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May 2005

Article selected from our quarterly magazine dedicated to the largest and most luxurious boats with information, interviews, technical articles, images and yachting news



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Article by
Angelo Sinisi


In previous articles on hull design we referred to metacentric height r-a (stability index) as an important value for transversal stability which should be sought through correct hull proportions.

In this article we shall try to explain more in depth the concept and conditions of vessel balance stability.


Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

The conditions that must be satisfied in order that a freely floating vessel be stably balanced may be deduced through ordinary and simple concepts.

If a body set on a horizontal plane is stationary we can say that it is in equilibrium. But if we wanted to ascertain whether its equilibrium were stable we would touch the body to move it from its initial position; and if after this action the body resumed its original position we would say that it was in stable equilibrium. Contrarily its equilibrium would be unstable.

If we consider that the body is in stable equilibrium it means that to our inclining action it opposes a stabilising action such as to bring the body back to its initial position when we cease applying pressure.

I shall attempt to illustrate this concept by considering a cylinder (Figure 1) of radius r and centre M set on a horizontal plane that has:

G = position of centre of gravity
P = weight of cylinder
S = support reaction
C = application point of support reaction
a = vertical distance of centre of gravity G from horizontal plane X
h = torque arm P and S

When the cylinder is stationary G is on the vertical Z and weight P is equal to support reaction S applied at point C.

If we apply an inclining force we obtain an inclining moment Fi in the direction of the arrow (Figure 2). To this moment the cylinder opposes moment Fs, equal and opposite to Fi. Moment Fs is generated by torque P and S, so we have Fs = P*h. It is clear that if moment Fi ceases, the cylinder will return to the initial position of Figure 1.

In fact moment Fs takes the cylinder back to its initial position and is annulled because h is zeroed as P and S are on the same vertical Z.

Since the new vertical position a1 of G from plane X is greater than the previous a, this means that during inclination the cylinder's centre of gravity is raised and that when balance is stable the centre of gravity is in the lower position. So if we want to raise the centre of gravity, which is to say if we want stable balance, all that is required is that centre M be above G, thus giving r-a > 0.

Let us now suppose that the cylinder is set as in Figure 3. It is clear that the cylinder is in balance since P and S are equal and G and M are on the same vertical Z, but the balance is unstable.

In fact by applying moment Fi (Figure 4) the centre of gravity G is lowered, torque P and S generates moment Fs with the same direction as moment Fi and contributes to overturning the cylinder.

Instability of balance may therefore be deduced from the consideration that since G is above M, we have r-a < 0. Subsequent to what has been said it is easy to intuit the required condition if a vessel is to remain stable in a given position.

In a vessel (Figure 5) M is the transversal metacentre, which is to say the centre of the curve that unites the centres of the hull at various stages of heeling, so if centre of gravity G is below the transversal metacentre we have r-a > 0. Whereas if centre of gravity G is above M we have r-a < 0 so balance is unstable. In this case, like the cylinder in Figure 4, the vessel will heel until centre of gravity G and the centre of the hull C are on the same vertical, perpendicular to the centre of buoyancy, and transversal metacentre M becomes greater than G. In this position the vessel remains heeled and is defined as "listing".

Stability then is chiefly influenced by barycentric position and by beam. But the most efficient of the two is beam.

In fact if we consider two prisms of the same length, as in Figure 6 and Figure 7, we see that in Figure 6, where r-a > 0, it is in stable balance but, with a value slightly above zero, does not have sufficient righting moment to counter even a weak heeling force. That is to say, if a vessel were in a similar condition it would be disastrous because it would capsize with the first wave.

Contrarily, the prism in Figure 7, being three times wider than the other, has a metacentric radius that is nine times greater. That is to say, the metacentric radius varies with the square of the ratio between the widths, having the same displacement.

Nevertheless, a metacentric height (r-a = 16.647) like that of the prism in Figure 7 is too much and therefore dangerous. In fact a great metacentric height would make life on board bothersome while under way. This because the vessel's period of oscillation T is linked to the transversal metacentric height by the relation T = K / √ r-a with K variable depending on the type of vessel. It may be deduced that with a high r-a value we have a very high righting moment, so the period of oscillation is less, with powerful gravity accelerations, and in a choppy sea the vessel behaves in a manner that is defined "too hard".

Contrarily, a vessel with a low r-a value has less righting moment so, in a rough sea, there will be very slow oscillations but so extensive that in extreme cases they may cause capsizing.

Another important element influencing the vessel's roll period T is included in the coefficient K and is the "mass inertia moment" of the "vessel-water" system with regard to the barycentric axis of rotation. Obviously its effect may be positive or negative as if it acted indirectly on the metacentric height value r-a. To tackle this problem would involve long and complex discussion for which we have neither space nor time.

Also with planing hulls the metacentric height r-a is enormously important because it also conditions dynamic stability. In fact with narrow and/or very high and fast vessels the edge of the deck comes frighteningly close to the water level when coming about.

So in order to have a sea-kindly vessel with regard to roll, we repeat the words we began with: transversal stability should be sought in correct hull proportions.