Boat Design - Stability Fundementals

8/6/2019 Boat Design - Stability Fundementals

1/12

36

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, Wand FB are in-line.

given a small angular disturbance, new buoyancy force line through B1intersects initial vertical line throughB at M.

Mis the meta-center and GMis the meta-centric height.

GMis 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 () WandFB are no longer in the samevertical line, they now form a couple referred to as the Righting Moment

(RM):

RM= . GZ = . GM sin . GM. ( 10)

GZis the righting arm

Mabove G,KM > KG, GM+, +RM stable (Positive stability)

Mbelow G, KM < KG, GM-, - RM unstable (Negative stability)


2/12


3/12

38

GM = KM - KG = (KB + BM) KGWhere:KG is a physical quantity, MandKB are geometrical quantities,

BM = metacentric radius

x

I

TBM =

xI = moment of inertia of waterplan area about thex-axis.

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 < 10:

(a) Upright and inclined WLs intersect on centerline.(b) Metacenter M remains fixed.

(c) Initial stability is measured by GM (righting arm GZ=GM sin ).

At large heel angles:

(a) Righting and inclined WLs do not intersect on the Centerline.

(b) MetacenterMis 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=BRBG sin

From figure 1-3, SinceBB1 || b1b2,BR || h1h2 || GZ1

vw = immersed wedge Volume = emerged wedge volume (since is

constant)


4/12

39

= buoyancy force of immersed wedgeb1, b2 = centroids of volume of merged and immersed wedges

h1, h2 = feet of perpendiculars from b1, b2 on to W1L1

Consequently, BR = h1h2

So righting moment on ship is:

Atwoods 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 ofbuoyancy force.

Sdepends only on ship geometry and can be determined for various

angles of heel and for various displacements independent of loading

condition.

Sis known when G is determined for a given ship loading, therefore:

GZ1 = SZ+ SG sin

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


5/12

40

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


6/12


7/12

42

The free surface correction (FSC) created by a tank within a ship is given

by the following equation:

ss

ttiFsc

=r

r

Where:

tr is the density of the fluid in the tank.

sr is the density of the water the ship is floating.

s is the underwater volume of the ship.

ti 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


8/12

43

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 ofthe 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 amuch 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


9/12

44

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


10/12

45

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 PerCentimeter 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 smalldistance for the length of the entire vessel

We assume that the weight is applied at F! This assures that thesinkage 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:


11/12

46

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


12/12

47

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) andany 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 ships 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 averagelocation 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 isbecause 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.

Documents

Boat Design - Stability Fundementals