Metacentric height
Hence, a sufficiently, but not excessively, high metacentric height is considered ideal for passenger ships.
It is the interplay of potential and kinetic energy that results in the ship having a natural rolling frequency.
For small angles, the metacentre, Mφ, moves with a lateral component so it is no longer directly over the centre of mass.
For example, when a perfectly cylindrical hull rolls, the centre of buoyancy stays on the axis of the cylinder at the same depth.
However, if the centre of mass is below the axis, it will move to one side and rise, creating potential energy.
When setting a common reference for the centres, the molded (within the plate or planking) line of the keel (K) is generally chosen; thus, the reference heights are: When a ship heels (rolls sideways), the centre of buoyancy of the ship moves laterally.
In the diagram above, the two Bs show the centres of buoyancy of a ship in the upright and heeled conditions.
It can be calculated using the formulae: Where KB is the centre of buoyancy (height above the keel), I is the second moment of area of the waterplane around the rotation axis in metres4, and V is the volume of displacement in metres3.
Very tender boats with very slow roll periods are at risk of overturning, but are comfortable for passengers.
However, vessels with a higher metacentric height are "excessively stable" with a short roll period resulting in high accelerations at the deck level.
In such vessels, the rolling motion is not uncomfortable because of the moment of inertia of the tall mast and the aerodynamic damping of the sails.
Depending on the geometry of the hull, naval architects must iteratively calculate the center of buoyancy at increasing angles of heel.
Because the vessel displacement is constant, common practice is to simply graph the righting arm vs the angle of heel.
As the displacement of the hull at any particular degree of list is not proportional, calculations can be difficult, and the concept was not introduced formally into naval architecture until about 1970.
An excessively low or negative GM increases the risk of a ship capsizing in rough weather, for example HMS Captain or the Vasa.
It also puts the vessel at risk of potential for large angles of heel if the cargo or ballast shifts, such as with the Cougar Ace.
A ship with low GM is less safe if damaged and partially flooded because the lower metacentric height leaves less safety margin.
A larger metacentric height on the other hand can cause a vessel to be too "stiff"; excessive stability is uncomfortable for passengers and crew.
When the vessel is inclined, the fluid in the flooded volume will move to the lower side, shifting its centre of gravity toward the list, further extending the heeling force.
The significance of this effect is proportional to the cube of the width of the tank or compartment, so two baffles separating the area into thirds will reduce the displacement of the centre of gravity of the fluid by a factor of 9.
Another worrying feature of free surface effect is that a positive feedback loop can be established, in which the period of the roll is equal or almost equal to the period of the motion of the centre of gravity in the fluid, resulting in each roll increasing in magnitude until the loop is broken or the ship capsizes.
Technically, there are different metacentric heights for any combination of pitch and roll motion, depending on the moment of inertia of the waterplane area of the ship around the axis of rotation under consideration, but they are normally only calculated and stated as specific values for the limiting pure pitch and roll motion.
The metacentric height is normally estimated during the design of a ship but can be determined by an inclining test once it has been built.
By means of the inclining experiment, the 'as-built' centre of gravity can be found; obtaining GM and KM by experiment measurement (by means of pendulum swing measurements and draft readings), the centre of gravity KG can be found.
As long as the load of a ship remains stable, G is fixed (relative to the ship). For small angles, M can also be considered to be fixed, while B moves as the ship heels.
