ship theory i - lemos.uni-rostock.de · xxx y yy z zz xyxy z xxz y zyz 2e (vvv)m 2[vsv s v s vs v s...
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![Page 1: Ship Theory I - lemos.uni-rostock.de · xxx y yy z zz xyxy z xxz y zyz 2E (VVV)m 2[VSV S V S VS V S V S] II I 2I2 I2 I (1.7) xzzyy zy y z x y zx x z xz z x y zx xyy yx x y z xz zyy](https://reader033.vdocument.in/reader033/viewer/2022041500/5e2138a62b1fd044400e17d4/html5/thumbnails/1.jpg)
University of Rostock Faculty of Mechanical Engineering and Marine Technology
Ship Theory I (ship manoeuvrability)
Prof. Dr.- Ing. Nikolai Kornev
Rostock
2010
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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)
![Page 3: Ship Theory I - lemos.uni-rostock.de · xxx y yy z zz xyxy z xxz y zyz 2E (VVV)m 2[VSV S V S VS V S V S] II I 2I2 I2 I (1.7) xzzyy zy y z x y zx x z xz z x y zx xyy yx x y z xz zyy](https://reader033.vdocument.in/reader033/viewer/2022041500/5e2138a62b1fd044400e17d4/html5/thumbnails/3.jpg)
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 .
![Page 4: Ship Theory I - lemos.uni-rostock.de · xxx y yy z zz xyxy z xxz y zyz 2E (VVV)m 2[VSV S V S VS V S V S] II I 2I2 I2 I (1.7) xzzyy zy y z x y zx x z xz z x y zx xyy yx x y z xz zyy](https://reader033.vdocument.in/reader033/viewer/2022041500/5e2138a62b1fd044400e17d4/html5/thumbnails/4.jpg)
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)
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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
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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)
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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)
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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)
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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)
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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”
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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)
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15. Rotating-arm basin
Fig. 12 Sketch of the rotating-arm facility [5].
16. Identification method
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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)
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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)
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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)
![Page 16: Ship Theory I - lemos.uni-rostock.de · xxx y yy z zz xyxy z xxz y zyz 2E (VVV)m 2[VSV S V S VS V S V S] II I 2I2 I2 I (1.7) xzzyy zy y z x y zx x z xz z x y zx xyy yx x y z xz zyy](https://reader033.vdocument.in/reader033/viewer/2022041500/5e2138a62b1fd044400e17d4/html5/thumbnails/16.jpg)
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)
![Page 17: Ship Theory I - lemos.uni-rostock.de · xxx y yy z zz xyxy z xxz y zyz 2E (VVV)m 2[VSV S V S VS V S V S] II I 2I2 I2 I (1.7) xzzyy zy y z x y zx x z xz z x y zx xyy yx x y z xz zyy](https://reader033.vdocument.in/reader033/viewer/2022041500/5e2138a62b1fd044400e17d4/html5/thumbnails/17.jpg)
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
![Page 18: Ship Theory I - lemos.uni-rostock.de · xxx y yy z zz xyxy z xxz y zyz 2E (VVV)m 2[VSV S V S VS V S V S] II I 2I2 I2 I (1.7) xzzyy zy y z x y zx x z xz z x y zx xyy yx x y z xz zyy](https://reader033.vdocument.in/reader033/viewer/2022041500/5e2138a62b1fd044400e17d4/html5/thumbnails/18.jpg)
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)
![Page 19: Ship Theory I - lemos.uni-rostock.de · xxx y yy z zz xyxy z xxz y zyz 2E (VVV)m 2[VSV S V S VS V S V S] II I 2I2 I2 I (1.7) xzzyy zy y z x y zx x z xz z x y zx xyy yx x y z xz zyy](https://reader033.vdocument.in/reader033/viewer/2022041500/5e2138a62b1fd044400e17d4/html5/thumbnails/19.jpg)
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)
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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)
![Page 21: Ship Theory I - lemos.uni-rostock.de · xxx y yy z zz xyxy z xxz y zyz 2E (VVV)m 2[VSV S V S VS V S V S] II I 2I2 I2 I (1.7) xzzyy zy y z x y zx x z xz z x y zx xyy yx x y z xz zyy](https://reader033.vdocument.in/reader033/viewer/2022041500/5e2138a62b1fd044400e17d4/html5/thumbnails/21.jpg)
* 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!
![Page 22: Ship Theory I - lemos.uni-rostock.de · xxx y yy z zz xyxy z xxz y zyz 2E (VVV)m 2[VSV S V S VS V S V S] II I 2I2 I2 I (1.7) xzzyy zy y z x y zx x z xz z x y zx xyy yx x y z xz zyy](https://reader033.vdocument.in/reader033/viewer/2022041500/5e2138a62b1fd044400e17d4/html5/thumbnails/22.jpg)
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27 Manoeuvrability diagram
Figure 8.5: Manoeuvrability diagram
28. Experimental manoeuvring tests
![Page 24: Ship Theory I - lemos.uni-rostock.de · xxx y yy z zz xyxy z xxz y zyz 2E (VVV)m 2[VSV S V S VS V S V S] II I 2I2 I2 I (1.7) xzzyy zy y z x y zx x z xz z x y zx xyy yx x y z xz zyy](https://reader033.vdocument.in/reader033/viewer/2022041500/5e2138a62b1fd044400e17d4/html5/thumbnails/24.jpg)
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)
![Page 25: Ship Theory I - lemos.uni-rostock.de · xxx y yy z zz xyxy z xxz y zyz 2E (VVV)m 2[VSV S V S VS V S V S] II I 2I2 I2 I (1.7) xzzyy zy y z x y zx x z xz z x y zx xyy yx x y z xz zyy](https://reader033.vdocument.in/reader033/viewer/2022041500/5e2138a62b1fd044400e17d4/html5/thumbnails/25.jpg)
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)
![Page 26: Ship Theory I - lemos.uni-rostock.de · xxx y yy z zz xyxy z xxz y zyz 2E (VVV)m 2[VSV S V S VS V S V S] II I 2I2 I2 I (1.7) xzzyy zy y z x y zx x z xz z x y zx xyy yx x y z xz zyy](https://reader033.vdocument.in/reader033/viewer/2022041500/5e2138a62b1fd044400e17d4/html5/thumbnails/26.jpg)
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
![Page 27: Ship Theory I - lemos.uni-rostock.de · xxx y yy z zz xyxy z xxz y zyz 2E (VVV)m 2[VSV S V S VS V S V S] II I 2I2 I2 I (1.7) xzzyy zy y z x y zx x z xz z x y zx xyy yx x y z xz zyy](https://reader033.vdocument.in/reader033/viewer/2022041500/5e2138a62b1fd044400e17d4/html5/thumbnails/27.jpg)
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
![Page 28: Ship Theory I - lemos.uni-rostock.de · xxx y yy z zz xyxy z xxz y zyz 2E (VVV)m 2[VSV S V S VS V S V S] II I 2I2 I2 I (1.7) xzzyy zy y z x y zx x z xz z x y zx xyy yx x y z xz zyy](https://reader033.vdocument.in/reader033/viewer/2022041500/5e2138a62b1fd044400e17d4/html5/thumbnails/28.jpg)
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