( P)

dt

(5.3) give

d(C

rse force2(C(x)T (

dxsmall sin

2 d(C(x)Vd

long the

verse force derivati

the chan

cosV .

cosV dt

d( P)

dx

2x)T (x))

dxn x coordi

2)) V sin

, co

2 (x))

hip lengt

e acting ove of the

ge of the

he time

dx

dt

dx

dt

2 sin cos

nate is:

os

s 1 , th

h

n the frammomentu

rame in t

erivative

cos

x

last expr

e of thein the

(5.2)

e cross

can be

(5.3)

(5.4)

(5.5)

ssion

(5.6)


42/102

Fig.21

The di

2C(x)T

full, i.e

The d

stern r

force

the for

The fo

At accord

This

what istead

Let usangul

Distribut

tribution

(x) has

. C(x) is

rivative d

gions. In

Y

x=0 ari

e scheme

ce arising

Y(

Hx HY(x )

ance withB

H

x

x

dYxdx

dxoment is

s quitemotion.

consider velocit

ion of C(

f2C(x)T

aximum i

maximum.2

(C(x)T (x)dx

the centr

es within

given in

within the

Bx

x

dYx) d

dx

2V C(

he paradB

H

x

x

d(

exactly e

xpectabl

a more. The sh

2)T (x) an

(x) and

the cent

In the b

) and the

l part alo

this regio

ig. 6.

ship lengt

2x V

2H H)T (x )

x of dAla2(x)T (x))

dx

ual to th

Munk

, becau

complicatip velocit

d of the t

length

Y

xalong

al part of

w and st

force dist

g the shi

n. This f

h from x t

Bx

x

d(C(x)T

dx

is zero

bert. O

2xdxV

Munk

22(x yV m

e no ot

ed shipy V is k

ansverse

the ship i

the ship

rn region

ribution d

length th

rce distri

the bow

2 (x))dx

because

the contrB

H

x

x

C(x) oment

211 2) V m

er mom

otion wipt const

force (5.

presente

here the

either C(

xare ma

e2C(x)T

ution is in

Bx is

2V C(x)

2HC(x )T (

ry, the m

2 2(x)dxV

nt can

h the drint. The

) along t

d in Fig.

frame sha

x) =0 or

ximal in b

(x) =const

accorda

2T (x)

H ) 0 whment is n

222m V

rise duri

ft anglepresence

e ship

1. The

pes are2(x) =0.

ow and

and no

ce with

(5.7)

at is in

ot zero

(5.8)

(5.9)

ng the

nd theof the


43/102

angulship

which

where

formulship

Y(x

Distrib

d(C(x)

dAnalyz

Bx 0

region

HY(x )

Fig.2

r velocity

hould be

(x) is te (5.6)- (

dY

dx

Bx

x

2

dY) dx

dx

V C(x)T

tions of2T (x)) x

L

ing the u

and Hx

. As in

0 . Again

2 Distribu

causes

dded to t

yV (x)

he effecti

.7) have t

2

2

dV

d(CV

x2

2

V

x(x) C

L

the tr

and C(x)

per pictu

0 . Ther

he case

, like in th

tion of th2(C(x)T (x

dx

dditional

x

e incomi

Vsin

e drift an

o be rewri

2

2

(C(x)T (x)

dx

x)T (x))

dx

B 2

2

d(C(x)T (

dx

(x)T (x)

nsverse

2 (x) alo

e, please

fored(

0 theprevious

transve) x

Land

velocity i

Yx V

R

g velocity

xV

L

le which

tten takin

(x))

d(C(x)

dx

Bx

x

x))dx

force c

g the sh

note that2(x)T (x))

dxfull trans

case

se force2C(x)T (x

n the eac

x

L

due to th

x( )

L

is variable

variabilit

2

2

d(C(V

(x)) x

L

2d(C(x)T (x

dx

mponent

ip length

the origi

x

Lis neg

erse forc

0, 0

ompone

) along th

h cross s

drift angl

V (x)

along th

of the dri

2

2

)T (x)(

dx

C(x)T (x

x)) x

dxL

proport

are pres

is in th

tive both

e is zero

the mom

ts propo

e ship le

ection al

ship len

ft angle al

x))

L

L

B

2

x

C(x)T (x)

ional to

nted in

ship cen

in bow a

when nt is not z

rtional to

gth

ng the

(5.10)

(5.11)

th. The

ong the

(5.12)

dxL

(5.13)terms

ig. 22.

tre, i.e.

d stern

0 , i.e.ero.

terms


44/102

5.1 Improvement of the slender body theory. Kutta conditions

At a glance the slender body theory is not practical because it is not capablepredicting the transverse force. This problem can be overcome in a way similar to the

famous Kutta condition introduced in the airfoil theory (see chapter 5.4 in [2]). Asexplained above the force arises due to a drastic change of the flow in the stern area(see Fig. 6b). This change is caused due to viscosity influence. The boundary layer isdeveloped starting from the bow. The vortices of the boundary layer shed from thestern alter the flow. As a result the force acts only on the front part of the ship.According to the slender body theory it is assumed that the force arises within theship section starting from the bow to the widest frame. This choice can beestablished using the similarity between the ship and the wing of small aspect ratio.

