Chapter IV (Ship Hydro-Statics

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Chapter IV(Ship Hydro-Statics Dynamics)
Floatation Stability
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• 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 (CB , CM , CP , CWL, vs. T)
• VCB (KB, ZB) Vertical distance of Center of
  Buoyancy (C.B) to the baseline vs. T
• LCB (LCF, XB) Longitudinal Distance of C.B or
  floatation center (C.F) to the midship vs. T

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• 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.

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• 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.

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• 4.2 How to Compute these curves
• Formulas for Area, Moments Moments of Inertia

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• 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

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• Red sheet will be studied in detail
• 1-6 Areas properties (F.C., Ic, etc) of W.L
• 7-11 Displacement, ZB , and XB up W.L., vertical
  integration.
• 12-15 Transverse station area, longitudinal
  integration for displacement, ZB , and XB
• 16-18 Specific Feature (wetted surface, MTI, etc.
• 19 Summary

24wl area
8wl area
16wl area
4wl area
Up to 8wl
Up to 4wl
32wl area
40wl area
Up to 24 40 wl
MTI
Disp. Up to 16wl
Disp. Up to 24wl
MTI
Wetted surface
Disp. Up to 40wl
Disp. Up to 32wl
Summary
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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.
C23 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.
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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 (ft2) 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 Simpsons
Coeff. C5. C4 x C3 C6. Arm distance between
this station and station 5 (midship) C7 C5 x
C6 f(M)
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Illustration of Table 9 C1. WL No. C2. f(V)
Notice first row up to 8. f(v) C3. Simpsons
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)
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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. C6h/3
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• 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.

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• 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
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• 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.

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For a displacement ship,
W.L
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.
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For submarines (immersed in water)
B
G
G
If B is above G, we have righting moment If B is
below G, we have heeling moment
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• 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

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Longitudinal Equilibrium For an undamaged
(intact) ship, we are usually only interested in
determining the ships 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.
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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.
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• Static Stability
• The initial stability (aka stability at small
  inclination) and,
• 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.

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• Initial stability
• Righting Arm A symmetric ship is inclined at a
  small angle dF. C.B has moved off the ships
  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.

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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
gt0. 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 dF.
How to determine the location of M?
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When a ship is inclined at small angle dF
WoLo Waterline (W.L) at upright position W1L1
Inclined W.L Bo C.B. at upright position,
B1 C.B. at inclined position - The
displacement (volume) of the ship v1, v2 The
volume of the emerged and immersed g1, g2 C.G.
of the emerged and immersed wedge, respectively
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Equivolume Inclination (v1 v2 )If the ship is
wall-sided with the range of inclinations of a
small angle dF, then the volume v1 and v2 , of
the two wedges between the two waterlines will be
same. Thus, the displacements under the
waterlines WoLo and W1L 1 will be same. This
inclination is called equivolume inclination.
Thus, the intersection of WoLo, and W1 L1 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.
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When a ship is at equivolume inclination, Accor
ding 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.
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Examples of computing KM
d
B
B
d
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Natural frequency of Rolling of A Ship
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• 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

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• The increase in the heeling moment due to the
  movement of C.G. of liquid

If there is no influence of free-surface liquids,
the righting moment of the ship at a small angle
dF is
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
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The reduced metacentric height GM
Comparing with the original GM, it is decreased
by an amount,
The decrease can also be viewed as an increase in
height of C.G. w.r.t. the baseline.

• How to decrease IOL
• Longitudinal subdivision reduce the width b, and
  thus reduces
• Anti rolling tank

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• 4.5 Effects of a suspended weight on the righting
  arm
• When a ship inclines at a small angle dF, the
  suspended object moves transversely
• Transverse movement of the weight h dF , where
  h is the distance between the suspended weight
  and the hanging point
• The increase in the heeling moment due to the
  transverse movement

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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
In other words, the C.G of a suspended object is
actually at its suspended point
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• Because the suspension weights liquid with free
  surface tend to decrease the righting arm, or
  decrease the initial stability, we should avoid
  them.
• Filling the liquid tank (in full) to get rid of
  the free surface. (creating a expandable volume)
• Make the inertial moment of the free surface as
  small as possible by adding the separation
  longitudinal plates (bulkhead).
• Fasten the weights to prevent them from moving
  transversely.

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• 4.6 The Inclining Experiment (Test)
• Purpose
• To obtain the vertical position of C.G (Center of
  Gravity) of the ship.
• It is required by International convention on
  Safety of Life at Sea. (Every passenger or
  cargo vessel newly built or rebuilt)

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• 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
where w is the total weight of the moving objects
and h is the moving distance.
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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. 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,

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4.6 The Inclining Experiment (Continue)
The metacenter height and vertical coordinate of
C.B have been calculated. Thus, C.G. can be
obtained.
Obtaining the longitudinal position of the
gravity center of a ship will be explained in
section 4.8.
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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.

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4.7 Effect of Ships Geometry on
Stability Transverse metacenter height GM BM
(ZG ZB)
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4.7 (Continue)
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• Conclusion to increase GM ( Transverse
  metacenter height)
• increasing the beam, B
• decreasing the draft, T
• lowering C.G (ZG)
• increasing the freeboard will increase the ZG,
  but will improve the stability at large
  inclination angle.
• Tumble home or flare will have effects on the
  stability at large inclination angle.
• Bilge keels, fin stabilizers, gyroscopic
  stabilizers, anti-rolling tank also improve the
  stability (at pp248-252).

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• 4.7 (Continue)
• Suitable metacenter height
• It should be large enough to satisfy the
  requirement of rules.
• Usually under full load condition, GM0.04B.
• 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)

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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 .
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• 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

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• Location of the Longitudinal Metacenter
• Using the same argument used in obtaining
  transverse metacenter.

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• Location of the Longitudinal Metacenter

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• Moment to Alter Trim One Inch (MTI)
• MTI (moment to alter (change) the ships trim
  per inch) at each waterline (or draft) is an
  important quantity. We may use the longitudinal
  metacenter to predict MTI

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• 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.,

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• MTI ( a function of draft)
• If the longitudinal inclination is small, MTI can
  be used to find out the longitudinal position of
  gravity center ( ).