Game Development Reference
In-Depth Information
Roll, pitch, and yaw are the terms also used for airplanes. The translation degrees of
freedom are called surge, heave, and sway. Surge, sway, and yaw are not that apparent
when vessels are moving forward, so it is acceptable to limit your model to heave, pitch,
and roll. Heave is the up-and-down motion of the boat caused by the change in elevation
of the water's surface as a wave passes. If a vessel is stationary, it would be referred to as
bobbing. Pitch is the rotation about the transverse axis of the vessel due to increased
buoyancy on one end of the ship as a wave passes. This motion is most pronounced
when the waves are traveling in the same direction (or 180 degrees) from the vessel. Roll
is like pitch, but about the longitudinal axis.
As stated before, heave is displacement in the vertical direction from the static equili‐
brium draft. This degree of freedom is straightforward to model as a hydrostatic spring
acting in the vertical direction. Assuming we have a barge that is 30 meters long and 10
meters wide, we'll develop an equation that can govern our heave simulation.
Commonly, a vessel's hydrostatics include something called tons per centimeter im‐
mersion (TPCM)—that is, for every centimeter you press the boat down, a certain
number of tons of buoyancy force is created. For our barge, this is a relatively straight‐
forward calculation.
Given that the water plane area is a constant 300 square meters, 1 centimeter of im‐
mersion would result in a volume of 3 cubic meters. As 1 cubic meter of saltwater weighs
1,027 kg, 3 cubic meters would be 3,081 kg, and (assuming this boat is on Earth), would
result in a buoyant force of 3,081 kg × 9.81 m/s 2 , or 30.2 kN. Therefore, 30.2 kN per cm
would make a good starting value for a spring constant to model the heave response of
this vessel in waves.
For us to simulate realistic roll motions, it is important that the ship take time to com‐
plete the motion. This time is called the roll period . This defines the angular velocity
that a ship rolled to one side will experience when it recovers. We can estimate it by the
following equation:
2 πk
T =
Where k is the radius of gyration and GM is the distance from the metacenter to the
center of gravity. A good estimate for the radius of gyration is often taken as 30% of the
beam of the vessel. A vessel with a shorter roll period will respond quicker to a wave
and try to assume the wave slope. This is known as being “stiff ” and can cause passenger
discomfort and damage via higher angular accelerations. Conversely, vessels with higher
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