Examination of the intact stability and the seakeeping behaviour

2.1 Description of the utilised seakeeping simulation method
the roll motion or the highly nonlinear restoring moments, the surge and roll motion have to
be simulated nonlinearly in the time domain. The linearisation of the roll motion for example,
would imply replacing the lever arm curve with a straight line having a gradient corresponding
to GM. It is easy to understand, that such a simplication is not permissible. Therefore the
roll and the surge motion are simulated nonlinearly in the time domain, based on the formulas
described in the following.
2.1.2.1 Roll motion
The roll motion is calculated in the time domain according to the equation of motion 2.1 shown
below:
¨ϕ =
[( ) ( ) ] (
I xz ¨ψ + ψ ϕ˙
2 cos ϕ − ¨ϑ + ϑ ϕ˙
2 sin ϕ − m g − ¨ζ
)
h s
I xx − I xz (ψ sin ϕ + ϑ cos ϕ)
+ M W ind + M Sway&Y aw + M W ave + M T ank − M D
I xx − I xz (ψ sin ϕ + ϑ cos ϕ)
(2.1)
with
ˆ ϕ, ϑ and ψ, the angles to describe roll, pitch and yaw as well as ζ describing the heave
direction which coincides with z of the hull-bound coordinate system.
ˆ h s , the righting lever in seaway according to Grim's [10] equivalent wave method.
ˆ m is the mass of the ship and g is the gravitational acceleration.
ˆ M W ind , M Sway&Y aw , M W ave and M T ank , the exiting roll moments due to wind, sway and
yaw, waves and uid in tanks or ooded compartments.
ˆ M D , the nonlinear damping moment depends on ship's speed. It is determined by using
damping coecients according to Blume [9].
ˆ I xx and I xz are the moment of inertia and the centrifugal moment.
Before the simulation is started, the cross curves of the ship are calculated, to avoid the timeconsuming
calculation of the actual righting lever in seaways for each time step of the simulation.
The actual value during simulation is interpolated from the pre-calculated righting levers using
Grim's [10] equivalent wave method.
2.1.2.2 Surge Motion
Finally, the surge motion is simulated based on the ship's resistance, speed, mass (including added
hydrodynamic mass) and surge-inducing wave forces. The wave force is calculated, assuming a
hydrostatic pressure distribution under the water surface at half of ship's draught. This means,
that at each frame the force of buoyancy is perpendicular to this line of equivalent pressure at
half draught. The surge motion is simulated based on the approach 2.2 below.
[ 2R (v0 ) R (v o )
¨ξ = − ˙ξ +
v 0 m ∗ v0 2 ξ˙
2 + △R ]
m∗ m ∗ (2.2)
with
ˆ R, representing the resistance curve in still water conditions
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