Chapter IV
(Ship
Hydro

Statics
& Dynamics)
Floatation & Stability
4.1 Important Hydro

Static Curves or
Relations
(see Fig. 4.11 at p44 & handout)
•
Displacement Curves
(displacement [molded, total]
vs. draft, weight [SW, FW] vs. draft (
T
))
•
Coefficients Curves
(
C
B
, C
M
, C
P
, C
WL
, vs.
T
)
•
VCB (KB,
Z
B
):
Vertical distance of Center of
Buoyancy (C.B) to the baseline vs.
T
•
LCB (LCF,
X
B
):
Longitudinal Distance of C.B or
floatation center (C.F) to the midship vs.
T
4.1 Important Hydro

Static Curves or
Relations (Continue)
•
TPI:
Tons per inch vs.
T
(increase in buoyancy due
to per inch increase in draft)
•
Bonbjean Curves
(p63

66)
a) Outline profile of a hull
b) Curves of areas of transverse sections (stations)
c) Drafts scales
d) Purpose: compute disp. & C.B., when the vessel
has 1) a large trim, or 2)is poised on a big wave crest or
trough.
How to use Bonjean Curves
•
Draw the given W.L.
•
Find the intersection of the W.L. & each station
•
Find the immersed area of each station
•
Use numerical integration to find the disp. and C.B.
4.2 How to Compute these curves
•
Formulas for Area, Moments & Moments of Inertia
0
0
2 2
2
2
0.0
0
) Area ,
) Moments ,
Center of Floatation /
) Moments of Inertia
,
L
A
L
M
I A
L
C F
a d ydx A ydx
b d xydx M xydx
x M A
c d x d x ydx
I x ydx I I x A
Examples of Hand Computation of
Displacement
Sheet
(Foundation for Numerical Programming)
•
Area, floatation, etc of 24’ WL (Waterplane)
•
Displacement (molded) up to 8’ WL
•
Displacement (molded) up to 24’ and 40 ‘WL (vertical
summation of waterplanes)
•
Displacement (molded) up to 24’ and 40 ‘WL (Longitudinal
summation of stations)
•
Wetted surface
•
Summary of results of Calculations
4’wl
area
8’wl
area
16’wl
area
24’wl
area
32’wl
area
40’wl
area
Up to
4’wl
Up to
8’wl
Up to
24’ & 40
wl
Disp. Up
to 24’wl
Disp. Up
to 40’wl
Disp. Up
to 16’wl
Disp. Up
to 32’wl
MTI
MTI
Wetted
surface
Summary
Red sheet will be studied in
detail
•
1

6 Areas & properties (F.C.,
Ic, etc) of W.L
•
7

11 Displacement,
Z
B
, and
X
B
up W.L., vertical
integration.
•
12

15 Transverse station area,
longitudinal integration for
displacement,
Z
B
, and
X
B
•
16

18 Specific Feature (wetted
surface, MTI, etc.
•
19 Summary
1
0 1
2 3
0 1
2
1 2
2
2
3
The distance
between the
two stations
2
Simpson's 1st
3
2 1
2
3 2
2
2 1
3 4
3
2..
4
2 Symmetric
m S
S
y y
y y
y y
y y
2
2
3
3
3
4
5 6
Formulas for the remaining coefficents
2
2
3
the distance between the two stations
2
Simpson's 1st rule coeff.; 2  Symmetr
ic
3
2
2
3
1 2 1
2 ( from )
3 3 3 3
2
2,
3 3
i
m
m S
S
m S
y
m S
h
m S m
2
2
2
3 3
h
S
Illustration of Table 4
:
C1 Station FP

0
AP

10 (half station)
C2 Half Ordinate copy from line drawing table ( 24’ WL).
(notice at FP. Modification of half ordinate)
C3 Simpson coefficient (Simpson rule 1) (1/2 because of half station)
C4 = C3 x C2
(area function) displacement
C5 = Arm (The distance between a station and station of 5 (Midship)
C6 = C5 x Function of
Longitudinal Moment
with respect to Midship (or station 5)
C7 = Arm (same as C5)
C8 = C6 x C7 Function of
Longitudinal moment of inertia
with respect to Midship.
C9.= [C2]
3
C10. Same as C3. (Simpson Coeff.)
C11. = C9 x C10. Transverse moment of inertia of WL about its centerline
Table 5 is similar to Table 4, except the additional computation of appendage.
Illustration of
Table 8
For low WLs, their change is large. Therefore, it is first to use planimeter or other
means to compute the half

