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:

• ${\displaystyle KM=KB+BM}$
• ${\displaystyle BM={\frac {I}{V}}\ }$

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.^[3]

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 or narrow and deep hulls have high transverse metacenters (relative to the keel), and the opposite have low metacenters . Ignoring the ballast, wide and shallow or narrow and deep means that the ship is very quick to roll and 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

Distance GZ is the righting arm: a notional lever through which the force of buoyancy acts

The metacentric height is an approximation for the vessel stability at a small angle (0-15 degrees) of heel. Beyond that range, the stability of the vessel is dominated by what is known as a righting moment. Depending on the geometry of the hull, naval architects must iteratively calculate the center of buoyancy at increasing angles of heel. They then calculate the righting moment at this angle, which is determined using the equation:

$${\displaystyle RM=GZ\cdot \Delta }$$

Where RM is the righting moment, GZ is the righting arm and Δ is the displacement. Because the vessel displacement is constant, common practice is to simply graph the righting arm vs the angle of heel. The righting arm (known also as GZ — see diagram): the horizontal distance between the lines of buoyancy and gravity.^[2]

• ${\displaystyle GZ=GM\cdot \sin \phi }$ ^[1] at small angles of heel

There are several important factors that must be determined with regards to righting arm/moment. These are known as the maximum righting arm/moment, the point of deck immersion, the downflooding angle, and the point of vanishing stability. The maximum righting moment is the maximum moment that could be applied to the vessel without causing it to capsize. The point of deck immersion is the angle at which the main deck will first encounter the sea. Similarly, the downflooding angle is the angle at which water will be able to flood deeper into the vessel. Finally, the point of vanishing stability is a point of unstable equilibrium. Any heel lesser than this angle will allow the vessel to right itself, while any heel greater than this angle will cause a negative righting moment (or heeling moment) and force the vessel to continue to roll over. When a vessel reaches a heel equal to its point of vanishing stability, any external force will cause the vessel to capsize.

Sailing vessels are designed to operate with a higher degree of heel than motorized vessels and the righting moment at extreme angles is of high importance.

Monohulled sailing vessels should be designed to have a positive righting arm (the limit of positive stability) to at least 120° of heel,^[4] although many sailing yachts have stability limits down to 90° (mast parallel to the water surface). 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.^[5]

GM and rolling period

The metacentre 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, 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. For this reason, maritime regulatory agencies such as the International Maritime Organization specify minimum safety margins for seagoing 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 and to cargo and may cause excessive roll in special circumstances where eigenperiod of wave coincide with eigenperiod of ship roll. Roll damping by bilge keels of sufficient size will reduce the hazard. Criteria for this dynamic stability effect remain to be developed. 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 12 seconds while a tanker or freighter might have a rolling period of 6 to 8 seconds.

The period of roll can be estimated from the following equation:^[1]

$${\displaystyle T={\frac {2\pi \,(a_{44}+k)}{\sqrt {g{\overline {GM}}}}}\ }$$

where g is the gravitational acceleration, a44 is the added radius of gyration and k is the radius of gyration about the longitudinal axis through the centre of gravity and ${\displaystyle {\overline {GM}}}$ is the stability index.

Damaged stability

If a ship floods, the loss of stability is caused by the increase in KB, the centre of buoyancy, and the loss of waterplane area - thus a loss of the waterplane moment of inertia - which decreases the metacentric height.^[1] This additional mass will also reduce freeboard (distance from water to the deck) and the ship's downflooding angle (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 flooded volume will move to the lower side, shifting its centre of gravity toward the list, further extending the heeling force. This is known as the free surface effect.