Let us consider the wing under a small angle of attack (Fig.23) and 0 .Because the aspect ratio is small the flow around of any transversal wing section is

nearly two dimensional like it is shown for section AB (see Fig.23 a and b). The

incident velocity is Vsin in each wing section. According to the vortex wing theory(see chapter 5 in [2]) each section contains transversal bound vortices generating thelift and free streamwise vortices which are necessary to make the transversal vorticesclosed at infinity. The traces of longitudinal free vortices in different cross sectionsalong the wing are shown in Fig. 23c. To understand the vortex scheme better thereader is referred to the section 5.9.2.2 in [2]. Both free and bound vortices induce

the downwash to counterbalance the incident velocity Vsin. If the aspect ratio issmall the contribution from the bound vortices can be neglected. Indeed, as saidabove, locally in each transversal section the wing acts on the fluid like a plate with

infinite chord. The problem is quasi two dimensional at each x. The bound vortices infront of the section induce negative downwash velocities whereas the bound vorticesbehind the section induce the positive up wash velocity. Since the cross section ischanged slowly the downwash and up wash contributions are nearly equal. Theresulting velocity is zero. On the contrary the contribution of free vortices issignificant. Each free vortex shed from the section at x propagates along the wingdownstream and influences the sections downstream. The total intensity of freevortices is growing along the wing chord. The no penetration condition is satisfied ineach section.

B(x)/2

B(x)/2

normal velocity componentinduced by free vortices

(y)

dy 2 Vsiny

(5.14)

The local span B(x) is changed along the x axis. Let us assume that the nopenetration condition was satisfied at x=x and we proceed to the next section at

x x x . The next section has the span from B(x x) / 2 to B(x x) / 2 consisting the old part from B(x)/2 to B(x)/2 and new winglets

B(x x) / 2, B(x) / 2 and B(x) / 2, B(x x) / 2 . The free vortices shed from the

section at x would be able to satisfy the no penetration condition within B(x)/2 to

B(x)/2 . But they are not sufficient to satisfy the no penetration condition on thewhole width B(x x) / 2 to B(x x) / 2 . New free vortices have to arise at


45/102

x x x . According to the fluid mechanics theorem the vortex lines should be theclosed lines. It means that the appearance of the longitudinal vortices leadsautomatically to the appearance of the transversal bound vortices. They generate thelift in the section x x x .

Let us consider now two sequential sections HG with the largest span and IJ. Thespan of IJ is either the same or smaller. No new winglets arise. It is assumed that theflow does not follow the wing contour rather than separates at the section with themaximum width. The free vortices coming from the section HG are able to satisfy theno penetration condition on the whole span in the section IJ because they were ableto do it on a larger span at x. No new free vortices are necessary. It means no boundvortices appear in the sections behind HG. Therefore, no lift is generated behind thesection HG.

Generalizing this analysis to a slender body theory, it is assumed that the transverseforce appears only on the ship part in front of the maximum width frame section at

maxx . This section can be identified as the section where the product 2C(x)T (x) is

maximal. Therefore, the first term in (5.12)2

2 d(C(x)T (x))Vdx

has to be

integrated from maxx to Bx .

Let us consider now the case 0, 0 . In this case the no penetration conditionreads:

B(x)/2

z

B(x)/2

normal velocity componentinduced by free vortices

(y)dy 2 x

y

(5.15)

The force arises due to two reasons. First, like in the case 0, 0 new freevortices arise in each section downstream due to change of the wing span. Second,the new free vortices appear because the right side of the equation (5.15) is changedalong the wing chord. The first effect is described by the second term in (5.12)

22 d(C(x)T (x)) xV

dx L , whereas the second effect by the third term 2C(x)T (x)

L

.

