Metacentre

In fluid dynamics, metacentre is the theoretical point where an imaginary vertical line passing through the centre of buoyancy and centre of gravity overlaps an unreal vertical line passing through a separate centre of buoyancy produced whenever the body is relocated, or tipped, throughout the water, however slightly.

As per the Metacenter definition, the centre of buoyancy of a ship goes laterally when it heels (rolls sideways). It may also rise or fall in relation to the water line. The metacentre is the point where a vertical line via the heeled center of buoyancy intersects with the line via the existing, vertical centre of buoyancy. By description, the metacentre stays exactly just above the centre of buoyancy.

Height of Metacenter

Metacentric meaning: The metacentric height (GM) is a calculation of a floating body's initial static stability. The distance between that ship's centre of gravity and its metacentre is computed. Considerably greater stability against overturning is associated with a higher metacentric height.

The natural time of rolling of a hull is also influenced by metacentric height, with quite massive metacentric heights typically linked with shorter durations of roll, that are inconvenient for passengers. As a result, passenger ships should have a reasonably large metacentric height, but not overly so.

Metacentric Height Formula: 

GM = KM - KG 

Where GM stands for Metacentric height

KM stands for distance from the keel to the metacentre

KG stands for Centre of Gravity.

Metacentric height is the result of subtracting KG from KM. The initial stability of the ship is measured by GM.

Different Centres

The centre of buoyancy is located at the centre of mass of the volume of water that is displaced by the hull. In naval architecture, this system is stated as B. The ship's centre of gravity is frequently known as point G or CG. Whenever a ship is in balance, the centre of buoyancy is vertically aligned with the ship's centre of gravity.

The metacentre is the place where all the lines of the upward buoyancy force of φ ± dφ intersect  (at angle φ). Whenever the ship becomes vertical, the metacentre is well above the centre of gravity, causing the ship to roll in the reverse direction of the heel.

GM is another abbreviation for this distance. The ship's centre of gravity remains relatively constant as it heels up because it is solely dependent on the location of the ship's load and cargo, however, the surface area expands, growing BMφ. To roll a stable hull, work needs to be done.

This is transformed into potential energy by increasing the hull's centre of mass above the water level, reducing the centre of buoyancy, or maybe both. This potential energy is emitted in an attempt to right the hull, with the stable attitude being where the amplitude is the smallest. The ship's natural rolling frequency is the outcome of the interplay of potential and kinetic energy. The metacentre, Mφ, goes with a lateral component for low angles, since this is no longer completely over the centre of mass.

Free Surface Effect

When tanks or spaces are partially filled with a fluid or semi-fluid (for example, fish, ice, or grain), the surface of the fluid or semi-fluid remains level as the tank is tilted. This causes the tank's or space's centre of gravity to shift relative to the overall centre of gravity. It has the same effect as holding a big flat tray of water. When an edge tips, water rushes to that side, exacerbating the tip even more.

Because the magnitude of this effect is proportional to the cube of the width of the tank or compartment, two baffles dividing the area into thirds would decrease the displacement of the fluid's centre of mass by a factor of nine.This is important in ship fuel tanks or ballast tanks, tanker cargo tanks, and damaged ship compartments that are flooded or partially flooded. 

Another concerning aspect of the free surface effect is the ability to create a positive feedback loop in which the duration of the roll is equal or nearly equal to the period of motion of the centre of gravity in the fluid, resulting in each roll rising in magnitude until the loop is broken or the ship capsizes.

Transverse and Longitudinal Metacentric Heights

The movements of the metacentre forward and aft as a ship pitches has a comparable consideration. Meta Centres are typically computed separately for transverse (side to side) rolling motion and longitudinal pitching motion.

Technically, various metacentric heights exist for any combination of pitch and roll motion, depending on the moment of inertia of the ship's waterplane area across the axis of rotation under consideration, but they are typically calculated and described as special values for the limiting pure pitch and roll motion.

Measurement

The metacentric height of a ship is generally estimated during its construction, but it can be calculated after it has been constructed using an inclining test. When a ship or offshore floating platform is in service, this can also be done. It can be computed using theoretical formulas based on the structure's shape.

The angle(s) measured during the inclining experiment are proportional to GM. The inclining experiment can be used to determine the 'as-built' centre of gravity; receiving GM and KM by experiment measurement (via pendulum swing measurements and draught readings), the centre of gravity KG can be determined. As a result, KM and GM become known factors during inclining.