The metacentric height (GM) is a measurement of the initial static stability of a floating body. It is calculated as the distance between the centre of gravity of a ship and its metacentre. A larger metacentric height implies greater initial stability against overturning. The metacentric height also influences the natural period of rolling of a hull, with very large metacentric heights being associated with shorter periods of roll which are uncomfortable for passengers. Hence, a sufficiently, but not excessively, high metacentric height is considered ideal for passenger ships.
Different centres
The centre of buoyancy is at the centre of mass of the volume of water that the hull displaces. This point is referred to as B in naval architecture. The centre of gravity of the ship is commonly denoted as point G or CG. When a ship is at equilibrium, the centre of buoyancy is vertically in line with the centre of gravity of the ship. The metacentre is the point where the lines intersect (at angle φ) of the upward force of buoyancy of φ ± dφ. When the ship is vertical, the metacentre lies above the centre of gravity and so moves in the opposite direction of heel as the ship rolls. This distance is also abbreviated as GM. As the ship heels over, the centre of gravity generally remains fixed with respect to the ship because it just depends on the position of the ship's weight and cargo, but the surface area increases, increasing BMφ. Work must be done to roll a stable hull. This is converted to potential energy by raising the centre of mass of the hull with respect to the water level or by lowering the centre of buoyancy or both. This potential energy will be released in order to right the hull and the stable attitude will be where it has the least magnitude. 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. The righting couple on the ship is proportional to the horizontal distance between two equal forces. These are gravity acting downwards at the centre of mass and the same magnitude force acting upwards through the centre of buoyancy, and through the metacentre above it. The righting couple is proportional to the metacentric height multiplied by the sine of the angle of heel, hence the importance of metacentric height to stability. As the hull rights, work is done either by its centre of mass falling, or by water falling to accommodate a rising centre of buoyancy, or both. 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. Conversely if a hull having a perfectly rectangular cross section has its centre of mass at the water line, the centre of mass stays at the same height, but the centre of buoyancy goes down as the hull heels, again storing 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:
KB – to Centre of Buoyancy KG – to Centre of Gravity KMT – to Transverse Metacentre
Metacentre When a ship heels (rolls sideways), the centre of buoyancy of the ship moves laterally. It might also move up or down with respect to the water line. 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 by definition. In the diagram above, the two Bs show the centres of buoyancy of a ship in the upright and heeled conditions. The metacentre, M, is considered to be fixed relative to the ship for small angles of heel; however, at larger angles the metacentre can no longer be considered fixed, and its actual location must be found to calculate the ship's stability. It can be calculated using the formulae:
K M = K B + B M {\displaystyle KM=KB+BM}
B M = I V {\displaystyle BM={\frac {I}{V}}\ }
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. KM is the distance from the keel to the metacentre. Stable floating objects have a natural rolling frequency, just like a weight on a spring, where the frequency is increased as the spring gets stiffer. In a boat, the equivalent of the spring stiffness is the distance called "GM" or "metacentric height", being the distance between two points: "G" the centre of gravity of the boat and "M", which is a point called the metacentre. Metacentre is determined by the ratio between the inertia resistance of the boat and 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 hulls have high transverse metacentres, whilst narrow and deep hulls have low metacentres . Ignoring the ballast, wide and shallow means that the ship is very quick to roll, and narrow and deep means that the ship is very hard to overturn and is stiff. "G", is the center of gravity. "GM", the stiffness parameter 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) or both. An ideal boat strikes a balance. 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. Sailing yachts, especially racing yachts, are designed to be stiff, meaning the distance between the centre of mass and the metacentre is very large in order to resist the heeling effect of the wind on the sails. 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.
Righting arm
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