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freedom in the 3D space, as presented in figure 1.
Figure 1: Ship motion is divided into six components in
the six degrees of freedom.
Surge, sway and heave correspond to the
translation motions along the x, y and z axis
respectively, while roll, pitch and yaw are the
rotation motions also in the x, y and z respectively.
Most of the ship motion simulators running on
desktop computers do not actually compute the
motions in real-time. Depending of the purpose of
the simulator, ship motions may be computed
previously, and the Virtual Environment (VE) works
as a post-processing visualization and interaction
tool for the simulation. Databases store the ship
motions’ parameters for different wave patterns, and
the data is then queried (in database) and applied in
real-time. It is the case of Daqaq (2003), in which
the ship motion must be realistic in order to test
loading and unloading operations.
Simulation models that compute ship motions in
real time such as Xiufeng et al. (2004) or Sutulo et
al. (2002), normally do not compute wave induced
motions and only the three degrees of freedom
corresponding to the surge, sway and yaw in the
manoeuvrability equations are considered. Such
tools are used to simulate port operations where the
motions induced by the sea waves (heave, roll and
pitch) are negligible. Although three of the ship
motions are not computed, these simulators include
manoeuvrability models to calculate ship trajectories
that consider the interaction between the water, the
sea bottom (shallow waters) and coastal structures
(walls, berths, etc.), which increases the
computational work required, Sutulo et al. (2010).
In an attempt to compute the six motion
components in real time, simplified models such as
Ueng et al. (2008) have been developed. They use a
moving grid attached to the ship generated by the
vertical projection of the ship’s body into the
horizontal plane. The bounding box of the projection
is used and a uniform grid of cells is superposed on
the bounding box. Then the height field of the sea
surface is evaluated at each grid point and the
average height field is multiplied by the area of the
grid to compute the excitation force of water. For
each ship motion component, resistance forces are
estimated and the net force is obtained by
subtraction. Accelerations are then calculated by
applying the Newton’s second law.
Another approach to estimate the ship motions
induced be sea waves is by computing commonly
called Response Amplitude Operators (RAOs). In
linear theory, ship motions are calculated as the
finite sum of sinusoidal components with a random
initial phase, Bhattacharyya (1978). The amplitude
of each of the components is given by RAOs that
define the relationship between ship motion and
wave height versus regular wave frequency, Lewis
(1988). RAOs may be obtained by wave tank
experiments with scale models of the ships or may
be computed by specialized software. The strip
theory is the common approach to compute the
RAOs of the ships for the six degrees of freedom,
Salvesen et al. (1970). This theory assumes the ship
as a slender body and includes ship motion
coefficients such as added mass and damping in
heave and pitch motions. Numerous comparison
studies showed that strip theory generally gives good
results for ship motions in low to moderate regular
waves in which the influence of nonlinearities is still
low. However, Fonseca and Guedes Soares (1998)
presented a generalization of the theory to deal with
large amplitude motions.
Pre-calculated RAOs, which are defined for each
particular ship hull with specific cargo conditions,
allows to compute the amplitudes of each ship
motion component in frequency domain for
predefined manoeuvrability conditions (ship speed
and heading) and sea states (frequency and direction
of propagation of the wave trains). The conversion
of the ship motion component from frequency to
time domain, which is the one of interest for the
interactive simulation, takes into consideration the
phase of each wave when it reaches the ship. In
order to achieve a consistent and physically correct
motion in time domain, the phase of each ship
motion component and the phase of the
corresponding wave train must maintain the
relationship given by the RAO. The final motion is
given by the sum of all the ship motions generated
by all the wave trains.
Each ship motion component is given as a
periodic function in time of the following type:
)
iiii
tax
+
sin
(1)
where i corresponds to one of the motion types, x
i
is
the ship motion of type i, a
i
is the amplitude of the
motion derived by the RAO, ω
i
is the frequency of
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