Page 117 - Introduction to Naval Architecture

104 SEAKEEPING

with the angular velocity. It always opposes the motion since energy is
being absorbed.
Neglecting still the effects of added mass, the equation for rolling in
still water becomes:

where B is a damping constant.
2 2
This has the form of the standard differential equation, d #>/dt +
2kft/0 d#>/dt + (ofyp - 0 where col = gGM T/k^ and k = Bg/2 ft> 0A^ and the
period of the motion is given by:

The other modes of oscillation can be dealt with in a similar way. When
damping is not proportional to the angular or linear velocity the
differential equation is not capable of easy solution. For more
background on these types of motion reference should be made to
standard textbooks.

MOTIONS IN REGULAR WAVES

It was seen in Chapter 5 that the apparently random surface of the
sea can be represented by the summation of a large number of
regular sinusoidal waves, each with its own length, height, direction
and phase. Further it was postulated that the response of the ship in
such a sea could be taken as the summation of its responses to all the
individual wave components. Hence the basic building block for the
general study of motions in a seaway is the response to a regular
sinusoidal wave.
For simplicity it is assumed that the pressure distribution within the
wave is unaffected by the presence of the ship. This is a common
assumption first made by R. E. Froude in his study of rolling and it is
often referred to as Fronde's hypothesis.

Rolling in a beam sea
The rolling a ship experiences is most severe in a beam sea. With
Froude's hypothesis, the equation for motion will be that for still water
with a forcing function added. This force arises from the changes in
pressures acting on the hull due to the wave and could be found by
integrating the pressures over the whole of the wetted surface.