Chapter 9 - Stability in Waves

Abstract

Longitudinal and quartering waves influence the stability of ships and other floating bodies by modifying the transverse moment of inertia of the waterplane as the wave passes along the ship. The modifications depend on the ship forms. Unfortunately, some new ship forms accentuate this effect and caused accidents. Usually, the stability reaches a minimum when the ship is on a wave crest, and a maximum when the ship is in a wave trough. This variation depends on the frequency of encounter, that is the frequency of waves that an observer on the ship can see. If we assume that this variation is proportional to the cosine of the frequency of encounter, we can transform the roll equation into a linear equation with periodic coefficients known as Mathieu equation. The stability of this equation depends on two parameters, one proportional to the own roll period of the ship, the other to the magnitude of the GM variation. The Strutt-Ince diagram divides the plane of the two parameters into regions of stable, and regions of unstable response. It is shown that the most dangerous situation is that in which the frequency of encounter is twice the own roll period of the ship. Experience and simulations confirm this conclusion. The chapter ends by showing how to simulate the Mathieu equation in a MATLAB environment.