areas of each stations up to No. 1 WL (8’ WL).
C1.
Station
C2.
Half area (ft
2
) of the given station
C3.
C3/(h/3) ( divided by h/3 is not meaningful, because it later multiplying by
h/3) (h = 8’ the distance between the two neighboring WLs)
C4.
½ Simpson’s Coeff.
C5.
C4 x C3
C6.
Arm
distance between this station and station 5 (midship)
C7
C5 x C6
f(M)
Illustration of
Table 9
C1.
WL No.
C2.
f(V) Notice first row up to 8’. f(v)
C3.
Simpson’s coeff.
C4. C2 x C3
C5. Vertical Arm above the base
C6.
C4 x C5.
f(m) vertical moment w.r.t. the Baseline.
*
Notice up the data in the first row is related to displacement up to 8’
WL. The Table just adding V)
Illustration of
Table 12
C1. Station No.
C2. under 8’ WL.
(From Table 8)
C3.
8’ WL x 1
C4.
16’ WL x ¼ (SM 1 + 4 + 1)
C5.
24’ WL x 1
C6.
(C2 + C3 + C4 + C5) Function of Area of Stations
C7.
Arm
(Distance between this station to midship)
C8.
C7 x C6
(Simpson rule)
C9.
C6*h/3
4.3 Stability
A floating body reaches to an
equilibrium
state, if
1) its weight = the buoyancy
2) the line of action of these two forces become
collinear
.
The equilibrium:
stable, or unstable or neutrally stable
.
•
Stable equilibrium
: if it is slightly displaced from its
equilibrium position and will return to that position.
•
Unstable equilibrium
: if it is slightly displaced form its
equilibrium position and tends to move farther away from
this position.
•
Neutral equilibrium
: if it is displaced slightly from this
position and will remain in the new position.
•
Motion of a Ship:
6 degrees of freedom

Surge

Sway

Heave

Roll

Pitch

Yaw
Axis
Translation
Rotation
x Longitudinal
Surge Neutral S.
Roll S. NS. US
y Transverse
Sway Neutral S.
Pitch S.
z Vertical
Heave S. (for sub, N.S.)
Yaw NS
Righting & Heeling Moments
A ship or a submarine is designed to float in the
upright
position
.
•
Righting Moment
: exists at any angle of inclination where
the forces of weight and buoyancy act to move the ship
toward the upright position.
•
Heeling Moment
: exists at any angle of inclination where
the forces of weight and buoyancy act to move the ship away
from the upright position.
G

Center of Gravity, B

Center of Buoyancy
M

Transverse Metacenter
, to be
defined later
.
If M is above G, we will have a
righting moment
, and
if M is below G, then we have a
heeling moment
.
W.L
For a displacement ship,
For submarines (immersed in water)
G
B
G
If B is above G, we have
righting moment
If B is below G, we have
heeling moment
Upsetting Forces (overturning moments)
•
Beam
wind
,
wave
&
current
pressure
•
Lifting a weight (when the ship is loading or unloading in
the harbor.)
•
Offside weight (C.G is no longer at the center line)
•
The loss of part of buoyancy due to damage (partially
flooded, C.B. is no longer at the center line)
•
Turning
•
Grounding
Longitudinal Equilibrium
For an undamaged (intact) ship, we are usually only
interested in determining the ship’s draft and trim regarding
the longitudinal equilibrium because the ship capsizing in the
longitudinal direction is
almost impossible
. We only study
the
initial stability for the longitudinal equilibrium
.
Static Stability & Dynamical Stability
Static Stability:
Studying the magnitude of the
righting moment given the inclination (angle) of the
ship*.
Dynamic Stability:
Calculating the amount of work
done by the righting moment given the inclination of
the ship.
The study of dynamic Stability is based on the study of
static stability.
•
Static Stability
1)
The
initial stability
(aka stability at small inclination) and,
2)
the
stability at large inclinations
.
•
The initial (or small angle) stability:
studies the
right
moments or right arm
at small inclination angles.
•
The stability at large inclination (angle):
computes the right
moments (or right arms) as function of the inclination angle, up
to a limit angle at which the ship
may lose
its stability
(capsizes).
Hence, the initial stability can be viewed as a special case of the
latter.
Initial stability
•
Righting Arm:
A symmetric ship is inclined at a small angle
dΦ. C.B has moved off the ship’s centerline as the result of the
inclination. The distance between the action of buoyancy and
weight,
GZ
, is called
righting arm
.
•
Transverse Metacenter:
A vertical line through the C.B
intersects the original vertical centerline at point,
M
.
sin
if 1
GZ GM d
GMd d
Location of the Transverse Metacenter
Transverse metacentric height
:
the distance between the
C.G.
and
M
(
GM
). It is important as an index of
transverse stability at small angles of inclination.
GZ
is
positive, if the moment is righting moment.
M
should be
above C.G, if
GZ
>0.
If we know the location of
M
, we may find
GM
, and thus the
righting arm
GZ
or righting moment can be determined
given a small angle dΦ.
How to determine the location of
M
?
When a ship is inclined
at small angle dΦ
WoLo
–
Waterline (W.L) at upright position
W
1
L
1
–
Inclined W.L
Bo
–
C.B. at upright position, B
1
–
C.B. at inclined position

