1. Introduction
Before entering in the details of Autohydro it would be useful to have an
overview of some of the basic principles of hydrostatics & stability.
1.1 Intact stability:
1.1.1 Initial stability:
Using the example of a surface ship (see figure 1-1):
Figure 1-1 stable ship
•in equilibrium: W = FB = ∆ (where FB = buoyancy force, w = weight,
∆ = ship's displacement)
•when upright, W and FB are in-line.
•given a small angular disturbance, new buoyancy force line through B1
intersects initial vertical line through B at M.
• M is the meta-center and GM is the meta-centric height.
• GM is a measure of the initial stability of a floating body or the ability
to resist initial heeling from the upright position.
• After inclining by small angle (φ) W and FB are no longer in the same
vertical line, they now form a couple referred to as the Righting Moment
(RM):
RM = ∆. GZ = ∆. GM sin φ ≈ ∆. GM. φ (φ ≤ 10°)
GZ is the righting arm
• M above G, KM > KG, GM+, +RM
stable (Positive stability)
• M below G, KM < KG, GM-, - RM
unstable (Negative stability)
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• M coincides with G, KM=KG, GM=0
neutral (Neutral stability)
See figure 1-2.
Figure 1-2 stability conditions
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•GM = KM - KG = (KB + BM) – KG
Where: KG is a physical quantity, M and KB are geometrical quantities,
BM = metacentric radius
I
BM  x
T
I = moment of inertia of waterplan area about the x-axis.
x
1.1.2 Stability at large angles of heel:
1.1.2.1 The major differences between Initial & large angles stability:
At small angles of heel, say φ < 10°:
(a) Upright and inclined WL’s intersect on centerline.
(b) Metacenter M remains fixed.
(c) Initial stability is measured by GM (righting arm GZ=GM sin φ).
At large heel angles:
(a) Righting and inclined WL’s do not intersect on the Centerline.
(b) Metacenter M is no longer a fixed point.
(c) Stability is measured in terms of the righting arm GZ.
1.1.2.2 GZ formula:
Fig 1-3 Stability at large angles
GZ1=BR – BG sin φ
From figure 1-3, Since BB1 || b1b2, BR || h1h2 || GZ1
vw = immersed wedge Volume = emerged wedge volume (since
constant)
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is
δ = buoyancy force of immersed wedge
b1, b2 = centroids of volume of merged and immersed wedges
h1, h2 = feet of perpendiculars from b1, b2 on to W1L1
Consequently, ∆BR = δ h1h2
So righting moment on ship is:
Atwood’s formula
1.1.2.3 The point S:
• G depends on ship loading which is not fixed; it's convenient to think of
a fixed point, S, and its perpendicular distance from line of action of
buoyancy force.
• S depends only on ship geometry and can be determined for various
angles of heel and for various displacements independent of loading
condition.
• S is known when G is determined for a given ship loading, therefore:
GZ1 = SZ + SG sinφ
1.1.3 Static stability curve:
You can have a complete picture of vessel stability from a plot of righting
moments versus angles of inclinations for several displacements (see
figure 1-4).
Figure 1-4 Static stability curve
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For a given displacement & center of gravity you can determine:
(1) Righting arm at any inclination,
(2) Angle of max righting Moment.
(3) Range of stability.
(4) Dynamic stability.
1.1.4 Cross curves of stability:
The cross curves of stability are a series of curves on a single set of axes.
The X-axis is the displacement of the ship in Tons. The Y-axis is the
righting arm of the ship in feet. Each curve is for one angle of heel.
Typically angles of heel are taken each 5 or 10 degrees (See Figure 1.5).
Figure 1-5a Cross curves of stability
Figure 1-5b curves of statical stability and cross curves of stability
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1.1.5 Dynamic Stability:
It's the energy available, by the ship righting moments, to resist any
external heeling energy from the position of equilibrium to any inclined
position. Mathematically, this work is,
∆∫GZ dφ
This is the product of the ship’s displacement with the area under the
curve of intact statical stability. The dynamic stability can’t be shown
directly on the curve of intact statical stability but the area under the
curve can be shown.
1.1.6 Free surface effect:
A free surface is fluid that is allowed to move freely, such as water in a
partially filled tank. As the ship lists, the fluid in the tank moves. The
fluid movement acts like a weight shift, causing the center of gravity of
the fluid to move which causes the ship's center of gravity to shift in both
the vertical and horizontal directions. The effect of the vertical shift is
negligible at small angles (φ < 5 to 7) and is discounted, but the
horizontal (transverse) shift of the center of gravity causes a decrease in
the righting arm (GZ).
It is shown graphically in Figure 1-6 that a vertical rise in the center of
gravity also causes a shortened righting arm.
Figure 1-6 free surface correction
The distance the center of gravity would have to rise to cause a reduction
in the righting arm equivalent to that caused by the actual transverse shift
is called the Free Surface Correction (FSC). The position of this new
center of gravity is called the "virtual" center of gravity (Gv). The
distance from the virtual center of gravity to the Metacenter is called the
Effective Metacentric Height (GMeff).
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The free surface correction (FSC) created by a tank within a ship is given
by the following equation:
i
Fsc  t t
 s s
Where:
 t is the density of the fluid in the tank.
 s is the density of the water the ship is floating.
 s is the underwater volume of the ship.
it is the transverse second moment of area of the tank's free surface area.
