University of Rostock Faculty of Mechanical Engineering and Marine Technology

Ship Theory I (ship manoeuvrability)

Prof. Dr.- Ing. Nikolai Kornev

Rostock

2010

1. Ship motion equations in the inertial reference system

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)

2 2 2k

m m m

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

(1.4)

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

(1.5)

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)

2 2 2k 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 I

2 I 2 I 2 I

(1.7)

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

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

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

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

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

d t d t d t d t d t

(1.8)

2 Ship motion equations in the ship-fixed reference system

Fig.1 Change of the linear and angular momentums due to displacement of the origin

of the ship fixed reference system from the point O to the point /O .

Fig.2 Change of the linear and angular momentums due to rotation at the angle t .

dP P F

dtd

D V P D Mdt

(1.11)

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

y z 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 dt

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

dt dtdVd d

I 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 dI S I V ( S S ) V S

dt dt dt

( I V S V S ) ( I

y z z xz zV S I ) M .

(1.13)

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

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 ,

dtdV

m( V V ) F ,dt

dVm( V V ) F ,

dtd

I (I I ) M ,dt

dI (I I ) M ,

dtd

I (I I ) M .dt

(1.16)

4 Forces and moments arising from acceleration through the water

6 6

Fl1 1

1

2 i k ik

i k

E V V m (1.17)

dSn

m k

S

iik

(1.19)

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)

5 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)

6 6 6yx

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

m V V m V V m Vdt

6 6

6k k z 5k k xk 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 dtd

m 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 zk 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)

6. Coordinate system, Aims of the ship manoeuvring theory, Main assumptions of the theory

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)

7. Equations in the ship-fixed coordinates with principle axes

26 xm S 0 (2.2)

2

/ ( ),

cos( , ) ( ),

cos( , ) (1),

/ (1),

(1).

y L O

n x O

n y O

x L O

O

(2.3)

2

2 2

2

2

( ) cos( , ) ( ) 0

cos( , ) cos( , ) 0

cos( , )

cos( , )

g gS m

gS 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

/ //22

22g

x m x mx

m m

(2.5)

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)

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)

8 Munk moment

11 22 z x

22 11 z y

zzz 66 z

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

(m m )( Vsin V cos ) (m m )V cos F ,

d(I m ) M .

dt

(2.11)

Fig.6. Illustration of the Munk moment. a)-inviscid fluid, b) viscous fluid.

9 Equations in terms of the drift angle and trajectory curvature

x

y

dV dV dcos Vsin V cos V sin ,

dt dt dtdV dV d

sin V cos Vsin V cos ,dt dt dt

(2.12)

z

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

R V (2.14)

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)

zz 6611 22x y

3L L L

I mm m m m, , .

A L A L A L2 2 2

(2.16)

/

/x x y x

//

y y x y

//

z

Vcos sin sin C ,

V

Vsin cos cos C ,

V

Vm .

V

(2.17)

10. Determination of added mass.

dSn

m k

S

iik

(3.1)

MN ii i 2

MNS

1 cos(n, R ) qV q dS 0

4 R 2 (3.2)

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)

ii 2 2 2

S

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

4 (x ) (y ) (z )

(3.4)

11. Added mass of the slender body.

/22 2 2 22

0 0

cos( , ) cos( , )L L

S C

m n y dS n y dCdL m dL (3.6)

66 6 60

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

S C

m x n y y nx dS x n y dCdL (3.7)

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)

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)

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

3

a bz

z z , (3.11)

/22

2

mC

T (3.12)

Fig. 8 Lewis coefficients depending on H 2T / B and spA /(BT) , where spA

is the frame area (taken from [1])

/ 222 22

0 0

( ) ( ) L L

m m dL C x T x dL , 2 / 2 266 22

0 0

( ) ( ) L L

m x m dL x C x T x dL , (3.13)

/ 226 22

0 0

( ) ( ) L L

m xm dL xC x T x dL

22 2 22 _ 66 3 66 _

2 22 _ 22 _ _

3 66 _ 66 _ _

( , ) , ( , ) ,

( , ) / ,

( , ) / .

slender slender

ellipsoid slender ellipsoid

ellipsoid slender ellipsoid

m R a b m m R a b m

R a b m m

R a b m m (3.14)

11 1 22 _( , ) slenderm R a b m . (3.15)

222 _ _

366 _ _

4,

34

15

slender ellipsoid

slender ellipsoid

m ab

m ab (3.16)

Fig. 9 Munk’s correction factors.

