PARAMETRIC ROLL RESONANCE IN SHIP STABILITY AND REMEDIES Deniz ÜNSALAN Kunsel IZET-ÜNSALAN

Many of the marine disasters of the past have occurred by the capsizing of the vessels in heavy weather. Capsizing usually occurs by the encounter of beam seas, resulting rolling to extreme angles.

The usually preferred action of many masters or officers in charge of a navigational watch is to change the course to take the seas from bow or stern.

Unfortunately, this action can also result the ship being caught into extreme rolling, by a mechanism called "parametric resonance" or "Mathieu effect", which also can eventually cause in the loss of the vessel.

Mathieu equation: Solution: Mathieu equation has regions of stable solutions and unstable solutions (solution is unbounded)

Strutt-Ince diagram:

The lines that separate the stable regions from the unstable regions intersect the a- axis at points where:

When a ship sailing at sea encounters waves, her underwater hull geometry changes. This is due to: Heave Pitch Change of underwater volume due to wave encounter

Metacentric height can be assumed to change periodically: Differential equation of undamped roll becomes: Substituting:

Undamped equation of roll becomes: This is the Mathieu equation, where:

When a = 1, parametric resonance happens. This is equivalent to: or, The ship rolls to extreme angles, which may result in direct capsizing, cargo shifting or water infusion.

Parametric resonance is basically transfer of energy of headstern waves to rolling mode. Therefore, the basic remedy to parametric roll resonance: Dissipate excess energy by damping. There are several ways of dissipating the roll energy by damping. Active devices (active fins, active roll tanks, gyrocompasses) are costly and occupying large space. Passive devices (passive tanks, bilge keels) also have drawbacks.

The most appropriate roll damper for a Passive roll tanks occupy large spaces and add to ship’s weight. The most appropriate roll damper for a merchant ship is the use of bilge keels, however, they add to the drag penalty.

A solution is a retractable roll fin- a passive device with little cost, and does not add to drag penalty when not needed. Bow section of the ship (usually has a large void space) can be used for the storage of passive retractable roll fins.

Bow fin acts by creating a lift force, which has a moment opposing to the direction of the rolling motion: Energy expended by the opposing moment to the roll shall then be, for a full (360 degree) cycle:

where Rt . Evaluating the integral, a relation for the energy of the opposing moment by fin becomes:

Conclusion A basic approach to parametric roll resonance is presented. A simple retractable passive device to reduce roll by damping is presented and governing equations are derived.