Heave and Pitch Motion of a Vessel in a Seaway

The ship motions of heave, pitch and roll are oscillatory in nature, this is due to the restoring force created by changes in buoyancy involved in these motions. The motions of a ship in response to waves, may be considered as a forced damped-spring-mass system. Seakeeper currently only deals with the coupled motions of pitch and heave. The two relevant equations of motion are for heave:

( 1 )

and for pitch:

( 2 )

where the variables are defined as follows:

M	mass of the vessel.
I₅	moment of inertia for pitch.
A₃₃	added mass coefficient for heave due to heave.
A₅₅	added mass coefficient for pitch due to pitch.
A₃₅	added mass coefficient for heave due to pitch.
A₅₃	added mass coefficient for pitch due to heave.
B₃₃	damping coefficient for heave due to heave.
B₅₅	damping coefficient for pitch due to pitch.
B₃₅	damping coefficient for heave due to pitch.
B₅₃	damping coefficient for pitch due to heave.
C₃₃	hydrostatic restoring coefficient for heave due to heave.
C₅₅	hydrostatic restoring coefficient for pitch due to pitch.
C₃₅	hydrostatic restoring coefficient for heave due to pitch.
C₅₃	hydrostatic restoring coefficient for pitch due to heave.
F₃	heave exciting force.
F₅	pitch exciting moment.
	instantaneous heave displacement.
	instantaneous heave velocity.
	instantaneous heave acceleration.
	instantaneous pitch displacement.
	instantaneous pitch velocity.
	instantaneous pitch acceleration.

In order to solve these equations it is necessary to obtain the coefficients and excitation force and moment. The procedure used is described in the following sections.

In Seakeeper the damping is calculated using inviscid flow theory. The user has the option of specifying additional heave and pitch damping to allow for damping not calculated by the inviscid flow modelled. The user is able to specify non-dimensional damping β₃₃ and β₅₅ from which the actual damping is calculated as follows.

with n = 3, 5.

These values are then added to the inviscid B₃₃ and B₅₅.