Equation of motion for roll

The vessel's roll motion may be represented by a second order differential equation, such as that describing a forced spring, mass and damper system.

( 94 )

where the variables are defined as follows:

I₄	moment of inertia for roll.
A₄₄	added inertia coefficient for roll.
B₄₄	damping coefficient for roll.
C₄₄	hydrostatic restoring coefficient for roll.
F₄	roll exciting moment at the encounter frequency .
η₄	instantaneous roll displacement.
	instantaneous roll velocity.
	instantaneous roll acceleration.

It may be shown that the solution to the above equation is given by:

where: is the phase lag relative to the forcing function:

This equation may be re-expressed in terms of the damping ratio, , the natural frequency of the system, , and the tuning factor, .

As an aside, it may be shown (by differentiation of the RAO function) that the damped natural frequency is given by:

The roll transfer function or response function is then assumed to be given by:

Strictly speaking this is the roll motion transfer function with regard to wave force and not wave slope, however, the two are assumed to be the same.

The RAO is then modified for wave heading and apparent wave slope so that the RAO at off head seas is given by:

thus the roll RAO is zero in head and following seas and has a maximum in beam seas.

In Seakeeper the required parameters are determined as follows:

I₄	mass inertia of vessel in roll , input by user
A₄₄	added inertia coefficient for roll , this is an average of values from Vugts (1968) and Lloyd (1998)
β₄₄	Non-dimensional damping coefficient for roll, input by user.
C₄₄	hydrostatic restoring coefficient for roll , VCG input by user

Roll free-decay test

If experimental facilities are available, the roll damping can be obtained from a free-decay test of the roll motions. The vessel is heeled over to one side and released, the roll amplitude is measured and plotted against time. The figure below shows the theoretical free-decay of two vessels with different damping coefficients.

Free-decay time series for two vessels released from an initial heel angle of 30 degrees

By plotting the value of one peak against the value of the next peak (the same can also be done for the troughs to obtain more data), the roll damping can be derived.

Peak/trough amplitudes for beta 0.075 vessel

beta0.075	peak/trough i	peak/trough i+1
trough 1	30.000	18.755
peak 1	23.752	14.808
trough 2	18.755	11.692
peak 2	14.808	9.231
trough 3	11.692	7.289
peak 3	9.231	5.754
trough 4	7.289	4.544
peak 4	5.754
trough 5	4.544

Peak/trough amplitudes for beta 0.1 vessel

beta0.1	peak/trough i	peak/trough i+1
trough 1	30.000	16.001
peak 1	21.745	11.646
trough 2	16.001	8.519
peak 2	11.646	6.183
trough 3	8.519	4.518
peak 3	6.183	3.285
trough 4	4.518	2.405
peak 4	3.285	1.747
trough 5	2.405	1.279
peak 5	1.747
trough 6	1.279

Plot of peak amplitude against peak amplitude of next peak. In this example, data for both peaks and troughs have been plotted

The non-dimensional roll damping parameter used in Seakeeper, β₄₄, is given by:

Thus for the beta0.075 vessel, the slope is 1.6023, giving a damping of 0.075 (as expected); similarly for the beta0.1 vessel, the slope is 1.8747 giving a damping of 0.100.

The free-decay roll test can be simulated in Seakeeper by choosing Roll decay simulation option in the Analysis | Calculate Wave Surface dialog, then choosing Display | Animate and saving the time-series to a file:

Simulation of free-decay roll test in Seakeeper