The displacement (volume) of the ship
v
1
, v
2
–
The volume of the emerged and immersed
g
1
, g
2
–
C.G. of the
emerged
and
immersed
wedge, respectively
Equivolume Inclination
(
v
1
=v
2
)
If the ship is
wall

sided
with the range of inclinations of a small
angle dΦ, then the volume
v
1
and
v
2
, of the two wedges between
the two waterlines will be
same
. Thus, the displacements under
the waterlines WoLo and W
1
L
1
will be same. This inclination is
called
equivolume inclination
.
Thus, the intersection of WoLo,
and W
1
L
1
is at the
longitudinal midsection
.
For most ships, while they may be
wall

sided
in the vicinity of
WL near their
midship section
, they are not
wall

sided near
their sterns and bows
. However, at a small angle of
inclination, we may still
approximately
treat them as
equivolume inclination
.
When a ship is at
equivolume inclination
,
According to a theorem from mechanics, if one of the bodies
constituting a system moves in a direction, the C.G. of the
whole system moves in the same direction parallel to the shift of
the C.G. of that body. The shift of the C.G. of the system and
the shift of the C.G of the shifted body are in the inverse ratio of
their weights.
1 2
0 1 1 2
,
vg g
B B v v v
3
0
0 1
1 2
0
3 3
1 2
0 0 0
0 0
2
3
,
tan( ) tan( )
1 2 2 2
( tan ) (2 ) tan,
2 3 3 3
the moment of inertia of WL w.r.t. the l
ongitudinal axis
L
x
L L L
x
x
y dx
B B I
vg g
B M
d d
vg g y y d y dx d y dx I y dx
I
For a ship inclined at a small angle , t
he location of
its transverse metacenter is approximate
ly above its
C.B. by , which is independent of .
.. (Metacenter measured from keel ), o
r
is
x
M
d
I
d
K M
H
.
the height of metacenter above the basel
ine.
,
where is the vertical coordinates of th
e C.B.
The vertical distance between the metace
nter & C.G,
x
M B
B
x
M G B G
I
K.M. = H = +Z
Z
I
GM H Z +Z Z
If we know the vertical position of the
C.G., and the
C.B., the righting arm at small angles
of inclination, ,
and the righting moment is
.
G
B
x
B G
x
w B g
Z
Z d
I
GZ GM d Z Z d
I
M Z Z d
Examples of
computing
KM
d
B
3
2
2
3
2
2
) Rectangular cross section
1
, ,
2 12
12
12 2
) Triangular cross section
2 1 1
, ,
3 12 2
6
2
6 3
B x
x
B
B x
x
B
a
d
Z I LB LBd
I
B
BM
d
B d
KM BM Z
d
b
d
Z I LB LBd
I
B
BM
d
B d
KM BM Z
d
d
B
Natural frequency of Rolling of A Ship
2
2
2
Free vibration
0
where is the inertia moment of the shi
p w.r.t. C.G.
A large leads to a higher natural freq.
X w
w
X
X
M GM
t
GM
M
M
GM
4.4Effects of free surfaces of
liquids on the righting arm
pp81