The contribution of the term

22 d(C(x)T (x)x)

V / L dx to the transverse force causedby the rotation is calculated as follows:

between maxx and Bx this term is realized in the full form2

2 d(C(x)T (x)x)V / Ldx

,

behind the maxx the second term in2 2

2 2 2d(C(x)T (x)x) dx d(C(x)T (x))V / L V / L C(x)T (x) xdx dx dx


46/102

is z

to

ass

Fig.23with flWith ttakes t

d

d

Theref

2

ero becau

low separ

umed, tha

Explanatw aroun

ese conshe form:

2V

re the tot

B

max

x

2

x

V

se the fra

ation at

the oscill

ion of forthe win

iderations

2d(C(x)T (x

dx

dY

dx

l transver

VY

2

2

d(C(x)Txdx

es of the

max (comp

ations of t

ce appeawith sm

the trans

)) d(C(

2 2C(x)T (x

se force a2

2maxCT

M

(x))dxL

modified

re sectio

e ship wa

rance onll aspect

verse for

2)T (x)) x

dx L

at xL

nd the tot

2 maxmax

xCT

L

B

H

x

z

x

dYx d

dx

B

max

x

2

x

d(x

ody are n

s HG, IJ

ke are ne

the slenratio.

e distrib

2C(x)T (x

maxx x

l momentmax

H

x

x

C(L

x

2

(x)T (x))dx

ot change

and KL i

lected at

er body

tion alon

) atL

are

2)T (x)dx

B

XH

x

x C(x)

d behind

Fig. 23c

small z .

using si

a slend

max Bx x

2T (x)xdx

max due

). It is

ilarity

r body

(5.16)

(5.17)

(5.18)

(5.19)


47/102

Here, additionally the factor is introduced because the force acting on the ship is a

half of the force acting on doubled body.2maxCT is the value of

2CT at maxx .

Introducing the force coefficients, we obtain from (5.18)2CYmax

2max

H

x / L

CY2 2 2Y y y max max

L L x / L

x xc c c CT CT CT d( )A A L L

These formulae take a very simple form for the case C const, T const :2

y y y y

22

z z z z z

y z

y z

VY c LT,c c c ,

2

VM m L T, m m m ,

2C C

C ,m ,2 4C C

C ,m .4 8

(5.20)

where2T

.L

the coefficients in (5.20) are nondimensionalized by use of the ship

length L and the lateral area LT . The moment is calculated around the ship

centre. Since zm 0 the moment component zm

is the stabilizing one, whereas

zm causes the instabiility.

The forces, obtained using the slender body theory, contain only the linearcomponents. The nonlinear components should be added additionally.

Chapter 6. Forces on ship rudders

The ship rudders are wings with the small and moderate aspect ratio which is variedin the range between 0.5 and 3.0. The relative thickness of rudders is between 10

and 30 per cent. The rudder area RA is chosen from the following two conditions:

stability of the motion (see chapter 7), required ship manoeuvrability.

The rudder design is performed in two stages. In the first design stage the rudderarea is chosen from the conditions of the motion stability and requiredmanoeuvrability. In the second stage the structure of the rudder and the torquemoment on the rudder stock are calculated. Typical ratios of the lateral ship area LT to the rudder area RA are presented in table 2.

Table 2. The ratio of the lateral ship area to the rudder area

Ship typeRLT/A

Cruise liner 85

Merchant ships 40-60

Sea tugs 30-40

River ships 12-22

Small boats 18-25


48/102

Det Norske Veritas (see [3]) recommends the following estimation for the ratio RA

LT:

2

RA B0.01 1 25LT L

The following aspects should be taken into account when hydrodynamics of therudder is considered: To increase the rudder efficiency a part of the rudder is located in the propeller

slipstream. In this case the rudder is also efficient at small and zero shipspeeds.

The slipstream induces not only the additional axial velocity but also additionaltransverse velocities on the rudder. As a result the local angle of attack of therudder is varied between zero and 15 degrees even for a non- deflectedrudder in the propeller slipstream.

The rudder is located in the ship wake. Its hydrodynamics is stronglyinfluenced by the wake.

The upper side of the rudder is located close to the ship hull.The ship hull has a positive effect on the transverse force arising on the rudder. If thegap between the hull and the rudder is zero, the effective aspect ratio of the rudder istwice the nominal value. The transverse force and the lift to drag ratio is gettinglarger. The explanation of this fact is illustrated in Fig.5.17 of the manuscript [2]. Isthe gap getting larger the positive effect quickly disappears. For customary gaps,the increase of the transverse force due to hull influence is only from five to ten percent.

The second fact motivated the engineers of the firm Becker Marine Systems [9] to

invent the twisted rudders (see Fig. 24): Conventional rudders are placed behind thepropeller with the rudder cross section arranged symmetrically about the verticalrudder centre plane. However, this arrangement does not consider the fact that thepropeller induces a strong rotational flow that impinges on the rudder blade. Thisresults in areas of low pressure on the blade that induce cavitation and associatederosion problems. To avoid cavitation and to improve the manoeuvrabilityperformance of a full spade rudder, Becker has enhanced the development of twistedleading edge rudder types,


49/102

FigurThe larVariouovervi

rudderrudderbearintype fowhat irarely,(Fig. 2(Fig. 2moderside fo(see Fi

Size atorquetransvusuallystock tTypicalis usua

24: Twi

gest rudd

rudderw of rudd

which isis rotated

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Ship Theory I