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Facu ty of Mec
Unive
hanical E
hip(ship
rof. Dr.-
rsity of R
gineerin
Thanoeuv
Ing. Niko
Rostock2010
ostock
and Ma
oryrability)
lai Korn
ine Tech
I
v
ology
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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
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)
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The symotion
1.2 Shi
The ssystem
system
Advant
inertia
To reestablireferenand an
Followi
V iV
instant
V t
. V
Fig.1
As see
in the
mome
aD (t
chang
stem (1.8)of the shi
ip motion
ip motionthe ship f
is mo
age of th
and static
rite the eh the relce systegular mo
ng to [1
yjV k
t both sys
ectors P
hange of
of the s
n in Fig.1
earth fix
tum is
at) D (t)
of the m
is the gein earth
equation
is sufficiixed refer
ing with
e ship-fix
moments
uations (tion betws. This rentums a
] let us
z and rot
tems are
nd D
are
the linear
ip fixed r
the linear
ed syste
hanged
(V t) P
mentums
eral systeonnected
in the s
ntly simplnce syste
velocitie
d coordi
are consta
1.1) in theen the linlation is fe kept co
consider
tion with
oincided.
also shift
and angul
ference s
momentu
due to
ue to c
D (V
due to tra
adP
d
m describreference
ip-fixed r
ified whem is used
xV iV
ates is t
nt in time,
ship-fixeear and aound undstant in t
consequ
angular v
At the tim
d from th
r moment
stem from
m vector t
translati
ange of
P) t
. Th
slation is:
_ tr dD0,t d
ing the sixsystem.
eference
instead. The origi
yjV kV
at the in
i.e. ijdI / d
referencgular mor conditioe ship-fix
ently the
locity
e t+t the
point O
ums due t
the point
ransferred
n aP (t
the arm
refore, t
_ tr V P
degree o
system
f the earof the sh
z and
rtia mom
i0,dS /d
systementums
n that ved referen
translati
x yi j
body is lo
o the poin
displace
O to the
by the sh
at) P (t)
of the li
e contrib
f freedom
h-fixed reip fixed re
xi j
ents, pro
t 0 .
it is neceritten in
tors of the system.
on with
zk
. At t
cated at t
t/O (see
ent of th
oint/O .
ip is not c
P
. The
inear mo
ution to
(6DOF)
ferenceference
y zk
ucts of
sary toifferent
e linear
velocity
he time
e point
ig.1).
origin
hanged
angular
entum
rate of
(1.9)
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Fig.2
In the
fixed r
of both
time t
Theref
Takingsystem
Here italso inseconangulasystemforces.
The eqThe s(xyz)axis ispositivof therest.
Change of
econd ste
ference s
vectors r
is
re, the co
(1.9) andis written
should bthe moviand third
r moment. The rig
uations (1
ip fixed rith x andthe longitto the p
origin lies
the linear
p, the mo
stem wit
main con
a
a
P (D
ntribution
(1.10) intin the for
noted thng refereterms in tums duet sides a
.11) were
ferencey lying inudinal cort side. Tat the lev
Fig.3 Sk
and angul
entums
the angu
stant. The
at) P (t t) D
to the rate
a _ rdP
dt
account:
dP
dt
d Ddt
at the forcce syste
he secondto translre respon
derived by
ystem isa horizonrdinate, pe origin il of the u
tch of the
ar momen
ectors ar
lar velocit
ir change
a
a
t) P(t) D
of chang
t P,
the mom
P F
V P
es F
and. The sequationtion andsible for
Kirchhoff in
he Carteal planeositive forin the pl
ndisturbe
ship fixed
ums due t
rotated d
xi
in the ear
( P)D (
of the m
a_rotD
dt
ntum the
D M
momentscond terdescribe trotationomentu
1869.
ian right-nd z vertiward, y isne of syfree surf
coordinat
o rotation
ring the r
y zj k
h fixed s
t,) t.
mentums
D.
rem in sh
M
havein the f
he changf the shi
change
handed ccal, positithe transmetry. Thce when
e system.
at the angl
otation of
. The ma
stem occ
due to rot
ip fixed re
to be detirst equati
of the linfixed redue to
oordinatee upward
verse cooe verticalthe ship i
e t
.
he ship
nitudes
rring in
tion is:
(1.10)
ference
(1.11)
rminedon andear andferencexternal
system. The xrdinate,location
at the
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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)
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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
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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
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/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)
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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.
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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.
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If the
motion
effect
Chap
2.1 Co
The cdesign
x0, y0x,y
V
R
2.2 AiThe sh
ship spee
and, vice
ccelerate
er 2. Eq
ordinate
oordinateations are
2 2x yV
S
a
s of the
ip manoeto keep thto changeto change
d is grow
versa, if
the ship
uations
ystem
Fig.5. Coo
system
oordinateoordinatehip speed
rift angle,ourse anudder anngle.
hip man
vring theoprescrib
the coursthe speed
dm
ing xdV
dt
he ship s
otion.
of ship
rdinate sy
sed in
in the inin ship-fi,
positive ifle, positivle, positi
euvring t
ry is intend course,to follow
.
xxF m
t
0 the flui
peed bec
anoeu
tem used
hip man
rtial coored coordi
the flow inif the ya
e if the r
heory
ed to inv
a prescrib
x11
dV
dt
d inertia
mes sma
ring.
in ship m
euvring
inate systnate syste
comes froing againdder cau
stigate th
ed traject
ffect dec
ller xdVdt
noeuvrin
s shown
m,m,
m the starst clockwises incre
ability of
ry and to
elerates t
0 the flui
in Fig.5
board side directiose of the
ship:
void obst
(1.44)
e ship
inertia
. The
,,course
acles,
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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
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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):
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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
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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)
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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
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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.
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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)
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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
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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)
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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
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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
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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:
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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)
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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.
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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
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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]).
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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
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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
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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
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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
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,
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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
.
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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
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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
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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:
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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
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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
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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)
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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)
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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
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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
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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
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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
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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)
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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
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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,
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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