L

211 1

0

L2

22 2

0

L2 2

66 3

0

1m R C(x)T (x)dx,

2

1m R C(x)T (x)dx,

2

1m R C(x)T (x)x dx.

2

(3.17)

13. Steady manoeuvring forces. Representation of forces

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 component1, 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

1 For the sake of brevity both force and moment are meant here and further under the term “force”

Hypothesis of quasi steady motion. Truncated forms. Cross flow drag principle.

Fig. 10. Typical dependence of the transverse force on the drift angle, q is the nonlinear part of the force.

y y yC ( ) C C (4.4)

2 2 2

//, .

2 4

z

y z

dM ddY dC m

V TL V TL (4.5)

3y y 3

y 3

dC d C1C ( ) ...

d 6 d

(4.6)

2y yC C (4.7)

y y yC ( ) C C (4.8)

14. The planar motion mechanism (PMM)

2 2 2

2 2x x xxi xi yi zi yi zi xi2 2

y 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 F

V 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)

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

15. Rotating-arm basin

Fig. 12 Sketch of the rotating-arm facility [5].

16. Identification method

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

zzz 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) 17 Calculation of steady manoeuvring forces using slender body theory

222 yP m V C(x)T (x)V sin x (5.1)

Fig.20 Active cross section along the ship length

ad( P)Y

dt

(5.2)

ad( P) d( P) dxY

dt dx dt

(5.3)

22d( P) dx d(C(x)T (x))

Y V sin cos xdx dt dx

(5.4)

22dY Y d(C(x)T (x))

V sin cosdx x dx

(5.5)

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

Vdx dx

(5.6)

Fig.21 Distribution of 2C(x)T (x) and of the transverse force (5.6) along the ship

length

B Bx x 22 2 2

x x

dY d(C(x)T (x))Y(x) dx V dx V C(x)T (x)

dx dx (5.7)

B B B

H H H

x x x22 2 2 2

22

x x x

dY d(C(x)T (x))xdx xdxV C(x)T (x)dxV m V

dx dx (5.8)

222 11 22( )Munk x yM V V m m V m (5.9)

Y xx x V

R L (5.10)

y

x xV (x) Vsin V V( ) V (x)

L L (5.11)

22

2 2

2 22 2

xd(C(x)T (x)( ))dY d(C(x)T (x) (x)) LV V

dx dx dx

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

dx dx L L

(5.12)

B B B Bx x x x2 2

2 2

x x x x

2 2 2

dY d(C(x)T (x)) d(C(x)T (x)) xY(x) dx V dx dx C(x)T (x)dx

dx dx dx L L

xV C(x)T (x) C(x)T (x)

L

(5.13)

Fig.22 Distribution of the transverse force components proportional to terms

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

dx L and 2C(x)T (x) along the ship length

18. Improvement of the slender body theory. Kutta conditions 19. Forces on ship rudders

2eff

R XR R

2eff

R yR R

2eff

ZR ZR R

VX C A ,

2

VY C A ,

2

VM m A C.

2

(6.1)

2eff 2

yR ReffR R

YR yR2 2L

L L

VC A VY A2C C

V V V AA A

2 2

(6.2)

2eff 2

XR ReffR R

XR XR2 2L

L L

VC A VX A2C C

V V V AA A

2 2

(6.3a)

2eff 2

ZR ReffZR R

ZR ZR2 2L

L L

Vm A C VM A C2m m

V V V A LA L A L

2 2

(6.3b)

2

eff RYR R YR

L

V AC ( ) C

V A

(6.4)

ZR R R R

LM ( ) Y ( ).

