Trans RINA, Vol 157, Part A3, Intl J Maritime Eng, Jul-Sep 2015
In the next sections the equation of motions of the ship and the flooding model
are presented. The first
applications carried out, are intended to test the reliability of the developed method by checking the transient stages and the equilibrium positions of the ship model, in still water. Moreover the heave and pitch
responses, in
longitudinal regular waves, obtained from the numerical simulation, have been compared to linear sea-keeping data for the same ship.
Finally the aforementioned damage stability applications are presented and discussed.
2. THE NUMERICAL MODEL
The equations of motion for a ship refer to a body fixed co-ordinate system, centred at the initial vessel’s centre of gravity G.
loads are evaluated, at each time, on the instantaneous ship wetted surface. Radiation and diffraction forces are derived using a linear model.
The intact ship numerical model, implemented in Matlab/Simulink, is based and derived on the same method reported in (Matusiak 2013). For the purpose of the application,
head sea, modeled with regular sinusoidal wave, is assumed.
The developed method for damage stability simulations, i.e. for grounding scenario and for not drained water in the garage compartment, is carried out in the next subsections. The radiation actions on the vessel are assumed to be constant, referring to the initial intact ship floating position also in case of damage.
2.1 DAMAGE ASSUMPTIONS
Ship grounding for a fast vessel represents a dangerous damage situation, as also mentioned by the rules (RINA 2009).
The damage simulation has been carried out by applying the water
loading method: the implemented model is Figure 1. Body fixed and inertial co-ordinate systems
The earth fixed axes, instead presents the X-Y plane coincident with the still water level. These two reference frames are shown in Figure 1.
The equations describing the 6DoF dynamic model for a damaged ship, regarded as a rigid body, are presented below (1):
Iωω Iω mr f i
mm ext
ext i
m u ω r ω u ω r i
ii i ii i
2 mmi i i
u ω uu u ω rf g
u fi ω uf g (1) (2) (3)
According to (Manderbacka et alt. 2011) ship motions are solved from the equations (1) and (2) based on the conservation of momentum, while the flood water is modelled as a lumped mass (3); the damage is simulated assuming the added water method.
In the external forces and moments, the radiant actions, i.e. added mass and damping terms, the non-linear buoyancy actions, and the wave loads, i.e. Froude-Krylov and diffraction actions, figure.
The simulation model works on a discrete representation of the
hull, using triangular panels. The non-linear restoring generalized forces and the Froude-Krylov wave A-154 ©2015: The Royal Institution of Naval Architects
capable to evaluate the flow rate of the water through the damaged hull as function of the instantaneous water height above the tank. The quantity of water mi that floods the interior compartment can be obtained from the following formula (Santos 2002) (Mironiuk 2010):
t mt A K g Z Z dt ic c
00 0
2( ) where the constant K is assumed equal to 0.6.
The abovementioned formula was developed from the Bernoulli theorem assuming p=0: 2
ghp v C
1 2
In this research work, the flow rate formula has been developed for analyzing the flooding condition of a damaged ship in wave. Assuming the presence of a regular sinusoidal wave, the wave pressure p should be taken into account by means of Froude-Krylov formula:
pg e
kZc
(5)
Therefore the equation (4) has been modified as follows and implemented within the numerical simulation:
Qt 00A K g e Z Z dt kZc
0 2 cc
(6)
t (4)
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