Question.

Explain how the period of roll varies with

The amplitude of roll

The radius of gyration

The intial metacentric height

The location of masses in the ship

The amplitude of roll. The time period of a ship is the time taken by the ship to roll from one side to the other and back again to the initial position.

T = C x Beam / √GM

Where

C is the constant determined by experience and the general value is between 0.7 to 0.8 for merchant vessel

When a ship is heeled by an external force and the force is suddenly removed, the vessel will roll to port and starboard with a rolling period.

The amplitude of the roll will depend upon the applied heeling moment and the stability of the ship. For angles of heel up to about 15° the rolling period does not vary with the angle of roll. The angle reduces slightly at the end of each swing and will eventually dampen out completely. This dampening is caused by the frictional resistance between the hull and the water, which causes a mass of entrained water to move with the ship.

TYPICAL DAMPING CURVE

The initial metacentric height or metacentre, M, is calculated based on the centre of buoyancy, KB, and the moment of inertia of the ship’s water plane, BM.

Then GM is the vertical distance between the centre of gravity and the metacentre and is linearly related to the righting arm by the sine of the heel angle.

Since the sine of a small angle is approximately equal to the angle and since the metacentre is relatively stationary for small angles of heel, GM can also be expressed as the initial slope of the righting arm curve.

As such, it provides a realistic approximation of the resistance of a vessel to small angles of heel in the manner of a linear spring constant, , where MH is the heeling moment, is the equilibrium heel angle, and is the displacement. 

The rolling period is inversely proportional to the metacentric height of the vessel.

Hence, the vessel with the large initial metacentric height will experience small period of roll.

This is due to the large righting lever at any angle of roll and hence offer considerable resistance to the rolling.

The ship is said to be stiff and very uncomfortable having a very small rolling period. It may sometimes lead to structural damage.

In contrast to the stiff ship, the ship with small metacentric height will have small righting lever and hence offer small resistance to rolling

The ship is said to be tender and will have long period of roll and smooth movement

The radius of gyration

Rolling period P= 2πk/√gGM seconds

where GM is the metacentric height

k is the radius of gyration of the loaded ship about a longitudinal polar axis. And is defined as a measure of bodys distribution of mass about its axis of raotation

The value of the radius of gyration will vary with the disposition of the cargo.

For dry cargo ships, where the cargo is stowed right across the ship, the radius of gyration varies only slightly with the condition of loading and is about 35% of the midship beam. It is difficult in this type of ship to alter the radius of gyration sufficiently to cause any significant change in the rolling period.

Variation in the period due to changes in metacentric height are easier to achieve.

In tankers and OBO vessels it is possible to change the radius of gyration and not as easy to change the metacentric height. If the cargo is concentrated in the centre compartments, with the wing tanks empty, the value of the radius of gyration is small, producing a small period of roll. If, however, the cargo is concentrated in the wing compartments, the radius of gyration increases, producing a slow rolling period.

This phenomenon is similar to a skater spinning on ice; as the arms are outstretched the spin is seen to be much slower.

Problems may occur in a ship which travels in a beam sea, if the period of encounter of the waves synchronizes with the natural frequency of roll. Even with small wave forces the amplitude of the roll may increase to alarming proportions. In such circumstances it may be necessary to change the ship's heading and alter the period of encounter of the waves.

The location of masses in the ship.

The time period of roll will change when the masses once placed on board and then shifted with the ship. This shift could effect both radius of gyration and metacentric height

If now masses located at a higher location is more, the GM will reduce and hence there will be increase in time period of roll.

Reverse of this phenomenon is also true that the rolling period will be small for lower location of masses with the ship.