Roll damping is ususally not linearly depending on the roll amplitude. However, when using potential strip theory calculation, only a linear roll damping coefficient can be used. One solution is thus to solve this iteratively for a given wave amplitude. Another solution is to use an equivalent linear roll damping coefficient calculated at reference amplitude. It is thus important to choose wisely the reference roll amplitude as the obtained response will only be valid at that amplitude. In most cases, the chosen reference amplitude is the roll criteria for operations such that the operational limit can be derived exactly. Finally, systematic linear calculations can also be used in combination with a post processing method depending on the sea condition. This method is presented further on.

Pre-processing

This preliminary part is done outside of RAOViewer using frequency domain tools and empirical roll damping formulations.

Firstly, a range of expected roll ampiltudes is chosen (rollAmpInp in SHIPMO), with at least one value under and one value above the expected extremes. This is important to avoid any need for later extrapolation. For each amplitude, an equivalent linear roll damping coefficient is provided. Usually, this equivalent damping is based on a linear and a quadratic roll damping model such that the damping work is the same for linearised and non-linear damping:

with:

Finally, the RAO of motion are computed for all the conditions and all the roll damping values. Each RAO is labeled with the roll amplitude used for the linearisation.

All signals that are affected by the roll motions have to be computed in the same way.

Post-processing

Once systematic linear calculations have been performed for various roll damping amplitudes, the data can be linearised for the waves of interest in RAOViewer. This has to be done for each wave or wave spectrum as the linearisation is based on the non-linear response.

Regular waves

The response of the ship in a given regular wave can be linearised quite simply. The non-linear roll response is the one where the input roll amplitude for the linearised roll damping coefficient is equal to the response. The following figure presents the response obtained for a range of reference amplitudes. The exact response is the one crossing the black line y=x (in this case 12 degrees).

For other signals, the same interpolation factors are then used. In this example, the response of a signal y is thus based on the RAO of y obtained for  at 10 and 20 degrees:

The amplitude of the wave is given in RAOViewer in meters in the Condition panel.

Irregular waves

For the case of irregular waves, the same approach is used but the regular wave is replaced by a spectrum and the roll response by a significant response. The response being a significant value, and not an ampiltude, it cannot be directly compared to the  value that was used for the roll damping. The following figure presents the SDA of roll for a given wave period and for various wave heights and roll dampings. Similarly to the regular wave case, each line is only valid at one point: the point where the relation between the SDA of roll and the B() if fixed.

Assuming a narrow banded roll response, the relation between the mean of the highest one Nth of the roll peaks x and the standard deviation is given by (see Ochi):

where  is the area under the spectral density spectrum, σ the standard deviation and:

where  is equal to the cumulative distribution function of the standardized normal distribution:

For instance, the following relations can be found:

If one assume that this coefficient can be used to compare a regular amplitude with the standard deviation, one can
linearise the response in a given sea state by comparing the RMS obtained from each RAO and .

The most common value for N is 3, meaning that the significant amplitude () is compared to the
amplitude used in the linearised roll damping coefficient.

N is given in RAOViewer in the Condition panel.

References

Ochi, M., and Bolton, W., 1973, “Statistics for prediction of ship performance in a seaway - Part 2”, International Shipbuilding Progress, Volume 20, pp. 89–121.