Metacentric height

The metacentric height (GM) is the distance between the center of gravity of a ship and its metacenter. The GM is used to calculate the stability of a ship and this must be done before it proceeds to sea. The GM must equal or exceed the minimum required GM for that ship for the duration of the forthcoming voyage. This is to ensure that the ship has adequate stability.

Metacenter


When a ship is tilted the center of buoyancy of the ship moves laterally. The point at which a vertical line through the tilted center of buoyancy crosses the line through the original, non-tilted center of buoyancy is the metacenter. In the diagram to the right the two Bs show the centers of buoyancy of a ship in the upright and tilted condition and M is the metacenter. The metacenter is considered to be fixed for small angles of heel however at larger angles of heel the metacenter can no longer be considered fixed and other means must be found to calculate the ship's stability. The metacenter can be calculated using the formula $$ BM =\frac{I}{V} \ $$ Where B is the center of buoyancy, I is the moment of inertia of the waterplane in meters4 and V is the volume of displacement in meters3.

Different centers
The center of buoyancy, is the center of gravity of the volume of water which the hull displaces. This point is referred to as B in naval architecture. The center of gravity of the ship itself is known as G in naval architecture. When a ship is upright, the center of buoyancy is directly below the center of gravity of the ship.

The metacenter is the point where the lines intersect (at angle φ) of the upward force of buoyancy of φ ± dφ. When the ship is vertical it lies above the center of gravity and so moves in the opposite direction of heel as the ship rolls. The metacenter is known as M in naval architecture.

The distance between the center of gravity and the metacenter is called the metacentric height, and is usually between one and two meters. This distance is also abbreviated as GM. As the ship heels over, the center of gravity generally remains fixed with respect to the ship because it just depends upon position of the ship's weight and cargo, but the surface area increases, increasing BMφ. The metacenter, Mφ, moves up and sideways in the opposite direction in which the ship has rolled and is no longer directly over the center of gravity.

The righting force on the ship is then caused by gravity pulling down on the hull, effectively acting on its center of gravity, and the buoyancy pushing the hull upwards; effectively acting along the vertical line passing through the center of buoyancy and the metacenter above it. This creates a torque which rotates the hull upright again and is proportional to the horizontal distance between the center of gravity and the metacenter. The metacentric height is important because the righting force is proportional to the metacentric height times the sine of the angle of heel.

Righting arm
Sailing vessels are designed to operate with a higher degree of heel than motorized vessels and the righting torque at extreme angles is of high importance. This is expressed as the righting arm (known also as GZ — see diagram): the horizontal distance between the center of buoyancy and the center of gravity.

Monohulled sailing vessels are designed to have a positive righting arm (the limit of positive stability) at anything up to 120º of heel, although as little as 90º (masts flat to the surface) is acceptable. 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.

Stability
GM and rolling period

GM has a direct relationship with a ship's rolling period. A ship with a small GM will be "tender" - have a long roll period - an excessively low or negative GM increases the risk of a ship capsizing in rough weather (see 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 (see Cougar Ace). A ship with low GM is less safe if damaged and partially flooded because the lower metacentric height leaves less safety margin. For this reason, maritime regulatory agencies such as the IMO specify minimum safety margins for  sea-going vessels.  A larger metacentric height, on the other hand can cause a vessel to be too "stiff"; excessive stability is uncomfortable for passengers and crew. This is  because the stiff vessel quickly responds to the sea  as it attempts to assume the slope of the wave. An overly stiff vessel rolls with a short period and high amplitude which results in high angular acceleration. This increases the risk of damage to the ship as well as the risk cargo may break loose or shift. In contrast a "tender" ship lags behind the motion of the waves and tends to roll at lesser amplitudes. A passenger ship will typically have a long rolling period for comfort, perhaps 28 seconds while a tanker or freighter might have a rolling period of 13 to 15 seconds.

Damaged Stability

If a ship floods, the loss of stability is due to the free surface effect, as the water accumulating in the hull will be in the bilges, lowering the centre of gravity and actually increasing the metacentric height. This additional mass will however reduce freeboard (distance from water to the deck) and the ship's angle of down flooding (minimum angle of heel at which water will be able to flow into the hull). The range of positive stability will be reduced to the angle of down flooding resulting in a reduced righting lever. When the vessel is inclined, the fluid in the bilge will move to the lower side, shifting its center of gravity toward the list, further extending the heeling force. This is known as the free surface effect (see below).

Free surface effect
In tanks or spaces that are partially filled with a fluid or semi-fluid (fish, ice or grain for example) as the tank is inclined the surface of the liquid, or semi-fluid, stays level. This results in a displacement of the centre of gravity of the tank or space. The effect is similar to that of carrying a large flat tray of water. When an edge is tipped, the water rushes to that side which exacerbates the tip even further.

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. This is always of significance in ship fuel tanks or ballast tanks, tanker cargo tanks, and in flooded or partially flooded compartments of damaged ships. 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.

Transverse and Longitudinal Metacentric heights
There is also a similar consideration in the movement of the metacentre forward and aft as a ship pitches. Metacenters are usually separately calculated for transverse (side to side) rolling motion and for lengthwise longitudinal pitching motion. These are variously known as $$\overline{GM_{T}}$$ and $$\overline{GM_{L}}$$, GM(t) and GM(l), or sometimes GMt and GMl.

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.

Measuring metacentric height
The metacentric height is normally estimated during the design of a ship but can be determined by an inclining experiment or Inclining test once it has been built. This can also be done when a ship or offshore floating platform is in service. It can be calculated by theoretical formulas based on the shape of the structure.

Stabilizer solutions
Because ships under way have their own characteristics and natural rolling periods, the amplitude can to a certain extent be reduced with the fitting of bilge keels, anti-rolling tanks, fin stabilizers, or a combination of these. Forces are generated, with the help of computers, equal and opposite to those of the sea. Most passenger ships make use of the equipment.