Metacentric Height - Metacentre

Metacentre

How quickly or slowly a boat rolls is like a pendulum or metronome, having a natural frequency. That frequency is determined (like with a metronome) by the amount of mass on some length of swing arm being pulled by gravity. Greater mass and/or arm length means a slower swing; and less mass and/or shorter arm length means a faster swing.

In a boat, the swing arm is a distance called "GM or metacentric height", being the distance between two points: "G" the center of gravity of the boat and "M", which is a point called the metacentre.

Metacenter is determined by a ratio of the inertia resistance of the boat divided by the volume of the boat. (The inertia resistance is a quantified description of how the waterline width of the boat resists overturning.) Wide and shallow or narrow and deep hulls have high transverse metacenters (relative to the keel), and the opposite have low metacenters; the extreme opposite is shaped like a log or round bottomed boat.

Ignoring the ballast, wide and shallow or narrow and deep means the ship is very quick to roll and very hard to overturn and is stiff. And log shaped round bottomed means slow rolls and easy to overturn and tender.

The bottom point of the swinging pendulum arm, "G", is the center of gravity. "GM", the swinging pendulum length of a boat, can be lengthened by lowering the center of gravity or changing the hull form (and thus changing the volume displaced and second moment of area of the waterplane.

An ideal boat strikes a balance. Very tender boats with very slow roll periods are at risk of overturning and have uncomfortable feel 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.

When a ship is heeled, the centre of buoyancy of the ship moves laterally. The point at which a vertical line through the heeled centre of buoyancy crosses the line through the original, vertical centre of buoyancy is the metacentre. The metacentre remains directly above the centre of buoyancy regardless of the tilt of a floating body, such as a ship. In the diagram to the right the two Bs show the centres of buoyancy of a ship in the upright and heeled condition, and M is the metacentre. The metacentre is considered to be fixed for small angles of heel; however, at larger angles of heel the metacentre can no longer be considered fixed, and other means must be found to calculate the ship's stability.
The metacentre 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 in metres4 and V is the volume of displacement in metres3. KM is the distance from the keel to the metacentre.