83
•
When a liquid tank in a ship is not full,
there is a free surface in this tank.
•
The effect of the free surface of liquids
on the initial stability of the ship is to
decrease the righting arm.
For a small parallel angle inclination,
the movement of C.G of liquid is
0 1
tan
OL
k
I
G G d
•
The increase in the
heeling moment
due to the movement
of C.G. of liquid
tan 0 1
heeling F k F OL
M G G I d
If there is no influence of free

surface liquids, the
righting
moment
of the ship at a small angle
dΦ
is:
ox
w B g w
I
M GM d Z Z d
In the presence of a free

surface liquid, the righting moment
is decreased due to a heeling moment of free

surface liquid.
The reduced
righting moment
M’
is
ox ol
F
heeling w B g
w
I I
M M M d Z Z
The reduced metacentric height
GM’:
OX F OL
B g
w
I I
GM Z Z
Comparing with the original
GM
, it is decreased by an
amount
,
.
OL
F
w
I
The decrease can also be viewed as an increase in height
of C.G. w.r.t. the baseline.
OL
F
g g
w
I
Z Z
How to decrease I
OL
:
•
Longitudinal subdivision: reduce the width
b,
and
thus reduces
Anti rolling tank
3
OL
I b l
4.5 Effects of a suspended
weight on the righting
arm
•
When a ship inclines at a
small
angle
dΦ
, the
suspended object
moves transversely
•
Transverse movement of the weight
=
h dΦ
, where
h
is the distance
between the suspended weight and
the hanging point
•
The increase in the heeling moment
due to the transverse movement
heeling
M w h d
In the presence of a suspended object, the righting moment &
righting arm are decreased due to a heeling moment of the
suspended object. The reduced
righting moment
M’
&
metacentric height
GM’
are:
ox
heeling w B g
w
ox
B g
w w
I
w
M M M d Z Z h
I
w w
GM GM h Z Z h
In other words, the C.G of a suspended object is actually at its
suspended point
Because the suspension weights & liquid with free
surface tend to
decrease the righting arm
, or
decrease the
initial stability
, we should
avoid
them.
1.
Filling the liquid tank (in full)
to get rid of the
free surface. (creating a expandable volume)
2.
Make the
inertial moment of the free surface as
small as possible by adding the
separation
longitudinal plates (bulkhead).
3.
Fasten the weights
to prevent them from moving
transversely.
4.6 The Inclining Experiment (Test)
Purpose
1.
To obtain the
vertical position of C.G
(Center of Gravity) of the ship.
2.
It is required by “International convention
on Safety of Life at Sea.” (Every
passenger or cargo vessel
newly built or
rebuilt
)
4.6 The Inclining Experiment (Continue)
Basic Principle
M:
Transverse
Metacenter (
A vertical
line through the C.B
intersects the original
vertical centerline at
point,
M
)
Due to the movement of
weights, the heeling
moment is
heeling
M wh
where
w
is the
total weight
of the moving objects and
h
is the
moving distance.
4.6 The Inclining Experiment (Continue)
The shift of the center of gravity is
where
W
is the total weight of the ship.
The righting moment = The heeling moment
1
wh
GG
W
1
tan cot( )
tan( )
wh
GM W wh GM GG
W
1.
w
and
h
are recorded and hence known.
2.
is measured by a pendulum known as stabilograph.
3.
The total weight
W
can be determined given the draft T. (at
FP, AP & midship, usually only a
very small trim
is allowed.)
4.
Thus
GM
can be calculated,
4.6 The Inclining Experiment (Continue)
,
x
M g M B
I
GM H Z H Z
The metacenter height and vertical coordinate of C.B have
been calculated. Thus, C.G. can be obtained.
g M
Z H GM
Obtaining the longitudinal position of the gravity center of
a ship will be explained in section 4.8.
4.6 The Inclining Experiment (Continue)
.
1.
The experiment should be carried out in
calm water & nice weather
. No
wind, no heavy rain, no tides.
2.
It is essential that the ship
be free to incline
(mooring ropes should be as
slack as possible, but be careful.)
3.
All weights capable of moving transversely should be
locked in position
and there should be
no loose fluids in tanks
.
4.
The ship in inclining test should be as
near completion
as possible.
5.
Keep as few people on board as possible.
6.
The angle of inclination should be
small enough
with the range of validity
of the theory.
7.
The ship in experiment
should not have a large trim.
4.7 Effect of Ship’s Geometry on Stability
Transverse metacenter height
GM = BM
–
(Z
G
–
Z
B
)
3 2 2
1 1
1
( )
where dpends on waterplane.
2
x
g B
x
B B
x
x
I
GM Z Z
I C LB C B B
C C
C LBT C T T
I
d
dB dT
I
B T
4.7 (Continue)
a) Increase only: 2 ( increases)
b) Decrease T only: ( decreases)
c) Change B & T but keep fixed:
x
B
x
x
x
x
x
I
d
dB
B C LBT
I
B
I
d
dT
I
T
dB dT
B T
I
d
I
3
dB
B
•
Conclusion
:
to increase
GM
(
Transverse metacenter
height
)
1.
increasing the beam,
B
2.
decreasing the draft,
T
3.
lowering
C.G (Z
G
)
4.
increasing the
freeboard
will increase the
Z
G
, but will
improve the
stability at
large inclination angle
.
5.
Tumble home or flare will have effects on the stability at
large inclination angle.
6.
Bilge keels
,
fin stabilizers
, gyroscopic stabilizers,
anti

