6.3.1
The same experimental data
used in the Alternative Procedure 1 above have been used in the application
of the PIT. Scope of this application is to predict the roll response
peak for the tested required steepness s = 0.0383 by
starting from available data at smaller steepnesses, i.e. 1/40 and
1/60. The following three calculations have been carried out:
-
.1 Calculation 1: prediction of φ
1r by fitting of the model on the steepness s =
1/60;
-
.2 Calculation 2: prediction of φ
1r by fitting of the model on the steepness s = 1/40;
-
.3 Calculation 3: prediction of φ
1r by fitting of the model on both the steepness s = 1/40 and
s = 1/60;
6.3.2
In the case of calculations
1 and 2, being only one steepness available, the reduced model (N-6.2)
has been used, and because of the linearity of the 
curve and because of the absence of any evident bending
in the response curve it has been assumed that γ
3 =
0 .
6.3.3
In the case of calculation
3, being two steepnesses available, additional terms have been added.
Two different analytical model have then been used: the first model
is exactly the same as that used for calculation 1 and 2, whereas
in the second model the linear damping coefficient μ has
been left free (see (N-6.3)). However, in both cases, the assumption
of linear restoring, i.e., γ
3 = 0 and γ
5 = 0 , has been kept.
6.3.4
In all cases the roll response
curve has been determined through an analytical approximate nonlinear
frequency domain approach where the response curve is obtained by
means of the harmonic balance technique [3].
6.3.5
The used analytical models
and the results obtained through the application of the PIT are summarized
in Table 6.1, while a global picture of the roll response curves is
given from figure 6.3 to figure 6.6.
6.3.6
From the analysis of the reported
exercise it seems that the PIT together with the proposed analytical
reduced models is able to reasonably predict the ship roll response
curve at the largest steepness by starting from the fitting of the
roll response curve(s) experimentally obtained at lower steepnesses.
The pure quadratic damping model allows for the achievement of good
predictions of the experimental peak, probably thanks to the presence
of bilge keels. In the case of linear+quadratic damping model, a negative
linear damping coefficient has been obtained, that is, of course,
physically meaningless. However, the equivalent linear damping in
the range of tested angles as given by the fitted model in Calculation
3-LQ is, of course, positive. The negative sign in the linear damping
coefficient is thus due to the fact that the equivalent linear damping
obtained from the fitted model in the range of tested rolling amplitudes
better fits the experimental data according to the minimization procedure.
If a series of experiments had been carried out at smaller steepnesses
with subsequent fitting, it would have increased the linear damping
coefficient, making it, probably, positive. Bearing in mind the theoretical
background of the PIT technique, negative linear damping coefficients
are often not a real practical problem, even if their presence usually
indicates that different types of analytical modelling for the damping
function could lead to a better representation of the real ship damping.
Analytical models used in the fitting and fitted parameters (model scale)
|
|
Calculation 1
|
Calculation 2
|
Calculation 3-Q
|
Calculation 3-LQ
|
| Steepness
used in the fitting
|
1/60
|
1/40
|
1/60 and 1/40
|
| Analytical model
|
|
|
| Fitted
coefficients
|
ω
0 = 3.344rad/s
β =
0.520rad
-1
α
0 = 0.873
|
ω
0 = 3.348rad/s
β =
0.518rad
-1
α
0 = 0.857
|
ω
0 = 3.346rad/s
β =
0.519rad
-1
α
0 = 0.864
|
ω
0 = 3.345rad/s
β = 0.684rad
-1
α
0 = 0.833
|
| Predicted value in
degrees of φ
1r for s = 0.0383
|
28.3
|
28.1
|
28.2
|
27.0
|
| Corresponding value
of φ
1 = 0.7 • φ
1r
|
19.8
|
19.7
|
19.7
|
18.9
|
| Experimentally determined φ
1 in degrees
|
19.3
|
Response curves for Calculation 1
Response curves for Calculation 2
Response curves for Calculation 3-Q
Response curves for Calculation 3-LQ
6.3.7
In order to better explain
this latter point, an additional calculation (Calculation 3-LQC) has
been carried out using experimental data for steepnesses s=1/60 and
s=1/40 in the fitting procedure together with a more flexible linear+quadratic+cubic
model for the damping, keeping the linear restoring assumption, i.e.:
6.3.8
The obtained parameters are
as follows:
|
= |
0.013 |
6.3.9
It can be seen that now the
negative linear damping has disappeared, and that the nonlinear damping
component is distributed among the quadratic and cubic term. Although
this result is more sound from a physical point of view, it is not
necessarily the best one in terms of the predicted roll peak at s
= 0.0383. The predicted peak of the roll response is, indeed, φ
1r = 26.6° leading to φ1 = 18.6°.
The reduction in the predicted roll peak is likely due to the introduction
of the cubic term. A summarizing plot is given in figure 6.7.
Response curves for Calculation 3-LQC