2 (6.5)

ZR R YR R

1m ( ) C ( ).

2 (6.6)

R

R,eff R

x( )

L (6.7)

RR,eff R

x( )

L (6.8)

20 Interaction between the rudder and propeller

2 2A B

B

point A point B

V Vp p

2 2

(6.10)

22CD

C

point D point C

VVp p

2 2

(6.11)

Figure 30: The streamline ABCD

2 2 2

2D A AC B A 2

A

V V (V u)p p V 1

2 2 2 V

(6.12)

S2A 0

Tc

V A2 (6.13)

eff A A SV V u V 1 c (6.15)

AV (1 w)V (6.16)

eff SV (1 w) 1 c V (6.17)

21 Yaw stability

/O( ), O( ),V / V O( ) (7.2) /

/x x y x

//

y y x y

//

z

Vcos sin sin C ,

V

Vsin cos cos C ,

V

Vm .

V

(7.3)

2 22

// 2 2

x x y x x

O( ) O( )O( )O( )

Vcos sin sin C C ....

V

22

//

y y x y y y y YR

O( ) O( )O( ) O( ) O( )O( )

Vsin cos cos C C C C .... C

V

2

2

//

z z z z zR

O( ) O( ) O( )O( )O( )

Vm m m m .... m

V

/

x

/x y y y

/z z

Vcos 0,

VC C ,

m m .

(7.4)

/ *y y y

/z z

C C 0,

m m 0.

(7.6)

/ / // /y y y y* *

y y

1 1C C

C C (7.7)

// / /zy y y y z* *

y y

*y y z y z z y// /

y y

mC C m 0

C C

C m C m m C0

// /2a b 0 (7.8)

where *

y y z y z z y

y y

C m C m m C2a , b

.

// /2a b 0 (7.9)

1 2

1 2

p p1 2

p p1 2

( ) e e ,

( ) e e .

(7.10)

1 2 1 2

1 2 1 2

p p p p/ // 2 21 1 2 2 1 1 2 2

p p p p/ // 2 21 1 2 2 1 1 2 2

( ) p e p e ( ) p e p e

( ) p e p e ( ) p e p e

(7.11)

1 2 *

1 2 */

1 1 2 2/

1 1 2 2

(0) ,

(0) ,

(0) p p 0,

(0) p p 0.

(7.12)

,1 2p p2 2 2

1 1 1 2 2 2 2

2 21 1 2 2

e (p 2ap b) p e (p 2ap b) 0

p 2ap b 0, p 2ap b 0.

(7.13)

1 2p p2 2 21 1 1 2 2 2 2

2 21 1 2 2

e (p 2ap b) p e (p 2ap b) 0,

p 2ap b 0, p 2ap b 0

(7.14)

2p 2ap b 0 2

1,2p a a b (7.15)

2 2 *y y z y y z y y z y z y2

2y

2 2 2* *y y z y y z y z y y y z y z y

2 2y y

C m 2C m 4 C m 4 m Ca b

4( )

C m 2C m 4 m C C m 4 m C0

4( ) 4( )

(7.16)

* *y z z y y z z y *

y z z yy y

C m m C C m m C0 0 C m m C 0

(7.19)

*

z y y zm C C m 0 (7.20)

*

z y y zm C C m (7.21)

z z*y y

m m

C C

(7.22)

z z*y y

m m

C C

(7.23)

or X X (7.24)

22 Influence of ship geometric parameters on the stability

B

C B C C CC 0

4 T 4 8 2

(7.25)

11x B

L L

m m m BC

TA L A L2 2

BC B1

2 CT (7.26)

23 Trajectory of a stable ship after perturbation

1 2 1 2p p p p1 21 2

1 20 0

( ) d ( e e )d (e 1) (e 1)p p

(7.27)

1 2 1 2

0

0

0

0 0

p p p p1 2 1 2 1 22 2

1 2 1 2 1 2

xcos( )d ,

L

ysin( )d ( )d

L

(e 1) (e 1) (e 1) (e 1)p p p p p p

(7.28)

0

0 1 2 1 2 1 22 2

1 2 1 2 1 2

x,

L

y.

