# Metacenter

The **metacenter of** a floating body, e.g. B. a ship, is important for the stability of the floating position of the ship. If the swimming position is disturbed by an external force, the old swimming position is restored after the disturbance has ended, when the metacenter is above the center of mass . The ship moves something like a cradle rolling around the circumference of a circle. The center of the circle corresponds to the metacenter.

## definition

The metacenter of a ship or, in general, of a floating body is the intersection of the lift vectors that belong to two adjacent angular positions. So there is a metacentre for every axis of rotation and every angular position. The latitude metacenter M or MB (for rotations around the longitudinal axis) and the length metacenter (for rotations around the transverse axis) ML are important and therefore also given a special name, whereby the ship is generally assumed to be floating upright.

## Differentiation in nautical science

In nautical science, the latitude metacenter dominates because it is a typical danger for a ship or boat to capsize over the long side , while capsizing over the bow or the stern is very rare. In the upright position of rest of the ship, the latitude metacenter lies in the midship plane (this is the mirror plane for the right and left sides of the ship). However, the metacenter defined above moves away from the midship plane with increasing heel angle, which is very disadvantageous for its use in ship navigation practice. For this reason it is called the *true metacenter* , but in practice it is left aside. Its counterpart, correctly called the *apparent metacenter* , is the point at which the lift vector intersects the midships plane. This apparent metacenter is the practicable size, the position of which in relation to the center of gravity G allows a direct statement about the stability of the ship with a given heel. That is why the "apparently" is usually left out, as is the indication that it is the *latitude* metacenter. In the rest position of the ship, the location of the *true* and the *apparent* metacenter coincide, this point is called the *initial* metacenter . With small heel angles (up to about 2 ° or 5 °), the true and apparent metacenter deviate only slightly from the initial metacenter in boats or ships with conventional hull shapes.

The figure above shows a cross-section through the hull of a freighter lying in the water, inclined by 15 °. The horizontal blue line shows the position of the water surface (water line). The points K (Kiel), B (center of gravity), G (center of mass) and M (apparent metacenter) are marked. The lines of force are also represented by G (weight force) and B (buoyancy force) and the direction of the forces.

The middle figure shows the same cross-section in an upright swimming position. A blue-green curve can be seen at the bottom of the fuselage, which shows which curve the center of gravity describes when the heel angle changes from −85 ° via 0 ° to + 85 ° (locus curve). Above it lies the purple curve that shows the locations of the true metacenter for the same angular range. You can clearly see that the true metacenter with increasing heel protrudes far out of the midship plane (green vertical in the middle).

The figure below shows an enlargement of the locus of B and M, in which it can be seen more clearly that the locus of M has a broken course, with changes in direction of almost 180 °. (That is quite typical and depends on the change in the curvature of the locus of B.) The heel angle range shown is from 0 ° to 85 °, the green vertical is the midship plane. The respective starting positions are marked with a small circle: violet the initial metacenter, blue-green the corresponding center of gravity in the rest position.

## calculation

The height of the metacenter above the keel K results from the addition of the distance Kiel - center of gravity B and the smallest area moment of inertia of the waterline area, divided by the displaced volume:

2 examples:

1.) A pontoon with length L and width B has a draft T.

The volume is ,

is the geometrical moment of inertia about the longitudinal axis ,

the center of gravity lies on .

Thus the route is

2.) A circular cylinder with the diameter D and the length L floats to the draft T. A calculation is not necessary here, as the lift vector always goes through the center of the circle. This applies to every angle of heel and every draft. So is .

## Metacentric height

The distance from the center of mass G to the metacenter M is called the metacentric height GM. The center of mass G of a floating body is located vertically below the metacenter, provided that no external forces or moments act on the body. That means: the body moves until this condition is met. The metacentric height is important for assessing stability at small heel angles. It can be determined by a heeling test so that the position of the center of mass can be determined. An estimate of the metacentric height can also be obtained from the roll period (roll test).

## Lever arm curve

Knowledge of the metacentric height is generally insufficient for assessing the stability of a ship. Rather, the entire course of the righting moment over the heel angle is important. In order to obtain a value that is independent of the size of the ship, one divides the righting moment by the weight of the ship to obtain the righting lever. It is equal to the distance between the center of mass and the lift vector. The metacentric height GM is equal to the slope of the tangent to the curve at the zero point.

## literature

- Henschke author collective VEB Verlag Technik Berlin, 1956
- Dietmar Gross, Werner Hauger, Walter Schnell: Technical Mechanics . tape 4 . Springer, Berlin 2004, ISBN 978-3-540-22099-2 .

## Quote

From Directive 2002/35 / EC on safety regulations for fishing vessels of 24 meters in length and more, section: Stability and seaworthiness:
*The metacentric starting height GM must not be less than 350 millimeters for single-deck vessels. For vehicles with a full body, the metacentric height can be reduced with the approval of the administration; however, it must not be less than 150 millimeters.*

## See also

## Web links

- Buoyancy and swimming stability, TU-Cottbus (PDF file; 55 kB)
- Lecture notes by Peter Junglas