The free surface correction is applied to the original metacentric height to
find the effective metacentric height:
GMeff = GM – Fsc = KM – KG – Fsc
1.2 Weight Additions, Removals and Shifts:
Shifting, adding or removing weight on a ship changes the location of G
on a ship. It is important for you to qualitatively understand which
direction the center of gravity will move when weight is shifted, added or
removed from a ship.
1.2.1 Weight Addition:
When weight is added to a ship the average location of the weight of the
ship must move towards the location of the weight addition.
Consequently, the Center of Gravity of the ship (G) will move in a
straight line from its current position toward the center of gravity of the
weight (g) being added. An example of this is shown in Figure 1-7.
Figure 1-7 the Effect of a Weight Addition upon the Center of Gravity of a Ship
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1.2.2 Weight Removal:
When weight is removed from a ship the average location of the weight
of the ship must move away from the location of the removal.
Consequently, the Center of Gravity of the ship (G) will move in a
straight line from its current position away from the center of gravity of
the weight (g) being removed. See Figure 1-9.
Figure 1-8 the Effect of a weight Removal upon the Center of Gravity of a Ship.
1.2.3 Weight Shift:
When a small weight is shifted onboard a ship the Center of Gravity of
the ship (G) will move in a direction parallel to the shift but through a
much smaller distance. G will not move as far as the weight being shifted
because the weight is only a small fraction of the total weight of the ship.
An example of this is shown in Figure 1-9.
Figure 1-9 the Effects of a Weight Shift on the Center of Gravity of a Ship
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1.2.4 General Vertical Weight Shift, Addition and Removal
Equation:
At this point we are ready to write the most general equation to quantify
all combinations of vertical shifts, additions, and removals of weight. We
should use a plus sign when weight is added and a minus sign when
weight is removed. The summation should have as many plus terms as
there are weights added and as many minus terms as there are weights
removed. The equation is shown below:
In applying this equation always write out the summation terms fully
showing each individual term used. This is necessary so that another
engineer can see the specific terms you are using and to check your work.
1.3 Trim:
Consider a ship floating on an even keel that is no list or trim, When a
weight, w, is added, it causes a change in draft (see figure 1- 10).
The ship will pivot about the center of flotation, F.
Figure 1-10-a Even keel ship
Fig 1-10-b the effect of longitudinal weight shift
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The change in draft will be evident in a change of draft forward and aft.
Figure 1-10-c Trim aft & fore
Graphically, it looks like this in figure 1-11:
Figure 1-11 graphical representation of trim
There are two aspects of draft to consider when finding the change in
draft:
1. Change due to the parallel sinkage of the vessel due to the added
weight, “w”:
TPC, Tons Per Centimeter Immersion is a geometric function of the
vessel at a given draft and is taken from the Curves of Form
• The added weight, w, will cause the vessel to “sink” a small
distance for the length of the entire vessel
•
We assume that the weight is applied at F! This assures that the
sinkage is uniform over the length of the ship
2. Change in draft due to the moment created by the added weight at a
distance from F:
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MT1C, or the Moment to Trim 1C, is also from the Curves of Form
• The weight, w, at a distance, l, from the center of flotation, F,
creates a moment that causes the ship to rotate about F.
• This rotation causes one end to sink and the other end to rise.
• The degree of rise or fall depends on the location of F with regard
to the entire length of the ship as given by Lpp.
The total change in trim fore and aft:
MT1C, or the Moment to Trim 1C, is also from the Curves of Form
• The weight, w, at a distance, l, from the center of flotation, F,
creates a moment that causes the ship to rotate about F.
• This rotation causes one end to sink and the other end to rise.
• The degree of rise or fall depends on the location of F with regard
to the entire length of the ship as given by Lpp.
1.4 Damage stability:
When a vessel is damaged, creating a gap or hole in the hull, water
will breech the ship. This results in:
• Increase in draft
• Change in trim
• Permanent angle of list
See figure 1-12
Figure 1-12 Change in draft due to hull damage
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The result of this flooding can be determined two ways:
1.4.1 Lost Buoyancy Method:
In the lost buoyancy method we analyze changes in buoyancy rather
than the center of gravity or displacement. Simply stated, the center of
gravity remains the same (the ship weight, metal etc is constant) and
any changes due to damage effect the distribution of the buoyancy
volume. The total buoyant volume must remain constant since the
weight of the ship is not changing. The draft will increase and the ship
will list and trim until the lost buoyant volume is regained.
The lost buoyancy method allows a damaged ship to be modeled
mathematically so that the final drafts, list, and trim can be determined
from assessed damage. The engineer can analyze every conceivable
damage scenario and produce a damage stability handbook that may
be used by the crew in the event of flooding. Using the lost buoyancy
method allows “a prior” knowledge of the resulting stability condition
of the ship so that appropriate procedures can be written and followed
in the event of a breach in the ship’s hull.
1.4.2 The Added Weight Method:
As the name suggests, in this technique, the ship is assumed
undamaged, but part of it is filled with the water the ship is floating in.
This is equivalent to a weight addition and can be modeled using the
techniques for shifts in the center of gravity of the ship.
Provided the volume of the damaged compartment, its average
location from the centerline, Keel, midship and the water density is
known, the shift in G can be predicted along with the consequences of
this shift upon the draft, trim and list of the ship.
 Permeability
An added complication to the analysis of a damaged ship is the
space available in a damaged compartment for the water to fill.
When a compartment is flooded, it is rare for the total volume of
this compartment to be completely filled with water. This is
because the compartment will already contain certain equipment or
stores depending upon its use. The ratio of the volume that can be
occupied by water to the total gross volume is called the
“permeability”.
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