rolling tank
also improve the stability (at pp248

252).
4.7 (Continue)
Suitable metacenter height
•
It should be large enough to satisfy the requirement of
rules.
•
Usually under full load condition,
GM
~0.04B.
•
However, too large
GM
will result in a very small rolling
period. Higher rolling frequency will cause the crew or
passenger uncomfortable. This also should be avoided.
(see page 37 of this notes)
4.8
Longitudinal Inclination
Longitudinal Metacenter:
Similar to the definition of the
transverse meta center, when a ship is inclined longitudinally
at a small angle, A vertical line through the center of
buoyancy intersects the vertical line through (before the
ship is inclined) at .
1
B
0
B
L
M
•
The Location of the Longitudinal Metacenter
For a small angle inclination
,
volumes of forward wedge immersed
in water and backward wedge emerged out of water are:
1
0
2 1 2
0
0 0
(2 ) ( tan ) where is the half breadth.
(2 ) ( tan ), .
Thus, 2 2, which indicates:
moment of area forward of = moment of a
rea after .
is the cente
l
L l
l L l
v y x dx y
v y x dx v v v
yxdx yxdx
F F
F
0 0
0 0
r of (mass) gravity of waterline , &
is called of . Therefore, for equal
volume longitudinal inclination the new
waterline always
passes through the (C.F
W L
W L
center of flotation
center of flotation
).
•
Location of the Longitudinal Metacenter
Using the same argument used in obtaining transverse metacenter
.
0 1 1 2
0
1 2
0
0
2 2
0
0 1 0
/,
(2 ) ( tan ) 2 tan
tan 2 2 tan
is the moment of inertia with respect t
o the
transverse axis passing the center of fl
otation.
t
l
L l
l
FC
L l
FC
B B vg g
vg g y x xdx yx xdx
yx dx yx dx I
I
B B B M
0
0
an, .
.
FC
FC FC
ML B B L B g
I
B M
I I
H B M Z Z GM Z Z
•
Location of the Longitudinal Metacenter
2
0,
0,
Usually Floatation Center (C.F) of a wat
erplane is not at the
midship,
,
where is the moment of inertia w.r.t. t
he transverse axis
at midship (or station 5) and is the d
istance from F
FC T
T
I I Ax
I
x
.C. to
the midship.
•
Moment to Alter Trim One Inch (MTI)
MTI: (moment to alter (change) the ship’s trim per inch) at
each waterline (or draft) is an important quantity. We may
use the longitudinal metacenter to predict MTI
•
MTI ( a function of draft)
Due to the movement of a weight, assume that the ship as 1” trim,
and floats at waterline W.L.,
0 1
0 1
0 1 0
1"1
tan, where is in feet.
12
Due to the movement of the weight, moves
to ,
,
tan tan
tan tan
1
12
w
L L G
FC
w L G w B G
FC
w B G
L
L L
G G
M w h G G
G G G M HM Z
I
M HM Z Z Z
I
Z Z
L
•
MTI ( a function of draft)
•
If the longitudinal inclination is small,
MTI
can be used to find
out the
longitudinal
position of gravity center ( ).
3
1
,
12 12
64 lb/ft, Long Ton = 2240 lb, MTI (ton
ft)
420
FC FC w FC
G B w
FC
w
I I I
Z Z M
L L
I
L
1
G
X
0 1
0 0 0
1 0 1
Trim
tan
tan,
Since is in the same vertical line as .
under ,
F A
FC FC FC
B G G B
FC
G B B
T T
T
L L L
I I I
T
G G Z Z Z Z
L
G C B W L
I
T
X X G G X
L
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