L p p p p p p

(7.29)

1 2

1 2

( )p p

(7.30)

Figure 7.1: The trajectory of the stable ship after perturbation

24 Steady ship motion in turning circle

x y y YR

z z zR

C C C ,

0 m m m .

(7.31)

*y zR z YR y zR z YR

c c* *z y z y z y z y

C m m C C m m C, .

m C m C m C m C

(7.32)

zc c

c c c

L V L L LR

V R V R

(7.33)

25 Regulation of the stability

cD Dc c

c

x xV( ) 0

L L

(7.34)

* By zcD B

c y z

B C CC

C 2mx C BT 4 4C CL C 2m CT2 2

2 (7.35)

Dx 1

L 2 (7.36)

Dx0.3 0.4

L (7.37)

26 Diagram

x y y YR

z z zR

C C C ,

0 m m m .

y YRf * *

y y

C C

C C

from the first equation for the y- force (8.3)

z zRm

z z

m m

m m

from the second equation for the z- moment

(8.4)

YRf *0

y

C0

C for positive rudder angles R 0 (8.5)

YRf *0

y

C0

C for negative rudder angles R 0 (8.6)

zRm 0

z

m0

m for positive rudder angles R 0 (8.7)

zRm 0

z

m0

m for negative rudder angles R 0 (8.8)

.

2 x B

BC

T . Please prove!

27 Manoeuvrability diagram

Figure 8.5: Manoeuvrability diagram

28. Experimental manoeuvring tests

29 Forces due to non-uniformity of the ship wake.

. 2 2 x1 x

1 y x x1 0

u UAY C (V (U u ) )( )

2 V

(9.1)

2 2 x2 x2 y x x2 0

u UAY C (V (U u ) )( )

2 V

(9.2)

x1,2uO( )

V

2 2x x x1

1 y 0

A(V U ) U uY C ( )

2 V V

(9.3)

2 2x x x2

2 y 0

A(V U ) U uY C ( )

2 V V

(9.4)

2 2x x x1

1 y 0 0

A(V U ) U uY C ( )sin

2 V V

(9.5)

2 2x x x2

2 y 0 0

A(V U ) U uY C ( )sin

2 V V

(9.6)

2 2 2 2x x1 x2 x x2 x1

1 2 y 0 0 0 y 0

A(V U ) u u A(V U ) u uY Y C ( )sin C sin

2 V V 2 V

(9.7)

30 Forces due to oblique flow.

2D

1

C AY ( r Vsin )

2 (9.8)

2D2

C AY ( r Vsin )

2 (9.9)

1 2 DY Y 2C AVsin ( r) 0 (9.10)

31. Shallow water effect. Influence of the wall on a mooring ship. Influence of the inclined wall or of inclined bottom

32 Criterion of the static stability of airplanes.

0zm

/0

/z

a

m

C

0X

33 CFD Calculation of yaw ship motion

1 2 31 2 3

0

1 2 31 2 3( ) ( ) ( )

dV dV dV dV di dj dki j k V V V

dt dt dt dt dt dt dt

dV dV dVi j k V i V j V k

dt dt dt

dVV

dt

(10.33)

00

dx dxx u u x

dt dt

(10.34)

20 0

0 20

2 ( )du du d x dx d

u x xdt dt dt dt dt

(10.35)

( )du u

u udt t

(10.36)

2

2

1( ) 2 ( )

u d x dx du u f p u x x

t dt dt dt

(10.37)

1( ) 2 ( )

uu u f p u u x

t

(10.38)

34. Overset or Chimera grids

Fig. 10.6 Chimera grid for tanker KVLCC2. Propeller is modeled using body forces distributed

along the propeller disc.

Fig. 10.7 Chimera grid for a container ship with propeller and rudder.

35 Morphing grids

1( )g

uu U u f p u

t

(10.39)

0g

U S

dU U ndSt

(10.40)

17-34 21-32